Lecture Notes in

Mathematics
A collection of informal reports and seminars
Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
Series: Mathematisches Institut der Universit~,t Bonn
Adviser: F. Hirzebruch

291

Peter Orlik
University of Wisconsin, Madison, Wl/USA

Seifert Manifolds

Springer-Verlag
A M S Subject Classifications (1970): Primary: 57-02, 55F55, 57A10, 57E15
Secondary: 14J15, 55A05, 57D85

I S B N 3-540-06014-6 Springer-Verlag Verlin • H e i d e l b e r g • N e w Y o r k

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to Artie
 Introduction

These are notes for a lecture series given at the University

of Os!o in 1971 -1972. Although the manifolds of the title were

constructed by $eifert [I] in 1933, considerable interest has

been devoted to them recently. The principal aim here is to sur-

vey the new results and to emphasize the variety of areas and

techniques involved.

The equivariant theory comprising the first four chapters

was initiated by Raymond [I], who discovered that two classes of

Seifert manifolds coincide with certain fixed point free 3-dimen-

sional St-manifolds. Chapter I contains Raymond's classifica-

tion of sl-actions on 3-manifolds. Chapter 2 describes equivar-

iant plumbing of D2-bumdles over 2-manifolds and identifies the

boundary 3-manifoids. This is used in chapter 3 to resolve sin-

gularities of complex algebraic surfaces with C*-action. The

technique is to compute the Seifert invariants of a suitable

neighborhood boundary of the singular point and use these to con-

struct an equivariant resolution following Orlik-Wagreich [1,2].

The equivariant fixed point free cobordism classification of

Seifert man~folds due to Ossa [12 is given in chapter 4.

The remaining chapters contain topological results. The

homeomorphism classification by Orlik-Vogt-Zieschang [I] using

fundamental groups is obtained in chapter 5. The known free

actions of finite groups on S3 are given in chapter 6 following

Seifert-Threlfall [I]. In chapter 7 we determimewhich Seifert

manifolds fiber over SI . The important results of Waldhausen

[1,2] are outlined in the last chapter together with a number of

 VI

other topics that we could not discuss in detail in the frame

of the lectures.

I would like to thank my friends Frank Raymond and Philip

Wagreich for teaching me directly or through collaboration much

of the contents of these notes; the mathematicians in Oslo in

general and Per Holm and Jon Reed in particular for their hospi-

tality; and Professor ~. Hirzebruch for inviting me to Bonn and

for recommending the publication of these notes. Thanks are also
due to Artie for thorough proofreadnig and to Mrs. Moller for
careful typing of the manuscript.

Oslo, April 1972.

Peter Orlik

*) Supported by grants from the National Science Foundation,

the University of Oslo and the University of Wisconsin.
 Contents

1. C i r c l e Actions on 3 - M a n i f o l d s . . . . . . . . . . . I

I • Manifolds and G r o u p s . . . . . . . . . . . . . 2
2. G-Manifolds . . . . . . . . . . . . . . . . . 4
3. G-Vector Bundles . . . . . . . . . . . . . . 5
4. Some Basic Results . . . . . . . . . . . . . . 7
5. The Circle Group . . . . . . . . . . . . . . . 8
6. Fixed Points . . . . . . . . . . . . . . . . . 10
7. Exceptional Orbits . . . . . . . . . . . . . . 11
8. Special Exceptional Orbits . . . . . . . . . . 13
9. The O r b i t Space . . . . . . . . . . . . . . . 13
10. The C l a s s i f i c a t i o n Theorem . . . . . . . . . 15
11. Remarks . . . . . . . . . . . . . . . . . . . 18

2. E q u i v a r i a n t Plumbing . . . . . . . . . . . . . . . 22

I. Plumbing . . . . . . . . . . . . . . . . . . 22
2. Equivariant Plumbing . . . . . . . . . . . . . 23
3. Quadratic Forms . . . . . . . . . . . . . . . 30

3. R e s o l u t i o n of S i n g u l a r i t i e s . . . . . . . . . . . . 32

I. Algebraic and A n a l y t i c Sets . . . . . . . . . 32

2. Intersections and C o v e r s . . . . . . . . . . . 36
3. M o n o i d a l T r a n s f o r m s and R e s o l u t i o n of
Singularities . . . . . . . . . . . . . . . . 39
4. Resolution and ~*-action . . . . . . . . . . 43
5. W e i g h t e d H o m o g e n e o u s P o l y n o m i a l s and
Good @*-action . . . . . . . . . . . . . . . 45
6. The Cone Over a Weighted Homogeneous Variety. 47
7. The Quotient of V- [£~ by ~* . . . . . . . 49
8. The C a n o n i c a l E q u i v a r i a n t R e s o l u t i o n of a
Surface . . . . . . . . . . . . . . . . . . . 50
9. The Seifert Invariants . . . . . . . . . . . 53
10. Surfaces in ~3 . . . . . . . . . . . . . . . 55
11. Milnor's Fibration Theorem . . . . . . . . . . 60
12. Non-isolated Singularities . . . . . . . . . . 63
 VIII

4. E q u i v a r i a n t Cobordism and the a-Invariant .... 66

I. Basic Results . . . . . . . . . . . . . 66
2. Fixed Point Free SI-Actions . . . . . . . . 68
3. 3-Manifolds . . . . . . . . . . . . . . . . 73
4. The a-Invariant . . . . . . . . . . . . . . 76

5. F u n d a m e n t a l Groups . . . . . . . . . . . . . . . . 82

I. Seifert Bundles . . . . . . . . . . . . . . . 82
2. Seifert Manifolds . . . . . . . . . . . . . . 86
3. Fundamental Groups . . . . . . . . . . . . . 90
4. Small Seifert Manifolds . . . . . . . . . . . 99

6. F r e e Actions of F i n i t e Groups on S3 . . . . . . . 103

1. Orthogonal Actions on S3 . . . . . . . . . . 103

2. Groups and O r b i t Spaces . . . . . . . . . . 109
3. Non-orthogonal Actions . . . . . . . . . . . 113

7. F i b e r i n g Over SI . . . . . . . . . . . . . . . . 115

1. Injective Toral Actions . . . . . . . . . . . 115

2. Fibering Seifert Manifolds over S I ..... 120
3. Non-uniqueness of the F i b e r . . . . . . . . . 126

8. F u r t h e r Topics . . . . . . . . . . . . . . . . 128

Io Waldhausen's Results . . . . . . . . . . . . 128

2. Flat Riemannian Manifolds . . . . . . . . . . 135
3. Solvable Fundamental Groups . . . . . . . . . 141
4. Finite Group Actions . . . . . . . . . . . . 143
5. Foliations . . . . . . . . . . . . . . . . . 145
6. Flows . . . . . . . . . . . . . . . . . . . . 148

References . . . . . . . . . . . . . . . . . . . 151
 I. Circle Actions on 3 - ~ T a n i f o l d s

In this chapter we i n t r o d u c e the n e c e s s a r y preliminary

material concerning the a c t i o n of a compact Lie group on a smooth

manifold. Some important standard results are stated without

proof.

We then p r o c e e d to the e q u i v a r i a n t classification of circle

actions on closed, connected, smooth 3-manifolds following Raymcnd

[I] and 0rlik and R a y m o n d [I]. This is done in terms of a w e i g h t -

ed 2 - m a n i f o l d (the orbit space together with information about the

orbit types). It may be s u m m a r i z e d as follows: the closed, con-

nected, smooth 3-manifold ~ with smooth S1 action is deter-

m i n e d up to e q u i v a r i a n t diffeomorphism (preserving the o r i e n t a t i o n

of the orbit space if it is orientable) by the f o l l o w i n g set of

invariants

M = [b; (e,g,h,t); (~l,~l),...,(~r,Sr) } .

Here e = o if the orbit space is orientable, e = n if not;

g is its genus; fLis the n u m b e r of c o m p o n e n t s of fixed points

in ~ ; t is the n u m b e r of components of orbits with isotropy

group ~2 and slice representation equivalent to r e f l e c t i o n about

a diameter in D 2 ; the r e l a t i v e l y prime pair of positive integers

(a,~) determines the orbit type of an orbit w i t h isotropy group

Ea ; and b is an i n t e g e r representing an o b s t r u c t i o n class sub-

ject to the c o n d i t i o n s that b = 0 if f+t > 0 , b 6 Z if

f+t = 0 and e = o , b 6 E2 if f+t = 0 and ¢ = n and b = 0

if f+t = 0 , ¢ = n and some aj = 2 .

 - 2 -

Manifolds with f+t = 0 belong to the classes 0,o and

N,nI of S e i f e r t EI] and together with the other Seifert manifolds

(introduced in chapter 5) w i l l be the m a i n topic of these notes.

1.1. Manifolds and Groups

A topological space X is a set w i t h certain subsets Ui

distinguished by b e i n g called open. The c o l l e c t i o n of open sets

~ is r e q u i r e d to s a t i s f y the following conditions:

(i) the empty set ~ E ~ and X E ~,

(iii) if Ui E ~ i E I then IJ U E SJ~for an a r b i t r a r y

' iEl i

index set I .

If x E X then an open n e i ~ h b o r h o p d of x is an element of

containing x . A basis for the topology of X is a subcollec-

tion of open sets, ~ so that each element of ~ is a u n i o n of

elements of D~ . X is a H a u s d o r f f spaoe if for a r b i t r a r y dis-

tinct points Xl,X 2 E X there are open n e i g h b o r h o o d s U1, U 2 so

that U I NU 2 = ~ . An open cover of X is a c o l l e c t i o n [Ui]iE I

of open sets so that i~IUi=X . A Hausdorff space is cqmpact if

for every open covering there exists a finite subcollection

[Uil,...~Uin) which is an open covering of X . A map f ~ X ~ Y

between topological spaces is continuous if the inverse image of

every open set is open. It is a h o m e o m o r p h i s m if there exists a

continuous map g: Y ~ X so that g of = id X , f ~ g = idy . A

space X is a t o p o l o ~ i g a l manifold of d i m e n s i o n n if it is a

Hausdorff space with a countable basis and every point x E X

has an open n e i g h b o r h o o d Ux homeemorphie to an open subset of

Euclidean n-space ~Rn . This h o m e o m o r p h i s m ~: Ux ~ n is called

 -3-

a coordinate system at x . Two coordinate systems ~ and

are d°° related if ~ o ~-I and ~° ~ -I are C°o functions

whenever defined. A set of coordinate systems ~ is a smooth

structure on the t o p o l o g i c a l manifold X if

(i) X is covered by the d o m a i n s of the c o o r d i n a t e systems

in ~ ,

(ii) any two coordinate systems in ~ are C~° related,

(iii) ~ is maximal with respect to (i) and (ii).

X is a s m o o t h manifold if it has a smooth structure. A map

f :X ~ Y between smooth m a n i f o l d s is called a smooth map if for

every two coordinate systems ~ on X and ~!, on Y the func-

tion ~o fo -I is of class C°o . A structure (topology, mani-

fold, smooth) on X and Y induces a corresponding structure on

the c a r t e s i a n product X x Y .

A group G is a t o p o l o g ~ a l group if G is a t o p o l o g i c a l

space and the group operations

-1
(gl,g2) ~ glg 2 and g ~ g

are c o n t i n u o u s maps. The topological group G is a Lie group

if G is a smooth m a n i f o l d and the above maps are smooth. Well

known examples are the g e n e r a l linear group GL(n;]R) of n x n

real invertible matrices, the o r t h o g o n a l group 0(n) of n x n

real orthonormal matrices and the special orthogonal group SO(n)

of n x n real orthonormal matrices with determinant +1 . Note

that GL(n;~) is an open submanifold of ~qn2 while O(n) and

SO(n) are c o m p a c t manifolds. A subgroup of a t o p o l o g i c a l group

is called closed if the c o r r e s p o n d i n g subset is closed in the

space of the group, i.e. its complement is open.

 -4-

1.2. G-I~ianifolds

Let G be a compact Lie group and M a smooth manifold.

A smooth (left) action of G on M is a smooth map

G × M ~ M

(g,x) ~ gx

satisfying

(i) g1(g2x) = (glg2)x

(ii) ex = x , where e E G is the i d e n t i t y element.

M together with the G action is called a G-manAfold. If M I

and M2 are G - m a n i f o l d s then the map ~ MI ~ M2 is called

equivariant provided for all g E G and x E MI we have g~(x) =

~(gx) Given x E M the subgroup of G defined by Gx =

[g I g x = x ] is called the i s o t r o p y or s t a b i l i t y group at x . The

action is effective if only e leaves every point fixed, i.e.

if gx = x fer all x E M then g = e . The subset of M de-

fined by Gx = [gx ! g E G] is called the orbit of x . The col-

lection of i s o t r o p y subgroups along Gx , [Ggx I g E G] is called

the orbit type. It is the c o n j u g a c y class of Gx in G since

Ggx gGx g-1 . Consider the e q u i v a l e n c e classes of orbits,

x ~y <~> ~g E G ~ y = gx . Let x* denote the e q u i v a l e n c e

class of x and M* the c o l l e c t i o n of e q u i v a l e n c e classes,

called the orbit space , ~* = i~I/G . Let ~ M ~* be the orbit

map. Topologize M* by the q u o t i e n t topology~ U is open in M*

if and only if w-1(U) is open in M .

Notice that M* is not a m a n i f o l d in general. An action is

transitive if for any two points x,y E M ~g E G O: y = gx ,

so all of M is one orbit and the orbit space is a single point.

A G-manifold with a transitive action is called a homogeneous

 -5-

space. A particularly important example of a h o m o g e n e o u s space is

obtained as follows: Let G be a c o m p a c t Lie g r o u p and H a

closed subgroup. The c o s e t space of H, G / H admits a natural

action of G by m u l t i p l i c a t i o n and the a c t i o n is c l e a r l y transi-

tive.

1.3. G- Vector Bundles

A fiber bundle ~ = (E,B,F,p) consist of a total space E ,

base space B , map p : E ~ B called bundle projection, a fiber

P , an open cover ~ and for each U E '~ a h o m e o m o r p h i s m

qO~T: ~ x S ~ p-1(~)

so that the c o m p o s ~ i o n p O~U is p r o j e c t i o n onto the first factor.

The s t r u c t u r e group G of a f i b e r b u n d l e is a g r o u p of h o m e o m o r -

phisms containing the h o m e o m o r p h i s m s F - p-l(b) defined by

x - ~ (b,x) , and their inverses, for every b E B . It is assu-

med that G acts on the above homeomorphisms transitively on the

right. A fiber bundle is p r i n c i p a l if the fiber is a t o p o l o g i c a l

group G which is also the structure group of the bundle. A

vector bundle is a fiber bridle with fiber a vector space and

structure group the g e n e r a l linear group of that v e c t o r space.

Thus a r e a l v e c t o r bundle has f i b e r ~n and g r o u p GL(n) .

Typical example of a v e c t o r bundle is the t a n g e n t bundle TM of

a smooth manifold Mn . The f i b e r at x E }~ , TM x = ~ n and the

total space of the bundle, TN is a s m o o t h m a n i f o l d of d i m e n s i o n

2n . A G-vector bundle is a G - m a n i f o l d Id and a v e c t o r bundle

with total space E over M so that there is a G - a c t i o n on E

compatible with the b u n d l e structure, i.e. the map f r o m E x =pl(x)

to E is an i s o m o r p h i s m m a k i n g the d i a g r a m below commutative.

gx
 - 6 -

G ×E > E

i id×p ~vp

G xM > M

Typical example is the tangent bundle TM of a G - m a n i f o l d M .

The m a p from TM x to TMg x is g i v e n by the d i f f e r e n t i a l of the

map g: M ~ M evaluated at x .

Given x E M the map gG x - gx defines an e q u i v a r i a n t em-

bedding G/G x ~ M with image Gx , the orbit of x . Thus we

may identify the G - m a n i f o l d s G/G x and Gx . Next we shall see

that the n o r m a l bundle of Gx in ~ is n a t u r a l l y a G-vector

bundle.

Let E ~ G/H be a G-vector bundle with base the h o m o g e n e o u s

space G/H . Let V denote the fiber at eH . Since h ~ H

leaves eH i n v a r i a n t , it l e s v e s V setwise fixed so V is an

H-module. Consider the p r i n c i p a l H bundle G ~ G/H and the

associated V bundle G ×H V over G/H obtained from G × V

by i d e n t i f y i n g [g,v~ = Egh,h-lv] . Let G act on G ×H V by

k E G k[g,v] = [kg,v] . Since V c E given g E G , v E V we

have gv E E , thus we h a v e a map [g,vl ~ gv consistent with

the identification, resulting in a m a p

G ×H V > E

which is c l e a r l y a G-vector bundle isomorphism. Thus a G vector

bundle over G/H is d e t e r m i n e d by the H - m o d u l e structure of the

fiber at eH .

Returning to the case when H = G x ~ the n o r m a l bundle at

x E Gx has fiber V x = TMx/(TGx) x . For each g E Gx the d i f -

ferential of g: M ~ M inducas a linear map Vx ~ Vx providing

a representation G x ~ GL(Vx) called the slice representation.

 - 7-

Its i m p o r t a n c e is g i v e n by the f o l l o w i n g theorem.

1.4. Some Basic Results

Slice theorem. Some G - i n v a r i a n t open n e i g h b o r h o o d o f the

zero section of G ×Gx V x mis e ~ u i v a r i a n t l y diffeomorphi~ t_~o ~

G-invariant tubular neighborhood of the orbit Gx i__nn M by the

map Eg,v] - gv so that the zero section G/G x maps onto the

orbit Gx .

A proof is g i v e n in J ~ n i c h Eli.

This gives at x E M a slice Sx with the f o l l o w i n g properties:

(i) Sx is i n v a r i a n t under Gx ,

(ii) if g E G , y,y' ~~ S x and g(y) = y' , then g E Gx ,

(iii) there exists a "cell n e i g h b o r h o o d " C of G/G x so that

C × Sx is h o m e o m o r p h i c to a n e i g h b o r h o o d of x . If F: C ~ G

is a local cross section in G/G x then the map F: C × S x ~ M

defined by F(x,s) = ~(c)s is a h o m e o m o r p h i s m of C × Sx onto

an open set c o n t a i n i n g Sx in M . In the d i f f e r e n t i a b l e case

we m a y choose Sx as a suitably small closed d i s k in Vx .

Another useful theorem from the g e n e r a l theory of t r a n s f o r -

mation groups is the f o l l o w i n g

Principal Orbit Type Theorem. Le_~t M be a G - m a n i f o l d and

assume that M/G is connected. Then there is an orbit type (H)

so that the orbits of this t2pe , ~(H) f o r m a dense subset of

and the s m o o t h manifold M(H)/G is conuected. The type (H) i_~s

called principal orbit type, an orbit is called a principal orbit

and the bundle M(H ) ~ M(H)/G is called the p r i n c i p a l orbit

bundle.

A proof is given in J ~ n i c h ~1].

 - 8 -

We shall also use the f o l l o w i n g result.

Conjugate Sub[roup Theorem. Let G be a q o m p a c t Lie g r o u p

actin~ on a m a n i f o l d M . If x E M and U c G is an open set

containing Gx then for y s u f f i c i e n t l y n e a ~ to x , Gy c U .

A proof is g i v e n in M o n t g o m e r y - Z i p p i n [I, p.2153.

1.5. The Circle Group

We are p a r t i c u l a r l y interested in the circle group G = SI .

Recall first that there are d i f f e r e n t ways of thinking of this

group:

(i) G = U(1) = [z E ~ Izl = I} , complex n u m b e r s of m o d u l u s I;

(ii) G = S0(2) , 2 x 2 real orthonormal matrices of d e t e r m i -

nant +I ;

(iii) G ~ T I =~/~ , reals modulo the integers. (When c o n v e n i e n t

we shall think of the e q u i v a l e n t form ~/2~ , i.e. e l e m e n t s

of G wi~ll be angles ~ where 0 ~ ~ < 2~ .)

Explicit isomorphisms are easily c o n s t r u c t e d and we shall use

these three forms of G interchar~y and w i t h o u t further warning.

The closed subgroups of SI are (e) , the cyclic groups Z and

SI and by the Conjugate Subgroup T h e o r e m the p r i n c i p a l orbit type

of an SI action is (e) . The purpose of this chapter is to

give an e q u i v a r i a n t classification of closed, connected 3-dimen-

sional St-manifolds. First c o n s i d e r some examples.

I) Let

S 5 = { Z l , Z 2 6 ¢2 ! z1~1 + z2~2 = 1}

and define an action of U(1) by t 6 U(1)

t(zl,z2) = (t~zl,t~z2) .
This a c t i o n is effective w h e n (~,~) = I . The orbit [z I = O,

z2~ 2 = I] has isotropy g r o u p ~ and the orbit [z 2 = O, z i ~ i = I]

has i s o t r o p y g r o u p ~ . All other orbits are principal. We

shall see l a t e r that fixed point free S] actions on $3 are in

one-to-one correspondance w i t h the pairs (~,~) .

2) Consider S3 as above w i t h the action

t(z 1,z 2) = (z 1,tz 2) •

The action has one circle of fixed points, [z 2 = O, ZlZ I = 1] and

all other orbits are principal. ~e shall see that this is the

only action on S3 w i t h fixed points.

3) Take any closed 2 - m a n i f o l d B and let M = B x SI . Define

an action of SI to be trivial in the first factor and the usual

one in the second. This gives a free S1 action w i t h orbit space

B .

4) Let V = D2 x SI be a solid torus w i t h SI action trivial

in the first factor and standard in the second. The subgroup

~2 c SI operates on the b o u n d a r y w i t h the p r i n c i p a l (antipodal)

action. If we collapse each of the orbits on the b o u n d a r y of V

by this ~2 action we obtain a closed m a n i f o l d N with 81

action. There are only p r i n c i p a l orbits (corresponding to the

interior of V ) and orbits w i t h i s o t r o p y g r o u p ~2 (correspon-

ding to the b o u n d a r y of V ) that are doubly covered by n e a r b y

principal orbits so that the local o r i e n t a t i o n is reversed. The

orbit space of the a c t i o n is a disk w i t h p r i n c i p a l orbits in the

interior and orbits with i s o t r o p y g r o u p E2 on the boundary.

The m a n i f o l d N is the n o n - t r i v i a l S2 bundle over SI called

the n o n - o r i e n t a b l e handle.

Before investigating the orbits w i t h n o n - t r i v i a l isotropy

 -10-

groups let us r e c a l l the o r i e n t a t i o n conventions of R a y m o n d [I]

and N e u m a n n [1]. Given an o r i e n t e d manifold M, its b o u n d a r y

~N is g i v e n the o r i e n t a t i o n which followed by an inward n o r m a l

coincides with the o r i e n t a t i o n of M . If M is an o r i e n t e d SI

manifold and N* is an o r i e n t a b l e manifold, then we orient M*

so that M* followed by the n a t u r a l orientation of the orbits

gives the o r i e n t a t i o n of M .

1.6. Fixed Points

Assume that Gx = S I so x is a fixed point. The slice

at x may be chosen as a s u f f i c i e n t l y small closed 3-ball D3

and the a c t i o n of Gx is an o r t h o g o n a l action of SI on D3

This is e q u i v a l e n t to the r o t a t i o n of D3 about an axis through

x . The orbit space of this action on D3 is a closed 2-disk

with x on the boundary. So fixed points lie on 1 - d i m e n s i o n a l

submanifolds and, by compactness~ circles. A sufficiently small

tubular neighborhood of one component of fixed points is t h e r e f o r e

a solid torus w i t h only fixed points and p r i n c i p a l orbits. If we

parametrize such a solid torus V = D 2 x SI by (r,y,8) O~r~1 ,

0 ~ y,8 ~ 2~ and let SI act by a d d i t i o n of angles, 0 ~ 0 < 2~,

then the a c t i o n is e q u i v a l e n t to

9(r,¥,6) = (r,¥+~,5) .

. . . . . . . .

/ , , / 4"'\ 5

Call the c o l l e c t i o n of fixed points F and the (finite)

number of components of fixed points f .

 -11-

1.7. E x c e p t i o n a l Orbits

Let Gx = ~ . The o r b i t is l - d i m e n s i o n a l and the slice m a y

be c h o s e n as a 2-disk, D2 . The actions of E on D2 are equi-

valent to r o t a t i o n (u>2) and r o t a t i o n or r e f l e c t i o n ( ~ = 2) .

Con~ider the r o t a t i o n s in this s e c t i o n and the r e f l e c t i o n in the

next. Let ~ = 2w/~ act on the u n i t disk as f o l l o w s

~(r,y) = (r,y+v~)

where (u,v) = I and 0 < v < W .

We call this the s t a n d a r d linear action of type [U,v] • Since

this is the a c t i o n in e a c h slice of such an e x c e p t i o n a l orbit

(called E - o r b i t ) , a small tubular neighborhood is a solid torus

V with action equivalent to

~(r,y,8) = (r,y + v@, & +~) .

The E - o r b i t corresponds to r = 0 and has isotropy group of order

. We call [~,v] the o r i e n t e d orbit invariants. The c o r r e s p o n -

ding oriented Seifert invariants (a,fl) are d e f i n e d by

a = ~ , ~v ~ 1 mod a , 0 < fl < ~ .

Their geometric interpretation is the f o l l o w i n g .

Given an o r i e n t a t i o n on V. orient the slice so that it f o l l o w e d

by the E - o r b i t gives the o r i e n t a t i o n of V . This orients the

boundary of tile slice mj a curve that is n u l l - h o m o t o p i e in V .

L~t 1 be a curve on $V homologous in V to the E - o r b i t and

so that the o r d e r e d pair m,1 gives the o r i e n t a t i o n on BV . Let

h be a n oriented principal orbit on ~V . Since the a c t i o n is

principal on all of ~V it admits a cross-section, q and any

other section, q' is r e l a t e d to q by

q' = ~ q + sh

for some s . Orient q so that the o r d e r e d pair q,h gives

 -12-

the same orientation as m,l . Then we have

m = a q + ~h

and a suitable choice of s reduces ~ to the interval 0<13 <m.

Similarly

! = - ~ q - ph

for some v and p so that

a B
= I
-v-p

thus ~ - I mod a .

Solving for q and h in the m,1 cystem we have

q = - p m - S1

h = vm + al

Since 1 may be changed by l' = l+ sm we can reduce v in the

range 0 < ~ < a . In this case

= (s',- 1)/o~ .

In the action above, the curve

q = [r = 1, y = p~p, 5 = ~, 0 ~ ~ < 2~) c 8V

oriented by decreasing ~ will satisfy the above conditions.

I}z i \
~./
"

Changing the orientation on the solid torus V, keeping the

action fixed results in a changed orientation for the slice and

 - 13-

hence the slice inveriants change to [~,~] = [~,~-v] . Similarly

the S e i f e r t invariants change to (~,8) = (a,a-~) . Thus the op-

posite o r i e n t a t i o n satisfies the c o n d i t i o n

~ ~-I mod a .

The latter was used in 0 r l i k - W a g r e i c h [1,2].

If there is no o r i e n t a t i o n specified on the solid torus V,

then the orbit invariants are only d e f i n e d as [W,v] , 0 < v ~ W / 2

and the Seifert invariants (a,~) , 0 < ~ ~ a/2 with v~ ~ ± 1

mod a . We shall call these the u n o r i e n t e d orbit and Seifert

invariants.

1.8. Special E x c e p t i o n a l Orbits

If Gx = E 2 and the a c t i o n in the slice is r e f l e c t i o n about

an a r c ~ t h e n the n e i g h b o r h o o d of such a special exceptional (SE)

orbit is easily seen to be d i f f e o m o r p h i c to the c a r t e s i a n p r o d u c t

of the M o e b i u s band w i t h an interval, the n o n - t r i v i a l D 2 bundle

over SI . All orbits i n t e r s e c t i n g the arc of r e f l e c t i o n are

SE-orbits,thus a component of SE-orbits is a torus. Let SE

stand for the c o l l e c t i o n of SE-orbits and t denote the (clearly

finite) n u m b e r of c o m p o n e n t s of SE .

1.9. The Orbit Space

As we have n o t e d in the last three sections, the orbit space

is a m a n i f o l d n e a r F*, E* and SE* . It is clearly a m a n i f o l d

near principal orbits, so we conclude:

Lemma 1. The orbit space M* is a compact 2-manifold with

boundary consisting of F* U S E * .

Let us associate the symbol ¢= o w i t h an orientable and

 - 14-

c = n with a non-orientable orbit space and let g denote the

genus in e i t h e r case. If e = o we assume that an o r i e n t a t i o n

of M* is given. Thus we m a y a s s o c i a t e the 4 - t u p l e (¢,g,f,t)

with M* where c =o or n , g ~ 0 , f ~ 0 is the n u m b e r of

boundary components in F* and t > 0 is the n u m b e r of b o u n d a r y

components in SE*

Lemma 2. If F I~ SE # ~ an__dd E = ~ then (e,g,f,t) is a

complete set of invariants for ~ up to e q u i v a r i a n t diffeomor-

phism (preservin~ the o r i e n t a t i o n of ~* if ¢ = o ) .

Proof. We show that the a c t i o n admits a cross-section.

Since E = ~ we have a principal bluudle over M * - F* U S E * and

since F* U S E * ~ ~ this b u n d l e is trivial. Choose a cross-sec-

tion to this bundle. It is n o w sufficient to extend this section

in the n e i g h b o r h o o d of each F-component and each SE-component.

By (1.6) the n e i g h b o r h o o d of an F-component is a solid torus V

in M . The given cross-section restricted to BV is a torus

knot of type (1,b) for some b and it is w e l l - k n o w n that there

is an annulus in V spanned by this knot and the "center curve"

(F-component) that extends the section. A similar argument ap-

plies to SE-components.

Next let us c o n s i d e r the somewhat more interesting case w h e n

F USE U E = ~ . Here all orbits are p r i n c i p a l and we have a

bundle over the closed 2-manifold H* . This bundle is classified

by a map M* - CP c° and hence by an element of H2(M*;Z). This

element is called the chern class or euler class of the bundle.

If c = o then H2(M*;E) = ~ and if c=n then H2(M*;~) = E2

so the o b s t r u c t i o n to the bundle being trivial is an integer b

where b E Z if e= o and b E ~2 if c =n .
 -15-

We m a y interpret this integer b as f o l l o w s : Remove the

interior of a s o l i d torus Vo from ~i . The r e m a i n i n g manifold,

Mo admits a cross-section ~g . Let qo be the cross-setion to

the action on the b o u n d a r y oriented as the b o u n d a r y of -M* The

o "
equivariant sewing of the solid torus V° into Mo is d e t e r m i n e d

up to e q u i v a r i a n t diffeomorphism by s p e c i f y i n g the curve on the

boundary of M
o
m = qo + bh

that is to b e c o m e nullhomotopic in V° . We h a v e obtained the

following:

Lemma 3. If E ~ F USE = ~ then Id i_~s d e t e r m i n e d u~ to

e~uivariant diffeomorphism by ¢ , g and b where b E • if

e = o and b E ~'2 if ¢ =n .

In case c = o the total space M is o r i e n t a b l e . A change

of o r i e n t a t i o n of M results in a c h a n g e of s i g n for b .

We n o w h a v e all the ingredients for the g e n e r a l case.

1.10. The Classification Theorem

Let SI act effectively and smoothly on a closed, connected

smooth 3-manifold M . Then the ~ o l l o w i n ~ orbit invariants

M = {b;(~,g,f,t); (a1,B1),...,(ar,Sr)}

subject to the conditions

(i) b = O if f+t >0 ,

b E~ if f+ t = O and c= o ,

b E2~ 2 if f+t = 0 and e=n ,

b = 0 if f+t = O , c=n and aj = 2 for some J ;

 -16-

(ii) 0 < ~j < aj , (aj,Sj) = I if e= o ,

0 < 8j _< aj/2 , (ej,Bj) = 1 __if ¢ =n ;

determine M up t__~oan equivariant diffeomorphism (which preserves

the orientation of X* if ¢ = o).

Proof. Given the above set of invariants a standard action

is constructed as follows: Remove from X* (r+l) disjoint open

disks D*o''''' D*r ' If F USE = ~ then the r e m a i n i n g manifold is

a trivial principal bundle over

X* - ~ D3 and admits a cross-
j=o a
section. If F~U S E ~ ~, remove the~ boundary components of
M* - rU D~, construct a cross-section and extend it to F* U SE*
j =O J

as in (1.9.2). Let Mr be the resulting manifold with (r+l)

N,,
boundary components and let X~ be the cross-section. Sew in

neighborhoods Vi of E-orbits with Seifert-invariant (aj,Bj)

j = 1,...,r next. Let Q be a boundary component of M~

I
and Q x S the corresponding boundary component of Mr . Let

Q x {0} be the cross-section. Now sew the solid torus V of

(1.7) equivariantly onto this boundary by mapping orbits onto

orbits and the cross-section q of V onto Q x [0] . Paramet-

rize Q x SI by [y,6} , where increasing y orients Q as a

boundary component of ~ .

Define the equivariant map

I
F:QxS ~8V

by

~(y,~) = (py+v~, ,sv+=5) .

Notice that

Ii =-I

and therefore ~ is o r i e n t a t i o n reversing as required. The

 - 17-

oriented c r o s s - s e c t i o n q of 8V maps onto the oriented curve

-Q.

The e q u i v a r i a n t sewing is therefore specified by the follow-

ing. G i v e n the c r o s s - s e c t i o n M~*r in Mr let qo,ql, "''qr be

cross-sectional curves in BM r oriented opposite to the induced

orientation as c o m p o n e n t s of ~i~ . The e q u i v a r i a n t sewing of

the solid torus Vj j = 1,...,r makes the curve mj = a j q j + Bjh

on the j-th component of 8~ null-homotopic in Vj .

If e= o then the pair (aj,~j) is d e t e r m i n e d in the inter-

val 0 ~ ~j ~ aj and if e =n only 0 ~ Bj ~ aj/2 since the

local o r i e n t a t i o n may be r e v e r s e d along a path in M* . We n o w

have a m a n i f o l d M° w i t h one torus b o u n d a r y and a c r o s s - s e c t i o n

qo to the action. We sew the last solid torus Vo fibered tri-

v i a l l y onto this b o u n d a r y so that the surve mo = qo + bh becomes

null-homotopic in V This gives a manifold M w i t h the re-

o
quired action.

Conversely, g i v e n an a c t i o n on M, we shall r e c o v e r its orbit

invariants as follows: Read off ¢,g,f,t f r o m the orbit space,

_v_

I~'~ . The e q u i v a r i a n t tubular n e i g h b o r h o o d s of E-orbits are iso-

lated. E a c h one is e q u i v a r i a n t l y diffeomorphic to a solid torus

V as d e s c r i b e d in (1.7) and the action is d e t e r m i n e d by the

Seifert invariants (a,~) , 0 ~ ~ ~ ~ . If c =n we use an iso-

topy of the tubular n e i g h b o r h o o d along a path r e v e r s i n g the orien-

tation on V* to reverse the o r i e n t a t i o n on V . This reduces

to O ~ ~ ~ a/2 . These pairs are invariants of V up to

equivariant (orientation preserving, resp. not) diffeomorphism,

specifying cross-sections ql,...,qr on the boundaries. If

USE ~ ~ these cross-sections may be e x t e n d e d to a g l o b a l

cross-section. If ~ USE = ~ and ~ = o we have an o b s t r u c t i o n

in
 - 1 8 -

H2(M *-int(V~ U c o .U V~) 9 5(V~ U . .i U V ;);~)

~" •

Its class is i d e n t i f i e d w i t h the integer b . If P USE =

and ~= n the above g r o u p equals ~2 and b may take on the

values 0 or I . A special argument shows that in the presence

of an E-orbit of type (2,1) the two actions are e q u i v a r i a n t l y

diffeomorphic, see Seifert [I, H i l f s a t z VIII.

It is easy to check that if M is orientable ( c = o and t = 0),

then a change of o r i e n t a t i o n results in the new orbit invariants

-M = [b';(o,g,f,0);(a1,~1-81),...,(ar,~r-Sr)]

where b' = 0 if f > 0 and b' = - b - r if f = 0 .

In order to facilitate the n o t a t i o n we shall not insist that

the Seifert invariants always be normalized. Writing M with

these invariants should cause no confusion since the n o r m a l i z a t i o n

is a well defined process.

Another notational c o n v e n t i o n will be the occasional use of

the orbit i n v a r i a n t s [W,~] instead of the a s s o c i a t e d Seifert

invaris~uts (a,~) Again, the c o n v e r s i o n is unique.

1.11. R e m a r k s

I. The e q u i v a r i a n t classification of (1.10) does not a n s w e r

the q u e s t i o n of w h e n two St-manifolds are h o m e u m o r p h i c i.e.,

what are the possible different actions on a g i v e n m a n i f o l d (c.f.

the examples in 1.5). We shall call this the " t o p o l o g i c a l classi-

f i c a t i o n problem".

(i) If F USE = ~ the m a n i f o l d s involved coincide w i t h

Seifert's classes 0,o and N,nl . These (together w i t h the

other Se[fert m a n i f o l d s introduced in chapter 5) are the central

objects of our c o n s i d e r a t i o n s and their m u t u a l h o m e o m o r p h i s m rela-

 -19-

tionship will be d i s c u s s e d in d e t a i l in chapters 5 and 7 • These

manifolds are irreducible with universal cover S3 or R3 °

(ii) If F ~ ~ then the i d e n t i f i c a t i o n of the m a n i f o l d s is

done u s i n g equivariant connected sums. An arc S* in the orbit

space w i t h end points on fixed point components and i n t e r i o r

points correspanding to p r i n c i p a l orbits has as inverse image u n d e r

the orbit m a p a 2-sphere, S . Using such arcs the m a n i f o l d is

decomposed as the e q u i v a r i a n t connected sum of 3-manifolds with

the f o l l o w i n g orbit spaces.

L* = / ~ = [o~o,o,l,o);(~,~)}

Clearly L is the r e s u l t of an e q u i v a r i a n t sewing of a solid

torus n e i g h b o r h o o d of P , V1 and a solid torus n e i g h b o r h o o d of

the E-orbit, V2 . Let hi and qi be the orbit and c r o s s - s e c -

tion in 8V i . Then we have the r e l a t i o n s for the b o u n d i n g curves

m I = h I, m 2 = aq 2 + ~h 2 The equivariant sewing is h2 ~ hI ,

q2 ~ -ql and g o i n g through the computations of (1.7) shows that

we obtain the lens space L(a,B) .

i~* = ~ O F ~ / " ,/ N, = [ 0 ; ( o , 0 , 2 , 0 }

Obviously ~¢ = $2 x S1 with the standard S1 action on the f i r s t

factor and trivial action on the second factor.

= [o;(o,o,1,1)}
 - 2 0 -

Similarly p = p2 x 81 with the standard S1 action on p2 and

trivial action on the second factor.

N = [0;(n,1,1,0)}

The manifold N is the non-orientable S2 bundle over 81 The

1
action is visualized by taking 82 x I with the usual S action

in the first factor and identifying 82 x 0 and S2 x I so that

the orbits are reflected about the equator of 82 .

We state the following result without proof, Raymond [1].

Theorem. Let

M = [b;(c,g,f,t); (a1,~l)9...,(ar,~r)}

and assume that f > O . Then M is equivariantly diffeomorphic

to the equivariant connected sum:

(a) S3 ~-(S2x $1)1 ~::...#(S2x81)2g+f_1{(p2x81~...#(p2xS1) t

# L(al,~ 1) #...~ L(ar,Sr) if (¢,g,f,t) = (o,g,f,t) , t ~ 0 ;

(b) (S2x sl)1 #...# (S2x S1)g+f_l ~ (p2x 81)1 {... ~ (p2xsl)t
# L(al,~1) #...~ L(ar,er ) if (c,g,f,t) = (n,g,f,t) , t > 0 ;

(c) N # (82x 81)1 # . ." # (S2x S1 )g+f-2 # L(¢l'fll) # ' ' ' #
L(~r,Br) if (¢,g,f,t) = (n,g,f,0) .

(iii) The case F = ~ , SE / ~ is handled using the classi-

fication of 8eifert manifolds. The action lifts to the orientable

double cover and commutes w i t h the covering transformation. For

details see 0rlik-Raymond [I].

 -21 -

2. We assume that M is a smooth manifold and SI acts

smoothly. It is known that all 3-manifolds are smoothable and

using somewhat more elaborate arguments all the results hold for

continuous SI actions on topological 3-manifolds, Raymond [I].

It follows from the discussion above that for the class of 3- ma-

nifolds with SI action the Poincar~ conjecture holds.

3. Raymond Eli also studies the case v~en M is not com-

pact. Allowing boundary makes the equivariant classification

more cumbersome but essentially the same.

4. The classification above provides us with examples of

manifolds that admit no SI action at all, e.g. any connected

sum not on the list of the theorem.

 - 22 -

2. E q u i v a r i a n t Plumbing

Plumbing is i n t r o d u c e d for b u i l d i n g blocks that are D2

bundles over closed, orientable 2-manifolds, where it e s s e n t i a l l y

consists of r e m o v i n g a D2 x D2 from each of the objects and iden-

tifying the r e s u l t i n g b o u n d a r i e s after an i n t e r c h a n g e of factors.

Prescribing an action of SI on the b u i l d i n g blocks we may re-

quire that the p l u m b i n g r e s p e c t this action. The r e s u l t i n g 4-

manifold with boundary is studied in terms of the g r a p h of the

plumbing. The b o u n d a r y is a closed, orientable 3-manifold w i t h

S I action and may be identified in terms of (1.10).

These ideas were first introduced by H i r z e b r u c h [1] and

yon R a n d o w [I]. The e q u i v a r i a n t analoo~e was n e e d e d in O r l i k and

Wagreich [I] to resolve singularities of algebraic surfaces w i t h

C* action. This a p p l i c a t i o n is presented in the next chapter.

The o r i e n t a t i o n convention adopted here is that of R a y m o n d

[I]. The opposite was used in O r l i k - W a g r e i c h [1,2], where the

letter b is also used differently.

2.1. P l u m b i n g

The p r i n ~ p a l S0(2) bundles over a closed, orientable 2-

manifold M are classified by H2(M;E) = ~ . Denote the associ-

ated D 2 bundles indexed by m E Z as ~ = (Ym,~,M) The com-

pact 4-manifold Ym has the h o m o t o p y type of M aud if we let

the zero s e c t i o n v: M ~ Y represent the positive generator

m
g E H2(Ym;E) J then its s e l f - i n t e r s e c t i o n number g.g = m is the

Euler class of Ym " It is c u s t o m a r y to let the bundle w i t h

E u l e r class m =- I over S 2 , ~ = (Y-I ,~,S 2 ) , be the disk

bundle whose boundary, S 3 , has the Hopf fibration.

 - 23 -

Given two such bundles ~I = (Ym1'~1'~1) and ~2 =

(Ym2,~2,M2) we plumb them together as follows. Choose 2-disks

B1 c MI and B2 c M2 and the bundles over them, ~I and ~2 "

Since they are trivial bundles there are natural identifications

~1: D 2 x D2 ~I , U2: D 2 x D 2 ~ ~2 " Consider the reflection

t: D 2 x D 2 - D 2 x D 2 , t(x,y) = (y,x) and define the h o m e o m o r p h i s m

f: ~1 ~ ~2 by f = ~2 t ~i-1 . Pasting ~I and ~2 together

along ~1 and ~2 by the map f is called plumbing. It yields

a topological 4-manifold with corners that may be smoothed. The

resulting smooth manifold is independent of the choices involved.

A graph r is a finite, 1-dimensional, connected simplicial

complex. Let Ao,...,A n denote its vertices. A star is a con-

tractible graph where at most one vertex, say A ° , is connected

with more that two other vertices. If there is such a vertex,

call it the center. A weighted graph is a graph where a non-nega-

tive integer gi (the genus) and an integer mi (the weight) is

associated with each v e r t e x A .

Given a weighted graph r we define a compact 4-manifold

P(r) as follows: For each v e r t e x (Ai,gi,mi) take the D 2 bun-

dle ~i = (Ymi'Wi'Mi) where Mi is a closed, orientable 2-mani-

fold of genus gi " If an edge connects Ai and Aj in r then

perform plumbing on ~i and qj . If Ai is connected with more

then one other vertex, choose pairwise disjoint disks on Mi to

perform the plumbing. Finally smooth the resulting manifold to

obtain P(r) .

2.2. E q u i v a r i a n t Plumbing

We shall now define S I actions on the building blocks ~ =

(Ym,~,~) . For g > 0 let SI act trivially in the base and by

 - 24 -

rotation in e a c h f i b e r . For g = 0 we d e f i n e actions on D =

( Y m , ~ , S 2) as f o l l o w s : Let S 2 = B 1U B 2 be the u n i o n of two

2-disks and Y m = B1 x D I U B 2 x D 2 . Parametrize D2 x D2 in

polar coordinates with radii r and s , 0 ~ r,s ~ I and angles

y, 6, o ~ y, 5 < 2n . The actions of SI on D2 are equivalent

to l i n e a r actions and we shall think of t h e m as a d d i t i o n of angles.

Let 8 6 S I , o < e < 2n m

. Define for i = 1,2

8i: D 2 x D 2 ~ D2 x D2

8i(r,y,s,5) = (r,y+ui%,s,8+vi8)

Now Ym is o b t a i n e d b y an e q u i v a r i a n t sewing

G: ~B I x D I ~ 8B 2 x D 2 .

Since the action is l i n e a r , G is d e t e r m i n e d by

F: ~B 1 x ~ D I ~ ~B 2 x D 2

which in t u r n is i s o t o p i c to a l i n e a r map of the torus. Let F

be d e f i n e d by

F(y) = xy+y6 , F(6) = zy+t8 .

Then F is e q u i v a r i a n t if

UlX+VlY = u2 and UlZ+Vlt = v2 .

In o r d e r that G be equivariant on 5B I x 0 - 8B 2 x 0 we n e e d in

addition that UlX = u 2 , thus y = 0 .

Since the determinant of F is -1 and the sewing results in a

total space with euler class m , we n e e d x =-I , t = 1 , z =-m.

Thus u 2 = -u I , v 2 =-mu I +v I . The action is e f f e c t i v e if a n d

only if (Ul,Vl) = I .

A plumbing is e q u i v a r i a n t if the identifying and trivializing

maps are e q u i v a r i a n t . Given a weighted graph F we say that

 - 25 -

P(F) is e q u i v a r i a n t if e a c h plumbing involved is e q u i v a r i a n t .

In that case the b o u n d a r y K(F) = ~P(F) is a 3-manifold with

SI action. We shall identify this m a n i f o l d for certain graphs.

For M = S2 we m a y think of the classifying element m as a

map S1 - SI of d e g r e e -m . As above, ~Ym is o b t a i n e d as the

equivariant union of two solid tori

8Yb = B~ × S1 F

where F has the m a t r i x

1 .

This is the sewing of two solid tori that results in the lens

space L(-m,1 ) . Due to the w e l l Imown diffeomorphisms L(p,q) =

- L(-p,q) = - L(p,p-q) , we m a y w r i t e

~Ym = L ( - m , 1 ) = L(m,m-1) .

Note also that the d i f f e r e n t actions on L(-m,1) are g i v e n by

the d i f f e r e n t pairs (Ul,Vl) . For example uI = 0 , vI = 1

(u 2 = 0 , v 2 = I) gives the free action

~(-m,1) = {-m;(o,o,o,o)} .

In case uI = I , vI = 0 we h a v e a circle of f i x e d points and

the orbit invariants are

L(-m,1) = [O;(o,O,1,0);(m,m-1)} .

Next consider the r e s u l t of ~ equivariant plumbing accord-

ing to the l i n e a r graph Fib I .... ,b s]

where each vertex has genus zero.

 - 26 -

Lemma I. The result of the e~uiyariant linear ~lumbing

according to the graph F[bl,...,bs] above is the lens space
LCps,p > where

Ps I
= bI ...,bs3
b2 I = [bl'

• I

bs

Proof. Decompose each base space as S i = Bi, I ~J Bi, 2 with

the corresponding trivializat!ons of the bundles. As we have

seen the first equivariant sewing requires Ul 2 = -Ul 1 and

9

Vl 2 = blUl I +vl I so it has matrix

9 9 ,

Since the plumbing is equivariant the actions of B1, 2 x $1, 2

and B2, I × $2, I are the same but the factors are reversed, i.e.

u2, I = vl, 2 and v2, I = Ul, 2 . The matrix of this map is

and we have that

Vl, I

The equivariant sewing of B2, I x $2, 1 and B2,2 x $2,2 has

matrix
 - 27 -

and the action on B2, 2 x $2, 2 is therefore expressed by

( u 2 , 2 , v 2 , 2) =
b2

Continuing the sewing results in the equation

Note that all orbits are principal with the possible excep-

tion of the center curves of BI, I × Si, 1 and Bs, 2 × Ss, 2 .

The orbit space of the complement of these two solid tori is an

annulus. Thus the total space is the result of the equivariant

sewing of two solid tori by the product matrix above. Let

(Us,2,Vs,2) = -I
Ps P~ / I, "
!
Then the total space equals the lens space L(Ps,Ps) , where
t
ps/Ps = [bl,b2,...,bs]° The latter fact follows from elementary
properties of continued fractions,vonRandow [I]. This completes the
proof.

In particular if the action on BI, I x $I, 7 has an orbit of

fixed points, Ul, I = I , Vl, I = 0 , then Bs, 2 x Ss, 2 has an E-

orbit with oriented orbit invariants [ps~-Ps_1 ] .
Next we shall show that equivariant plumbing imposes a strong

condition on the shape of the graph provided the weights are nega-

tive. This will be the case for the applications in the next
chapter.

Lemma 2. Let F be a weighted graph and assume that P(F)

is equivariant. If
 - 28 -

(a) F has a v e r t e x (Ao,go,mo) where the a c t i o n is t r i v i a l in

the base,

(b) for each vertex (Ai,gi,mi) we h a v e m i ~ -I , and

(c) for e a c h v e r t e x (Ai,O,-1) connected with (Aj,gj,mj) we

have gj > 0 o__rr mj ~ -2 (or both) then

(i) gi = 0 for all v e r t i c e s i > 0 ,

(ii) F is a w e i g h t e d star w i t h c e n t e r Ao ,

(iii) the a c t i o n is n o n - t r i v i a l on the b a s e for i > 0 .

Proof: Since we p l u m b around a fixed point, 0 x 0 c D 2 × D 2,

a vertex connected with more than two v e r t i c e s must have trivial

action in the base. Thus if AI is plumbed into Ao, it has n o n -

trivial action in the base, hence gl = 0 and ul I = I ,v I I =0"

F r o m above we get Ul, 2 = -I , Vl, 2 = -m I . Define inductively

Po = I , Pl =-ml ' P2 = - m 2 P l - P o ' Pj = - m j P j - I - P j - 2 ' j = 2,...,r.

T h e n the a c t i o n has uj,2 = -Pj-1 ' vj,2 = Pj " We d e f i n e the
! !
auxiliary parameters Po' = 0 ' Pl' = 1 ' P2' = -m2 ' P3 = -m3P2 -Pl '

Pj' = - m jPj-1
' - P j '- 2 ' j = 3,...,r . Then induction shows
! !
1) PjPj-1 - P j - l P j = -1 for 0 < j ~ r ,

2) (pj,p~) = I , (pj,pj_l) = I , (pj,pj_l) = I for 0 < j ~ r ,

3) if -mj ~ 1 for 0 < j ~ r and if -mj = I then -mj± 1 > I

!
implies that we h a v e pj / 0 and 0 < pj < pj .

This p r o v e s the lemma.

Lemma 5. Consider the star S below ~ith each bi, j ~ 2

and gi,j = 0 except for the center.

 - 29 -

The result of the equivariant boundary plumbing K(S) has

Seifert invariants

K(S) = {b;(o,g,0,O)~(~1~l),...,(~r,~r)}
where

aj Sj
aj- = [bj ,I,"" ,bj ,sj] , j = 1,...,r .

Proof: By Lemma I each linear branch gives rise to a sew-

ing of an E-orbit with orbit invariants [Psj'- Psj-1 ]" Since
Psi > 0 , aj = Psj and ~ = -Psj-1 •
!
From (1.7) and equation I) above we have P = Psj-1 and before
normalization ~ = -p~j . According to 3) the normalized =

This proves the assertion that

a+ = a -Psi
Psi 6.
P~ ~j_-~j = [bj,1,...,bj,sj] •
J
The Seifert invariants of the manifold before normalization equal
!
K(S) = {b+ r;(o,g,0,O);(PSl ' - P ! Sl )'''''(Ps r' -Ps r ))
 - 30 -

and normalization gives the required Seifert invariants.

Lemma 4. Given relatively prime integers (a,B) with

0 < 8 < a the fraction a/a- 8 ma 2 be obtained as a unique

continued fraction

= [bl,b2,...,bs]

where b i _> 2 , i = 1,...,s .

Proof: Repeated application of the I~clidean algorithm.

Corollary 5. Every Seifert manifold

K = {bl(o,g,O,O)l(a1,~l) .... ,(ar,~r)]

is the result of an e~uivariant plumbing according to a star S(K)

as in Lemma 3.

2.3. Quadratic Forms

Given a connected, oriented 4k-dimensional manifold M, a

quadratic form SM may be associated with it by homology inter-

sections. Let V = H2k(M;~) / torsion and define

by intersection of representative cycles. This is a well defined

symmetric bilinear pairing, hence it induces a quadratic form on

V , called SM . As usual, the form may be diagonalized over the

reals. Let p+ denote the number of positive entries and p_

the number of negative entries. The integer

• (M) = T(s M) = p + - p _
 - 31 -

is called the signature of the quadratic form (manifold). It is

called positive (negative) definite if p+ (p_) equals the rank

of V .

We want to compute the quadratic form of the compact 4-

manifold P(F). It is clear from the remarks of (2.1) that the

graph F contains all necessary information. We may choose a

basis for V consisting of one generator for each vertex (A,g,m)

of F with self-intersection number mp and any two vertices

connected in r have intersection number I.

In particular the star corresponding to the Seifert manifold

K = [b;(o,g,0,0);(a1,81),...,(ar, Br)]

S(K) provided in (2.2.5) has quadratic form with matrix below

where each unfilled entry equals zero.

-b-r I I I

1 -bl I I

1 -hl , 2

' 1
1 -hl,Sl

-b2,1 1

1 " I
SM =

-br, I I

I
I -br,sr

Since bi, j _> 2 for all i,j this matrix is easily seen to be

negative definite if and only if

-b-r < O.
 - 32 -

3. Resolution of S i n g u l a r i t i e s

This chapter describes some results from 0rlik-Wagreich rl,21.

Many of the ideas go b a c k to IIirzebruch [I].

Given a complex algebraic surface with singularities, V ad-

mitting a "good" action of @~ , the m u l t i p l i c a t i v e group of com-

p l e x members, we obtain a resolution of the s i n g u l a r i t i e s of V

by the f o l l o w i n g method. If V has an i s o l a t e d singularity, then

a small n e i g h b o r h o o d boundary Sc invariant under the action of

U(1) c @* intersects it in K = VmS¢ , a smooth, orientable,

closed 3-manifold with S 1 action. Given the orbit invariants

of K (1.10) we prove that the c o r r e s p o n d i n g star (2.2.5) is

the dual g r a p h of a (canonical equivariant) resolution of the iso-

lated singularity of V . If the s i n g u l a r i t y is n o t isolated then

a normalization must preceed the above construction.

I~ather than g i v i n g all the details as published, the e m p h a s i s

here is on a survey of the b a c k g r o u n d material, motivation and

examples.

3.1. Algebraic and A n a l y t i c Sets

~e shall define the n e c e s s a r y terminology as g i v e n in F u l t o n

~I~ and G u n n i n g [17. Let R be a c o m m u t a t i v e ring w i t h unit.

Let R[XI,...,Xn] denote the ring of pol~fnomials in n variables

over R . A polynomial F E ....

r~rx1,...,Xn~ is h o m o g e n e o u s of de-

gree d if each m o n o m i a l of F has degree d . An element a ER

is i r r e d u c i b l e if a = b. c implies that b or c is a unit.

A ring R is a d o m a i n if a.b = 0 implies a = 0 or b = 0 .

R is a UFD if every element has a unique factorization up to

units and order. If R is a UFD so is R[X] . In p a r t i c u l a r

k~X1,... ,X n 7
•- ~
is a UFD for any f i e l d ~ . The quotient field
 - 33 -

of k [ X I g . . . , X n] is the field of rational functions, k(X],°°.,Xn).

An idea I I c R is proper if I ~ R , maximal if it is contained

in no larger proper ideal and prime if ab E I implies either

a E I or b E I . An ideal is p ~ i n c i p a l if it is generated by

one element. A principal ideal domain (PID) is a domain where

every ideal is principal. The residue classes of elements in R,

modulo an ideal I, form a ring R/I and the natural map ~ :R

R/I is a ring homomorphism. In particular k[Xl,.°.,Xn]/I is

a vector space over k . Given an ideal I, define its radical

by radI = [aER! an E I for some integer n ~ O~ .

Let Gn be the affine complex n-space. If S is a set of

polynomials in C[ZI,...~Zn] let V($) = [~ E ¢ n F(~) = O for all

F ES] . Clearly V(S) =

N V(F) . A subset X E ~n is algebra-
F ~S
i__~c if X = V(S) for some S . Note the following properties~

(i) if I is the ideal in C[Z] ..... Zn] generated by S then

V(S) = V(1) , so every algebraic set is equal to V(I) for some

ideal I ;

(ii) if [I ] is any collection of ideals, then V(~ Ia) = anV(I ),

so the intersection of any collection of algebraic sets is an al-

gebraic set;

(iii) V ( F . G) = V(F)!!V(G) , so any finite union of algebraic sets

is an algebraic set;

(iv) if I defines an algebraic set then I = tad I .

A ring is N o e t h e r i a n if every ideal is finitely generated.

In particular the Hilbert Basis Theorem shows that C[Zl,...,Z n]

is Noetherian.

Pr__~ective complex n-space C~ n is defined as all lines

through the origin ~ E ~n+1 Any point z = (Zo, Zn) ~ O

defines a unique line [kZo~ .... kZn! k E C*] and two points ~, ~'
 - 34 -

determine the same line if and only if there is a k ~ C* so that

z i = X z!1 for all i We let the e q u i v a l e n c e class of these

points EZo:Z1:...:Zn] be the h o m e o g e n e o u s coordinates of a p o i n t

in C~ n . A projective algebraic set X is defined by h o m o g e n -

eous polynomials. It is i r r e d u c i b l e if its ideal I(X) is prime.

In that case the residue ring R X = C[Xo, .... X n ] / I ( X ) is a d o m a i n

called the h o m o ~ e n e g u s coordinate rin~ of X .

The ring of germs of h o l o m o r p h i e functions in n variables

at ~ E ~n is denoted ~a " It is identified with the ring of

convergent complex power series @~z I - a1~...,z n - a n ] . For ~=2

call the ring simply ~.~ . Note that for any two points ~, _b the

rings ~a an~ ~b are c a n o n i c a l l y isomorphic. The ring ~is

a Noetherian UFD . Its quotient field ~fY~ is the field of germs

of m e r o m o r p h i c functions at ~ . The units of ~ are h o l o m o r p h i c

germs not zero at 0 . The ideal i of n o n - u n i t s in ~ is m a x i -

mal and ~ is called a local

r ~ . Note that ~/I ~ C .

The sheaf of ~erms of h o l o ~ o r ~ h i c functions in n variables

is also d e n o t e d ,J . For any open set U c Cn there is a n a t u r a l

identification of the sections F ( U , C v) w i t h the ring ~U of

holomorphic functions over U . For any p o i n t a E @n the stalk

of ~ at ~ is n a t u r a l l y the ring (~a d e f i n e d above.

analytic sheaf ~ over an open set U c ~n is a sheaf

of m o d u l e s over the r e s t r i c t i o n ~U . It is f i n i t e l y g e n e r a t e d

over U if there are f i n i t e l y many sections of ~ over U which

generate the stalk ~a as an Q~ a module at each point --

a E U "

An a n a l y t i c subvariety X of an open set U c @n is a sub-

set of U w h i c h in some open n e i g h b o r h o o d of each p o i n t of U is

the set of common zeros of a finite number of functions defined

and h o l o m o r p h i c in that n e i g h b o r h o o d . Two such pairs (XI,UI) ,

 - 35-

(X2,U2) are e q u i v a l e n t if there is an open n e i g h b o r h o o d W c

U I 0U 2 so that WOX I = ~ X2 . The equivalence class is called

a ~ e r m of an analytic subvariet~. The ideal of the s u b v a r i e t y at

a p o i n t is defined for the origin by I(X) = [f E (~o I ~ analytic

subvariety X of U c Cn representing the germ X and an analy-

tic f u n c t i o n f E ~JU representing the grem f with f!x ~ 0] .

A germ X is said to be r e d u c i b l e at ~ if X = X I UX 2 where

Xi are also germs of analytic subvarieties at ~ ; otherwise it

is i r r e d u c i b l e at a .
u

An analytic v a r i e t y is a IIausdorff space V with a distin-

guished subsheaf ~V of the sheaf of germs of continuous complex

valued f u n c t i o n s on V so that at each point ~ E V the germ of

V together w i t h the stalk ((~V) a is called the sheaf of germs

of h o l o m o r p h i c functions on V . A ~ o q ~ h i s m b e t w e e n analytic va-

rieties V and V' is a continuous mapping ~ V - V' so that

~*(~V, ) c ~V ' A point in an ~ualytic v a r i e t y V is r e g u l a r

(simple) if the g e r m of V at that point is e q u i v a l e n t to the

g e r m of Cn for some n . The set of all r e g u l a r points is the

regular locus of V . It is ~i analytic manifold, not n e c e s s a r i l y

connected or pure dimensional. Its c o m p l e m e n t in V is called

the singular locus and a point on it a ~ingular point. The varie%v

is called n o n - s i n ~ u l a r if the singular locus is empty. A singular

point ~ is isolated if there is a germ at ~ with no other sin-

g u l a r points.

Notice that if V is algebraic in Cn then I(V) is fini-

tely generated, say I(V) = (gl,...,gr) . The J a c o b i a n m a t r i x

J(v) = ~(gl,'",gr)
~(z1,...,z m)
has m a x i m a l rank, rk J(V) = m - n at r e g u l a r points and at singular

points rkJ(V) < m-n .

 - 36 -

3.2. Intersections and C o v e r s

Let V be a n o n - s i n g u l a r complex analytic surface. The al-

gebraic intersection pairing

is d e f i n e d using Poincar@ duality

8: H2(V) - H2(V ) •

For X,Y E H2(V) define the p a i r i n g by

(X,Y) - (X .Y) : X(LY) .

Recall that in case V is not compact we u s e homology with closed

supports in the d e f i n i t i o n of ~ .

A map 0~ V' - V is said to be p r o p e r if the inverse image

of a c o m p a c t set is compact. If ~ is a p r o p e r surjective map

of a n a l y t i c spaces of d i m e n s i o n n , then there is a p o s i t i v e in-

teger d and an open subset U c V so that ~-1(v) consists of

d points for all v E U . We call d the degree of ~ . If V

and V' are complex surfaces~ ~ is a m a p of degree d and DI

and D2 are elements of H2(V), then (~*(DI) • ~*(D2)) = d(~.D2).

Let X,X' be curves in a n o n - s i n g u l a r surface V and x E

X0X ~ . ~e say that X meets X ~ normall~ at x if there is a

coordinate neighborhood U of x and l o c a l coordinates zI and

z2 so that X n U is the locus zI = 0 and X' o U is the lo-

cus z2 = 0 . It is w e l l k n o ~ that if X # X' and ( X . X') = I

then X meets X ~ normally at p r e c i s e l y one point.

We say that ~ is a f i n i t e map if ~ is p r o p e r and ~-1(v)

consists of a finite n u m b e r of p o i n t s for all v E V . Suppose

moreover that ~ is s u r j e c t i v e . The set B of p o i n t s v E V,

so that ~-1(v) consists of fewer than d = degree~ points, is

 - 37 -

called the b r a n c h locus of ~ . It is w e l l k n o v ~ that if V is

non-singular then B is the u n i o n of a f i n i t e number of i r r e d u -

cible subvarieties e a c h of c o m p l e x codimension I ("purity of the

branch locus").

Suppose X is a curve on a surface V . If x E X we r e -

call that X is l o c a l l y irreducibl~ at x if for e v e r y suffici-

ently small n e i g h b o r h o o d U of x in V there is a u n i q u e irre-

ducible component of X n U containing x . If x E X then

there is a n e i g h b o r h o o d U of x in V so that XOU = X~ U . .

.. L!Xr , where each Xi is a curve ~ h i c h is l o c a l l y irreducible

at x . The Xi are called the b r a n c h e s of Xi ~ x .

Definition I. Suppose ~ ~ V' ~ V is a f i n i t e map of n o n - s i n g u l a r

surfaces or curves, B is the b r a n c h locus of ~ and ~(v') =

v E B . Let Xi be a b r a n c h of ~-I(B) passing through v' (in

the case of curves this is just v~). There is a n e i g h b o r h o o d U

of v in V and a h o l o m o r p h i c function f in U having a zero

of order I along B ~ U and no other zeros. Let e(Xi) equal

the o r d e r of the zero of f ~ ~? along Xi . This is c a l l e d the

ramification index of rp along the b r a n c h Xi at v' . Now

v' J ~ - 1 ( v ) e(Xi) : degree c?

v' E X i

where we let Xi range over all b r a n c h e s of ~-I(B) through v'.

If there is a u n i q u e branch of ~-I(B) through v', we d e n o t e

e(Xi) by e(v') . In this case we get ~ I e(v') = d e g r e e ~.

v, E ~- (v)
Note that v E B if and only if e(v') ~ I f o r some v' E ~ - 1 ( v ) .

If X is an i r r e d u c i b l e curve on a n o n - s i n g u l a r analytic sur-

face V, t h e n there is an open dense subset Y c X with the p r o -

perty that X is l o c a l l y irreducible at all p o i n t s of Y .

 - 38 -

Suppose ~-I(x) = XII~... U X r where the Xi are irreducible.

Then there is an open dense subset Y' of X so that Y' ~ Y ,

Xi 0 - l ( y , ) is l o c a l l y irreducible and for any V l , V 2 E X in~-1(T)

we have e(vl) = e(v2) . Call this integer e(Xi) , the r a m i f i -

cation index of X. over X . It follows immediately from the

i

definition of ~* that

r
~*(x) = z e(Xi):: i ~ ~12(v ') .
i= i

We can use the r a m i f i c a t i o n index to get a u s e f u l relation

between the genus of an analytic curve and the genus of a finite

cover of that curve.

Proposition 2. (Hurwitz formula) Le~ ~ ~ X' - X be a finite

morphism of compact non-singular complex curves. Let

2gx=dim H I ( x , ~ ) ,2gx, = dim H I ( x ', ~ ) . Then

(2-2gx,) : (degree~)(2-2g x) - Z (e(x')-1) .

x'EX'

Proof. Triangulate X so that the points of the b r a n c h

locus are v e r t i c e s of triangles and no two are c o n n e c t ~ b y a l-sim-

plex. The E u l e r number of the triangulation is 2 - 2g X . It can

be lifted to a t r i a n g u l a t i o n of X' by means of ~ since outside

of B the map ~ is a local homeomorphism. This m u l ~ p l i e s the

number of faces and edges by degree ~ . if x E X is a v e r t e x

and x ~ B , then there are degree ~ vertices above x . But if

x E B , then there are degree ~ - Z (e(x')-1) vertices

~(x'):x
above x . This proves the formula.
 - 39-

3.3. M o n o i d a l Transforms and R e s o l u t i o n of S i n g u l a r i t i e s

Definition I. Suppose V is an analytic space, ~V is the

sheaf of h o l o m o r p h i c functions on V and I c ~V is an ideal

sheaf. The m o n o i d a l t r a n s f o r m w i t h center I is a pair (~,V')

with w~ V ~ - V and

(i) I~P~V , is l o c a l l y p r i n c i p a l i.e. ~v £ V' the stalk

(I~v,) v is g e n e r a t e d by one function,

(ii) for every ~o : Vo ~ V satisfying "I ~ o is locally

principal" there is a unique d : V O ~ V' with ~a=~ O

.

The m o n o i d a l t r a n s f o r m exists, Hironaka [ I ,p. 129], and is

unique by (ii). If X is a subspaee of V and IX is the sheaf

of f u n c t i o n s v a n i s h i n g on X , then the m o n o i d a l transform with

center X is just the m o n o i d a l t r a n s f o r m w i t h center Ix .

~e can construct the m o n o i d a l transform as follows. Suppose

v E V . Then there is a n e i g h b o r h o o d U of v and h o l o m o r p h i c

functions fo,...,fr on U so that the r e s t r i c t i o n of I to U

is g e n e r a t e d by fo,...,fr . Let X be the set of common zeros

of the fi " These functions define a map

~:U- X ~ C~ r

by ~(u) = [ f o ( U ) : . . . : f r ( U ) ] . Let

F c (U-X) x CP r

be the g r a p h of ~ , let V~ be the closure of F in U x CP r

and let

~U : V U' ~ U

be the p r o j e c t i o n map. Then (Wu~V~) is the m o n o i d a l transform

with center IIU . If we choose an open cover [Ui] of V where

the Ui are as above, then the u n i v e r s a l property of m o n o i d a l

 - 40 -

transforms guarantees that the (~Ui ,V ~ ) piece together to give

Ui
(~,V') . Note that if Y is the set of c o m m o n zeros of the func-

tions in I , then V - Y is an open dense subset of V and

-I
~: ~ ( V - Y) - V - Y is an isomorphism. The m o n o i d a l transform

with center [v} is also called the a-transform with center at v .

Definition 2. Suppose V is an a n a l y t i c space and X c V

is the set of s i n g u l a r points of V . We say that ~ : V' - V is

a resolution of the singularities of V if

(I) ~ is proper,

(2) V' is n o n - s i n g u l a r ,

(3) w induces an i s o m o r p h i s m between V' - ~-I(x) and V - X .

Remark. It is known, Hironaka [I], that if V is an alge-

braic surface, then there is a r e s o l u t i o n ~ which is a composite

of m o n o i d a l transforms. For an isolated singularity we shall con-

struct a "canonical" resolution but first we n e e d a definition.

Definition 3. An analytic space V is said to be n o r m a l at

v E V if for every n e i g h b o r h o o d U of v and m e r o m o r p h i c func-

tion f on U and h o l o m o r p h i c functions [a i) on U , the equa-

tion
fn+ &n_l:~ "-t+ . o • + a0 = 0

implies that f is h o l o m o r p h i c . V is said to be n o r m a l if V

is n o r m a l at every v E V . A curve is n o r m a l if and only if it

is n o n - s i n g u l a r . On a n o r m a l variety V the singular locus has

codimension > 2 . If v E V is a simple point, then v is a

normal point. F o r any a n a l y t i c variety V there is a unique pair

(~,V) so that ~ :~ ~ V , ~ is n o r m a l and for any n o r m a l variety

 - 41 -

V' and w: V' - V there is a u n i q u e map ~ ~ V' ~ ~ with w ~ =

w' . The pair (w,~) is called the n o r m a l i z a t i o n of V . The

map w is finite and it is an i s o m o r p h i s m over an open dense sub-

set of V .

Suppose V is a c o m p l e x algebriac surface with an isolated

singular point v . There is a finite sequence of maps

wi: V i ~ Vi_ I ' i = 1,...,n so that Vo = V , Vn is n o n - s i n g u -

lar; ~i is a n o r m a l i z a t i o n if i is even and wi is the m o n o i -

dal t r a n s f o r m w i t h center at the (isolated) s i n g u l a r points of

Vi_ I . Thus Vn is a r e s o l u t i o n of v E V but w-1(V) may be

r a t h e r complicated.

In order to improve w-1(v) we p e r f o r m a further sequence

of m o n o i d a l transformations Wn+j: Vn+j ~ Vn+j-1 so that the com-

posite w = wl "'" Wn+k satisfies

(*) w-1(v) = Xl ~!..o ~fXr , the Xi are n o n - s i n g u l a r irreducible

curves, (X i. Xj) = 0 or I for i # j and ~oXj 0X k=

for d i s t i n c t i, j, k .

Let ~i = w1° . oW.i .Then . we can

. choose Wn+~. so that it is the

monoidal t r a n s f o r m w i t h center x E Vn+j_ I where either

(I) x is a singular p o i n t of some component of Cn~j_1(v)

(2) x is a point of X i 0 Xj and Xi and Xj do not meet

normally at x ,

(3) x is a point of X i q Xj and X i ~ Xj consists of more

than one point,

(4) x E X i o Xj n X k , where i, j~ k are distinct.

Definition 4. Given a resolution ~ of the isolated singu-

larity v E V , w: ~ ~ V satisfying the conditions of (*) we

 - 42 -

-I
associate a graph F to ~ as follows: To each Xi in (v)
assign a vertex (Ai,gi,mi) where gi is the genus of Xi and

mi its s e l f - i n t e r s e c t i o n number. ~e join Ai to Aj by an edge

if Xi meets Xj Let Sc be a small sphere around v and

K : V~S . Clearly w-1(K) is h o m e o m o r p h i c to K and it is the

boundary of a t u b u l a r n e i g h b o r h o o d of w-1(v) . Hence K is a

singular SI fibration over w-1(v) . In f a c t it is o b t a i n e d by

plumbing according to the g r a p h F .

One can ask if there is a b e s t r e s o l u t i o n .

Definition 5. A resolution ~ ~ ~ ~ of an i s o l a t e d singu-

larity v E V is called m i n i m a l if for any r e s o l u t i o n w' : V ' ~ V

there is a u n i q u e map ~ : V' ~ ~ with w ~ = ~' . Of course the

minimal resolution is unique. Brieskorn FI~ p r o v e d that the m i n i -

mal resolution exists if V is a surface.

Remark 6. There is a simple criterion for a r e s o l u t i o n of a

surface to be minimal. Suppose V is a n o n - s i n g u l a r surface and

o
X c V° is a c o m p a c t irreducible curve. Then there is a n o n - s i n -

gular surface VI and a p r o p e r m o r p h i s m w~ V ° ~ V I so that

w(X) = v E VI and w induces an i s o m o r p h i s m between V° - X and

VI - [v~ if and only if X is a n a l y t i c a l l y isomorphic to CP I

and (X- X) = -I . This is k n o v ~ as C a s t e l n u o v o ' s criterion. A

curve X satisfying the above is called exceptional of the f i r s t

kind. A resolution ~ ~~ ~ V of an i s o l a t e d singularity v E V

is m i n i m a l if and only if no c o m p o n e n t of ~-1(v) is e x c e p t i o n a l

of the f i r s t kind. Note that in g e n e r a l if w is the m i n i m a l

resolution, then it w i l l n o t n e c e s s a r i l y satisfy the conditions

of (.) .
 - 43 -

Suppose w: V ~ ¥ is a r e s o l u t i o n of a n o r m a l singularity

v E V and w-1(v) = XIU...~!X r , where the Xi are irreducible

curves. Then the m a t r i x A = ((X i • Xj)) is an i m p o r t a n t invari-

ant of w . One can see w i t h o u t difficulty, Mumford [I~, that A

is n e g a t i v e definite, the d i a g o n a l entries are n e g a t i v e and the

off d i a g o n a l s are > O . It is r e m a r k a b l e that the converse of

this theorem is true.

Theorem (Grauert). Suppose V° is a n o n - s i n g u l a r analytic

surfac_~e, X = X I U . . . U X r , where Xi are compact irreducible

curves and ((X i. Xj)) is n e g a t i v e definite. Then there is an

analytic surface VI and a morp_h_ism ~ : v° ~ vI so that w(X) =

v E VI and w induces an i s o m o r p h i s m between V° - X and ~- Iv].

It is i n t e r e s t i n g to n o t e that if V is a l g e b r a i c VI need
o
not be algebraic.

3,4. Resolution and C*-action

In this section we show that if V is a s u r f a c e with a C*-

action,then there is an e q u i v a r i a n t resolution ~: ~ ~ V i.e. we

can c h o o s e (w, ~) so that the C* a c t i o n on V extends to ~ .

Definition I. Suppose G is a c o m p l e x Lie g r o u p and V is

an a n a l y t i c space. An a c t i o n ~ of G on V is a m o r p h i s m of

analytic spaces

~ G x V ~ V

so that ~(gg', v)=~(g,o(gV,v)) and ~(1,v ) = v .

We shall denote ~(g, v ) by gv when there is no danger of con-

fusion. Recall that the action is said to be effective if gv =v

for all v implies g = I .
 - 44 -

Proposition 2. Suppose a is an a c t i o n of G o_~n V ,

I c ~V is an i d e a l sheaf and ~: V' - V is the m o n o i d a l trans-

form with center I . If ~(g)*(1) = I for all g E G then

ther%is a unique action of G on V' compatible with the a c t i o n

on_n V . In p a r t i c u l a r if X c V is i n v a r i a n t under the action of

G and N is the m o n o i d a l transform with center X then the

above conclusion holds.

Proof. If g E G then g defines an a u t o m o r p h i s m c(g)

of V . The u n i v e r s a l property of m o n o i d a l transform (3.3) implies

that if I is i n v a r i a n t under g there is a u n i q u e map T(g) :

V' ~ V' so that N o T(g) = G(g) o N . By the u n i q u e n e s s we see

that T defines an a c t i o n . To be m o r e precise we m u s t check that

the m a p T: G X V' - V' is a n a l y t i c . Consider the diagram

G × V' T > V'

T

~ V
G xV > V
G

where ~o = i d G X ~ . Let p2~ G x V ~ V be the p r o j e c t i o n of

G × V on V . Then ~(g)(I) = I for all g £ V implies ~*(I)=

p~(I) . Now one can e a s i l y check that n° is the m o n o i d a l trans-

form with center p~(I) . Thus (~ O ~ o ) * ( I ) is l o c a l l y principal

and there is a u n i q u e map T ~ G x V' ~ V~ making the diagram com-

mutative. This is the same as our T above.

Proposition 3. Suppose a is an a c t i o n of G o_~n V . Then

t h,ere, is a u n i q u e extension of ~ to the n o r m a l i z a t i o n ~ of V.

Proof. Just use the u n i v e r s a l property of n o r m a l i z a t i o n .

 - 45 -

Proposition 4. Suppose G is a connected algebraic grou9

and ~ is an a c t i o n of G on a surface V . Then ~ leaves the

following invariant:

(I) an i s o l a t e d s i n g u l a r point,

(2) an e x c e p t i o n a l curve,

(3) a s i n g u l a r point of a n _ 3 x c e p t i o n a l curve,

(4) a point x 6 V where two or more components of the excep-

tional locus meet.

Proof. E v e r y element t 6 G acts as an a u t o m o r p h i s m of V .

Hence if v satisfies any of the above properties, then so does

tv . But if tv / v then the set of points satisfying that pro-

perty is p o s i t i v e dimensional and this is impossible. If X c V

is an e x c e p t i o n a l curve and t(X) / X , then V is covered by

exceptional curves. But there are only a finite n u m b e r of such

~urves.

3.5. W e i g h t e d H o m o g e n e o u s Pqll~omials and Good @*-action

Definition I. Suppose (w ° ..... v~) are n o n - z e r o rational

numbers. A polynomial h(Z ° ..... Zn) is w e i g h t e d h o m o g e n e o u s of

type (Wo,...,wn) if it can be expressed as a linear c o m b i n a t i o n

of m o n o m i a l s Z l° ... Z in for w h i c h
o n

i° + in
.+-- I
W W
o n

This is e q u i v a l e n t to r e q u i r i n g that there exist n o n - z e r o integers

qo,...,qn and a positive integer d so that h(tq°zo,...~nZn ) =

tdh(z o .... ,Zn) In fact if h is w e i g h t e d h o m o g e n e o u s of type

 - 46 -

(Wo,...,Wn) then let <Wo,...,Wn> denote the smallest positive

integer d so that for each i there exists an integer qi with

qiwi = d . These are the qi and d above.

Let V be the variety defined by weighted homogeneous poly-

nomials h I ..... h r with exponents (qo,...,qn) Then there is

a natural C* action

a(t,(z o, .... Zn)) : (tq°zo .... ,tqnzn )

~le call this action good if it is effective and qi > 0 for all i.

Proposition 2. Suppose V c @n+1 is an irreducible analytic

variety and ~ is a ~ood C* action leaving V invariant,

~(t~(z o .... ,Zn)) : (tq°zo , .... tqnzn )

Then V is algebraic and the ideal of polynomials in C[Zo,...,Z n]

yanishing on v is ~enerated by weighted homo6eneous polynomials.

Proof. Let f belong to C[Zo~...,Z n} the ring of conver-

gent power series. We let fi denote the unique polynomials so

that

f( tq°zo ,...,t qnZn) : ~°tifi

E (Zo,..., Zn) .
i=o
The power series on the right converges for sufficiently small t.

New suppose f vanishes on V near ~ . Then v 6 V implies

OO
E tmf (v) = 0 for all sufficiently small t . Hence fi(v) = 0
i=o 1
for all i and all v 6 V near 0 Let f(1) f(r) gen-

erate the ideal I(V) of all functions in C[Zo,...,Z n] vanish-

ing on V . Let J be the ideal generated by [(f(J))i ] . Clearly

J c I(V) . No~ if v I V •As within the radius of convergence of
f(J) for all j then there is some fi(J) ~o that f[J)(v) ~ 0 " Hence
the locus of zeros of J is V and hence the radical of J is I~). Let
 - 47 -

jT be the ideal generated by ~(f(J))i ) in @[Zo,...,Zn] and

let I' be the radical of J' . Then I'C[Zo,...,Zn~=radJ=I(V).

Therefore I(V) is generated by polynomials.

Now let I'(V) be the ideal of V in C[Zo,...,Zn~ . If

f E I'(V) then fi E I'(V) . If f is a polynomial, then there

are only a finite number of integers i with fi # 0 . Therefore

if f(1),...,f(r) generate I'(V), then the weighted homogeneous

polynomials [f~J)~ generate I'(V) .

Proposition 3. If V c Cm .is an algebraic variety and there

is a ~* action o on V defined by a m 0 r p h i s m a : @*× V ~ V

of algebraic varieties then

(i) there is an embeddin~ j: V ~ C n+1 for some n and a @*

action ~ o_~n @n+1 so that j(V) is invariant and ~ induces

on V ,

(ii) by a suitable choice of coordinates in C n+1 we may write

~(t, Zo~...,Zn) = (t qo Zo,...,t qnzn ) where qi E Z ,

(iki) if the action is fixed point free on V - ~) then we may

choose qi ~ 0 for all i .

Proof. (i) is a special case of R o s e n l i c h t [1,Lemma 2]~ (ii)

is proved in Chevalley rl, expos@ ~i, s@minaire I] and (iii) fol-

lows from Prill rl].

3.6. The Cone 0ver a W e i g h t e d IIomo~eneous V a r i e t y

Henceforth we shall assume that V c C n+1 and ~ is a good

C* action leaving V invariant.

 - 48 -

Definition 1. Let co : C n+1 ~ C n+1 be defined by

c0(Zo,...,Zn)
qo qn
= (z ° ,...,z n ) and let V' =
-1 (V) . Then V'

has a natural @* action defined by

~(t,(z o, .... Zn)) = (tZo,...,tz n)

and ~ commutes with the C* action. We call (~,V') the cone

over V . Note that V is the quotient of V'

by Eqo×'''X~qn
acting on @n+1 coordinatewise.

Proposition 2. The cone is. a ~ e n e r i o a l l y non-sin6ular vari-

ety, i.e. there is an o p e n algebraic (hence dense) subset U o c V'

so that if

I = fi(Zo .... ,Zn)) i = I .... ,r

is the ideal of polynomials vanishin~ on V and

gi(Zo, ,Zn) fi(Z~ ° Znqn) i = I, r

then

for all w E Uo where s = dim~V .

Proof. We may assume that V is not contained in any coor-

dinate hyperplane [Zi= 0] . Now V is a variety, hence it is

generically non-singular i.e.

/~fi \
rank (~-~j} v = n - s + I for v 6U,open dense inV.

Then
 - 49 -

!% / ~fi )( /5(zqJ~ \
j,t
5 Z k } (Z O,-..,z n) "( ~Zk / .
?zj qo nqn
Z0 9..,~ z )

There exists a point (Zo,...,z n) E V with zi / O for all i ,

so that the matrix on the right is invertible at this point. Hence

rank \~Zk I ( Z o ' ' ' ' ' Z n ) = n-s+1 .

But this property holds on some open algebraic subset and the sub-

set is non-empty. This proves the assertion.

3.7. The Quotient of V - [~} b_.yy ~*

The cone V' above V is defined by homogeneous polynomials

gl,...,g r . These polynomials define a projective variety X' c C ~ n.

In fact X' is precisely the algebraic quotient of V' - [~} by

~* . The analogue is true for V , I~umford [2, chapter 2].

Proposition I. There is a pro iective variety X and an

algebraic morphism w : v - [0} - x so that

(1) the fibers of w are precisely the orbits of the action,

(2) the topology of X is the quotient topology,

(3) for any open algebraic subset U c X the al~ebraic func-

tions on U are precisely the invariant functions oB -I(~o

The map w' : V' - [~) ~ X' has fibers @* . We would like

to add a zero section to get a map with fiber @ . Let

r~, c ( v , - [ o ~ ) × x'
be the graph of ~' ~ let F' be the closure of F in V' × X'

and let T' : F ' ~ X' be the map i n d u c e d by projection on t h e

 - 50-

second factor. We have obtained ~' from V' by b l o w i n g up the

origin y' : F' ~ V' Clearly ~'(x') = (O,x') gives the zero

section of (~',F') This pair is just the h y p e r p l a n e bundle of

X' . Now the action of G = ~qoX...x~ on V' induces an ac-

tion on F' . Let F be the q u o t i e n t of F' by this action.

Note that F is just the closure of rw in~-~)XX . The actions

of C* and G on V' commute, hence X is the quotient of X'

by G . We have the c o m m u t a t i v e diagram

G
F' >F

v v/
G
X' >X

~.here the h o r i s o n t a l maps are q u o t i e n t s by the action of G , ~'

is the zero section, ~ is the map induced by ~' and , is

the map induced by ,' . Let y :F ~ V be the map induced by y'.

3.8. The C a n o n i c a l Equivariant Resolution of a Surface

Suppose dimcV = 2 and V has an i s o l a t e d singularity at 2.

Then by P r o p o s i t i o n (3.6.2) there is an open dense subset Uo of

V' so that every point of V' is simple. Hence there is an open

dense subset U c X' with the same property. Now (~',F') is a

line b u n d l e , h e n c e ~-I(U) is n o n - s i n g u l a r . Clearly G is a fi-

nite map r a m i f i e d along a finite number of fibers of T' . Hence

there is an open subset UI c X so that ,-I(uI) is n o n - s i n g u l a r .

Now F -ix(X) is n o n - s i n g u l a r , hence F has only a finite number

of s i n g u l a r points along ~(X) , all w i t h n e i g h b o r h o o d s of the

form C2/~ for some a . Let Po : ~7 ~ F be the m i n i m a l resolu-

tion of these singular points. Then the C* action extends to

(since there is an e q u i v a r i a n t resolution dominating ~) . The

 - 51 -

composite map .o ~ ~ Po > F "y> V is a r e s o l u t i o n of the singu-

larity of V . We shall say that p is the c a n o n i c a l equivariant

resolution of V . Since p is e q u i v a r i a n t given a small U(1)-

invariant disk De at £ , the m a n i f o l d p-1(D e) is a U(1)-inva-

riant subset obtained by e q u i v a r i a n t plumbing of D2 bundles by

the g~aph of p-I(£) . Its boundary, K is therefore a smooth,

orientable 3-manifold with 81 action and F trSE = ~ .

The p r o 2 e r transform X° of X c F is the unique irreducible

curve in ~ so that 0o(Xo) = X . Note that the C* action is

trivial both on X and X . It is easily proved that the other

o
curves of the r e s o l u t i o n have no i s o t r o p y groups. It also f o l l o w s

directly from the fact that the s i n g u l a r i t y is i s o l a t e d that X

and X° are i s o m o r p h i c non-singular projective curves.

Theorem I. Let P- 1 (2) = X o U ''" U X r , where X.z is an irre-

ducible curve and X is the p r o p e r transform of X . Then

O

(I) Xi is n o n - s i n g u l a r for_all i , Xi meets Xj at no more

than one point, Xi crosses Xj normally at that point and

XiDXj0X k = @ for d i s t i n c t i,j,k ,

(2) the a c t i o n is trivial on X

O '

(3) the a c t i o n is n o n - t r i v i a l on X i , i > 0 , and gi = 0, i > O ,

(4) F is a w e i g h t e d star w i t h center Ao ,

(5) m. < -2 , for all i > 0 .

1

Proof: By (3.4.4) we can p e r f o r m a sequence of m o n o i d a l

transforms with centers at fixed points of the action so that the

composite p': V' ~ ~ satisfies

 - 52 -

(a) the action extends to V~

(b) V' and p ~ p' satisfy (I).

Let (p o p,)-1(O)
B
= X'O !J , a .
UX~, and let F' be the graph asso-

ciated to 0 o 9' . Now F' satisfies (2.2.2.a) and (X~. X!)l < 0

as noted in (3.3). Finally, if X! and X~ have genus zero~ X!

l j l
meets X~j and (X~. X!)l = (Xi " X~)3 = -I then the intersection

matrix ((X~. Xi) ) cannot be negative definite. Applying (2.2.2)

we see that gi' = 0 for ± > 0 and ~' is a weighted star with

center A o' . Thus F' satisfies (I) - (4). Let s be the num-

ber of m i = -I . We will prove by descending induction on s

that (I) - (4) are satisfied for any resolution between V' and

. Suppose X! is a rational curve with n o n - t r i v i a l action and

I

(X~ • X!)l = - I Then by Castelnuovo~s criterion (3.3.6) there is

a manifold V" and a map f~ V' ~ V ~' so that f(X~) is a point

and f is an isomorphism outside of X~ . Now X!l meets at most

two other curves~say X~ and X~ . It meets each at one point

and with normal crossings there. Let Xj = f(Xi) . Then

XI ' X2 = f*(X1 ) " f*(X2 ) = (X~ + X ~ ) . (X~ + X ~ ) = I . Thus XI meets

X2 normally at one point. Thus V" satisfies (I) - (4). Pro-

ceeding inductively we see that V satisfies (I) - (4). But

is a minimal resolution of F, hence (X i. Xi) ~ -2 . This com-

pletes the proof.

Combining the above theorem with the results of (2.2) we ob-

tain the main resolution theorem.

Theorem 2. The we.ighted graph associated to the can£nical

eq___uivariant resolution of the isolated s in~ularity of V at the

orisin is the star of K , S(K) .

Thus in order to obtain this resolution it is sufficient to
find the Seifert invariants of K from the algebraic descr~tdznofV.
 - 53 -

3.9. The Seifert Invariants

Assume now that V is an a l g e b r a i c surface w i t h an i s o l a t e d

singularity given as the locus of zeros of some p o l y n o m i a l s in

@n+l and it is i n v a r i a n t under a good ~* action. We shall de-

scribe h o w to find the S e i f e r t invariants of K . More specific

results for h y p e r s u r f a c e s in ¢3 are g i v e n in the n e x t section.

I. Finding aj . If all c o o r d i n a t e s of a point z = (Zo,...,Zn)

are d i f f e r e n t from zero,then z is on a p r i n c i p a l orbit since

( qo ,..., qn ) = I . The point _z in the h y p e r p l a n e H = { zi I : ' "

• .=z. = O} w i t h all other coordinates non-zero has isotropy

Ik
group of order a = (qo''''~qi l'''''qi
^ k''''' qn) . The n u m b e r of

orbits with isotropy group ~a lying in H equals the n u m b e r of

those components of V 0 H that are not in any smaller coordinate

hyperplane.

2. Finding Bj • Let ~ be an orbit of K with isotropy group

~ , ~ > I . F o r an analytic slice D2 in K through x E S we

can find an analytic isomorphism ~ : ~ = [u C ¢ I !u! < I ) ~ D so

that the induced E action ~ on ,~ is a s t a n d a r d linear

action. For 0 = exp(2~i/a) and for some 0 ~ ~, < a we have

T(p,u) = p~u . Then G~ ~ I mod a and 0 ~ B < a . (Notice that

the o r i e n t a t i o n adopted in O r l i k - W a g r e i c h [1~2~ is the opposite of

this. )

3. Finding b . Suppose V is i n v a r i a n t under the good

action

~(t,Zo, o..,Zn) = (tq°zo .... ,tqnzn )

and d is the degree of the cone over V as defined in (3.6).

Making adjustments for the p r e s e n t orientation convention we oh-

 - 54 -

tain the following formula

d r ~
b =
qoq1"''quq j=1 mj

Rather than repeating the p r o o f as g i v e n in O r l i k - W a g r e i c h

[I] we shall only outline the arg,~ment. If V is d e f i n e d by

homogeneous polynomials of d e g r e e d, then qo = ' ' ' = qn = I and

there are no E-orbits. In this case V - [0} is a C*-bundle

over X induced by the C* b u n d l e C n+1 - {0} ~ CP n . The l a t t e r

has chern class -I . The fact that X has degree d means that

the m a p

H2(cpn; ~) ~ H2(X;Z)

induced by i n c l u s i o n is m u l t i p l i c a t i o n by d so the chern class

of the b u n d l e over X is -d and therefore b = d satisfying

the f o r m u l a in this case. The g e n e r a l formula is o b t a i n e d as

follows. Let ~ : V' ~ V be the covering of V by its cone,

V = V'/G , G = ~qo®...®~q~. and F,X,F~,X ' as in (3.7). Since

V ~ may have non-isolated singularities the curve X' may be sin-

gular. Let H: Y~ ~ X' be its d e s i n g u l a r i z a t i o n and F = F' ×Y'.

o X'
Since F' is a C-bundle over X ~ of d e g r e e -d the same h o l d s

for F° over Y' and (Y''Y')Po = -d . Let ~ be the canonical

equivariant resolution and X the center curve. We w a n t to c o m -

pute (X. X)~ . First one constructs non-singular varieties Wo

and VI and b i r a t i o n a l maps T: ~o ~o and pI ~ VI ~ ~ and a

map ~ : Wo ~ V I so that the d i a g r a m below is c o m m u t a t i v e

W ~ -->
o o

v v
Pl
v1 >~ ~>F
 - 55 -

Here W is the blowing up of the fixed points of the action of

o
G on Y, c F ° . Then G acts freely on Wo and ~ is the

quotient map.
J~ JL N

Let Yo = T~(Y') ' X = ~'7(X) ~ XI = p~(X) . The degree of the

map is qoql...qn and it is easily seen that

(qo...qn)(X1"Xl) V = ( ~ X 1 • ~*XI) ~ : (Yo'Yo)w

0 0

The second part of the argument shows how the maps Pl and T

change these intersection numbers. Specifically one proves that

(xl.Xl)vl= (X.~)V
and

r aj-Oj = (Y"Y')~
(Yo-Yo)w + qo...qn Z • ~J
o j=1 0

giving the formula as asserted.

4. Finding g . This computation is purely algebraic. The non-

singular curve X has arithmetic (and topological) genus Pa(X) =

dim H I ( X , ~ x ) which is the constant term of the Hilbert polyno-

mial of the homogeneous coordinate ring, RX . Now X' is de-

fined by homogeneous polynomials so its coordimate ring~ RX, is

known. One proves that R X = (R~,) (m) where m = qo...qn and

( )G denotes the subring fixed by G . There are technical dif-

ficulties because the ring R~, is not generated by forms of de-

gree I and therefore the Hilbert polynomial is not defined, see

Orlik- W a g r e i c h [2]. An alternate method is given in (3.11) for

hypersurfaces in @3 .

3.10. Surfaces in @3

Suppose that V is a surface in C3 having an isolated sin-

gularity and admitting a good @* action. It follows from (3.5.2)
 - 56 -

that V is defined by a w e i g h t e d h o m o g e n e o u s polynomial,

h ( Z o , Z l , Z 2) . U s i n g the ~* a c t i o n it is shown in O r l i k - W a g r e i c h
[1] that there is an e q u i v a r i a n t analytic deformation of V into

a surface d e f i n e d by one of the f o l l o w i n g six classes of p o l y n o -

mials
(1) aI a2
Z ao
o + Z I + Z2

aI a2
(if) Z ao
o + Z I + ZIZ 2

aI a2
(ill) ao + Z 1 Z 2 + Z 2 Z 1
Zo

aI a2
(Iv) Zoa° + ZoZ I + Z IZ2

(v) a aI a2
Zo°Z 1 + Z 1 Z 2 + ZoZ 2
a
(vl) Z o ° + Z iZ2

inducing an e q u i v a r i a n t diffeomorphism of r e s p e c t i v e neighborhood

bounderies of the isolated singularity at the origin.

Thus it is sufficient to study these six classes of p o l y n o -

a
mials. The p o l y n o m i a l ZoO + ZIZ 2 is ~ l ~ l y t i c a l l y isomorphic to
ao + 2
Z° Z1 + Z so it may be treated as a subclass of I .

Assuming that the weights equal wi , i = 0,1,2 and they

are r e d u c e d as a f r a c t i o n to w i = u i / v i , we introduce auxiliary

integers

c = (Uo,Ul,U2)

co = (Ul,U2)/o , cI = (Uo,U2)/c , c2 = (Uo,Ul)/C , Cl,2

Uo/CClC 2 , Co, 2 = U l / C C o C 2 , Co, I = u2/CCoC 1 Note that Co,

Cl,C 2 are pairwise relatively prime, Co,l,Co, 2 and Cl,2 are

pairwise relatively prime and (ei,cj,k) = I if i,j and k are

distinct.

The i n t e g e r d defined as the least common m u l t i p l e of the ui

equal s
 - 57 -

d = 0CoCiC2Co,iCo,2Ci, 2

and f r o m this we compute qi = d/wi as qo = V o C o C o , l C o , 2 '

ql = v i C i C o , I C l , 2 , q2 = v 2 C 2 0 o , 2 C I , 2 "

I. Orbits w i t h n o n - t r i v i a l i s o t r o p y groups are in the h y p e r p l a n e

sections. The n u m b e r of orbits in a g i v e n h y p e r p l a n e section is

the n u m b e r of i r r e d u c i b l e components of the curve of intersection.

~or example in class I the subset

aI a2
[z o = O, z I + z 2 = O} N S p

has isotropy g r o u p ~ m o = ~ ( q 1 ' q 2 ) = ~ci ,2 . It c o n s i s t s of

n o = (al,a2) = cc° orbits. S i m i l a r arguments yield the f o l l o w i n g

table where So, al, a 2 are the three possible isotropy groups in

the three h y p e r p l a n e sections and no, nl, n 2 are the number of

orbits in each.

so no aI nI a2 n2
I I I
I [
I
:I 01,2 i cc° i Co~2 cc I 0o,1
I
j cc 2
i
[I 01 ,2 (CCo-1)/V2 'V2Cl ,2 I c
o,1
O

t
[II ci ,2 (CCo-Vl-V2)/VlV2 F~v2cI ,2 1 VlCl, 2 1
i
[V °o,1 ! (c-1)/Vl ! v2 1 VlCo, 1 1

V VO ~ 1 I v1 1 v 2 It 1

2. In order to compute 8 we note that a s u f f i c i e n t l y close

slice in V maps d i f f e o m o r p h i c a l l y onto a slice in K so we may

consider the former. All orbits in the same h y p e r p l a n e section

have the same orbit type since so does the whole hyperplane. Con-

sider for example an orbit w i t h isotropy group E in class I

So
as above. Let ~ = exp(2~i/~o) . The action of ~ in ¢3 is
 - 58 -

~(Zo,Zl,Z 2) = (~q°zo,Zl,Z2) .

Considering the z° plane as a slice the action is the standard

action of type [ao,qo] and hence Bo is defined by the congru-
ence

qoBo ~ 1(mod ~o) .

Notice that this is the orientation convention of (1.1.7) and the

opposite of that used in Orlik-Wagreich [1,2]. For an orbit on

the intersection of two hyperplanes, e.g. in class II

[z o = z I = O, Iz212= 1~
ao aI
the slice at z2 = I is the curve [z o + z I + z I = O) . This
curve near (0,0,1) may be "approximated" by changing it by an

analytic automorphism

~(Zo~Z I) = (z o + h o ( z o ~ z l ) , z I +hl(Zo,Zl))

where h i E @[Zo,Z1) have all terms of degree ~ 2 . The curve

a
[zoo + z I = 0 ~ is an approximation and if g = exp(2~i/al) the
action in the slice is approximated by

a i ) = ( qOz ° ,-~-q o a o zoa ° ' I ) =( qOzo _z2O ,1).

~(Zo,_zoO

So we have Vl = qo and hence

Blq ° ~ l(mod a I) .

The table below gives the vj , j = 0,1,2

Since Bj~j ~ 1(mod aj) and 0 ~ Bj < aj this determines the

Pj •
 - 59 -

Vo Vl v2

I qo ql q2

II qo qo q2

III qo qo qo

IV q2 qo q2

V q2 qo ql

3. As we have mentioned earlier b is given by the formula

b = d
qoqlq2 j=1 aj

4. Finally the construction of the previous section gives the

following expression for g , Orlik-\,ragreich [1,(3.5.1);2,(5.4)]

d2 d(qo, q I ) d(q I ,q2) d(q2, q o)

2g =
qoqlq2 qoql qlq2 q2qo

(d,q o) (d,q I ) (d,q 2)

+ - - + - - + - - I .
qo ql q2

We shall give an alternate way of obtaining this formula

using the fibration theorem of M i l n o r [I~ in the next section.

Pirst consider an example.

Let a v a r i e t y V in ~3 be defined by the weighted homo-

geneous polynomial of class III, h(Z) = Z15+ZIZ 2+Z~z I . It

O

has an isolated singularity at the origin. We find w ° = 15 ,

w I = 9/2 , w 2 = 9 , d = 45 , qo = 3 , ql = 10 , q2 = 5 , c = 3 ,

co = 3 , c I 2 = 5 and the other c-s equal 1 . The locus

Y

[z O =0, z~ + z ~ = 0 ] 0 S5 consists of 3 orbits with stability

group of order a o = (ql,q2) = 5 . There is one orbit

~Z = = = = =
o Zl = Ol 0 S 5 with al q2 5 and one orbit [z o z2 O)NS 5
 - 60 -

with a2 = ql = 10 . The 9 o r r e s p o n d i n g ~o = ~I = ~2 = qo so

8o = 2 , 81 = 2 and 82 = 7 • The f o r m u l a for b gives b = -I

and the f o r m u l a for g gives g = 3 . Thus

K = [-1;(o,3,0,0);(5,2),(5,2),(5,2),(5,2),(10,7)]

and the star of K

is the dual of the g r a p h of the c a n o n i c a l equivariant resolution

of the s i n g u l a r i t y of V

3.11. Milnor's Fibration Theorem

Let V be an algebraic hypersurface in ~n+1 defined by

the zeros of a polynomial, V = [~ ! f ( ~ ) = O] Let ~ be an

arbitrary point on V and S ~ sufficiently small sphere cen-

tered at x Let K = V o S The f o l l o w i n g fibration theorem

-- C

is due to ~ i l n o r [1].

Theorem. The m a p p i n g

from S c-K t_~o S 1 is the p r o j e c t i o n m a ~ of a smooth fiber

 - 6] -

bundle. Each fiber

F~ = ~-1(ei~) c S¢- K

is a smooth ~ara!lelizable 2n-manifold.

For an isolated singularity there is additional information.

Theorem. If ~ is an isolated critical point of f then

each fiber F~ has the homotopy type of a bouquet snv...vS n of

n-s~heres. Their number, u is strictly positive. Each fiber

can be considered as the interior of a smooth compact manifold

with b o u n d a r y

closure(F~) = P~ U K

vqhere the common boundary K is an (n-2)-connected smooth (2n-I)-

manifold.

The complement of K in S¢ ~ Sc - K is therefore obtained

from ~ × [0,2w~ by identifying Po and F2~ by a h o m e o m o r p h i s m

h~ F - F,

called the characteristic map. The W a n g sequence associated to

this f i b r a t i o n is according to I~ilnor FI~8.4]

h -I
...~ Hj+I(S ¢- K) ~ HjF :~ *> HjF ~ H j ( S c - K ) 4...

v~here I is the identity map of F . in case ~ is an isolated

singularity we can use the information on the connectivity of F

and K , Alexander duality and Poincar@ duality to see that for

n ~. 2 the V/ang sequence reduces to the short exact sequence

h.-I.
0 ~ HnK - HnF > HnF ~ Hn_ IK ~ 0 .

Let A(t) = det(tl.-h.) denote the characteristic polynomialof~

transformation h. : HnF ~ HnP .
 - 62 -

If f(~) is a w e i g h t e d h o m o g e n e o u s polynomial of type

(We, .... Wn) then M i l n o r shows furthermore that F is d i f f e o m o r -

phic to the n o n - s i n g u l a r algebraic variety

F, : [~ ! f(£)= ~}

and the c h a r a c t e r i s t i c map h may be chosen

h(z o, .... z n) = (~ q °Zo,...~ gqnzn )

where ~ = exp(2wi/d) In p a r t i c u l a r h is of finite order di-

visible by d . Thus the m i n i m a l polynomial of h. divides

(t d - I) and hence it is a s q u a r e - f r e e polynomial. This implies

in turn that the rank of the k e r n e l and cokernel of (h.-I.)

equals the exponent ~ of (t-l) in A(t) . An e x p r e s s i o n for

was obtained by M i l n o r - O r l i k [I] in terms of the weights. Let

w i = ui/v i , i = O,...,n be in i r r e d u c i b l e form. Given integers

ao,...,a k denote their least common multiple by [ao,...,ak].

We have

c )n-s lo
~(Wo, . . . . Wn) = L (-1 ~...,u.
[ui ° as

where the sum is taken over the 2 n+1 subsets {io,...,i s] of

{0 .... ,n} .

In the case of a surface in C3 we already k n o w HIK in

terms of g e n e r a t o r s and relations. There are 2g free g e n e r a t o r s

from the partial cross section to~ether w i t h the g e n e r a t o r s

q o , q l , . . . ~ q r ,h
satisfying the relations:

qo + ql + ' ' " + qr = 0

qo + bh = 0
~ j q j + 8jh = 0 j = I, .... r .
 - 63 -

The first comes from the partial cross section and the r e m a i n i n g

ones from the sewings of the solid torus n e i g h b o r h o o d s of the b-

obstruction and the E-orbits. The d e t e r m i n a n t of the r e l a t i o n

m a t r i x equals P = b~'1 ''' a r + ~ l a 2 "'" a r + ' ' ' + a ! ~ 2 "'' ~r

p r
= b+ Z
~I ''" ~r j=1 ~j

On the other h a n d from the e x p r e s s i o n for b (3.10.3) we obtain

r d
b+ ~
j=1 aj qoqlq2

so we see that p > 0 and therefore the g e n e r a t o r s q o ' ' ' ' ' q r 'h

are torsion elements of HI K . Thus

7~(Wo,W I,w2) = rank HIK = 2g .

Substituting w i = d/q i ~ i = 0, I,2 in ~t(Wo~Wl,W2) yields

3.10.4).

Although this proof is correct it is somewhat u n s a t i s f a c t o r y

in that the essential r e a s o n for p > 0 is h i d d e n in the proof

of the f o r m u l a for b . Examining that proof one observes that

p > 0 is e q u i v a l e n t to the n e g a t i v e definiteness of the quadratic

form of the resolution.

F i n a l l y note that this a p p r o a c h is v a l i d only for h y p e r s u r -

faces. Por h i g h e r e m b e d d i n g dimensions the algebraic method men-

tioned in (3.9) has no t o p o l o g i c a l replacement at present.

3.12. N o n - i s o l a t e d Singularities

Rather than giving a d e t a i l e d account of the r e s o l u t i o n of

non-isolated singularities of surfaces w i t h a good ~* action as

in O r l i k - W a g r e i e h [21 we shall point out the a d d i t i o n a l difficul-

ties compared w i t h the isolated case.

 - 64 -

I. Let 8: ~ - V be the n o r m a l i z a t i o n (3.3.3) of V , where

V c C n+1 is a surface invariant under a good @~ action. We are

interested in the r e s o l u t i o n of the isolated singularities of

using the m e t h o d s already developed. The fact that V is g i v e n

with a good @* action is of little help, however, because the

same m a y n o t be assumed of ~ . A canonical equivariant resolution

of the s i n g u l a r i t i e s of ~ m a y be c o n s t r u c t e d as follows: Let V'

be the cone over V in ~n+1 and V'-0/@* = X' c C p n . Let

~: X' ~ X' be the n o r m a l i z a t i o n (resolution) of the p r o j e c t i v e

curve X' . Let F' denote the h y p e r p l a n e (Hopf) bundle of CP n

restricted to X' . Since the degree of F' is n e g a t i v e Grauert's

Theorem (3.3) assures that there is a b i r a t i o n a l map j': ~' ~ V'

collapsing the zero section. Let F' = ~*(P') and 7' = F' ~ ~'

be the map collapsing the zero section. Now ~' maps into the

normalization of V' and it is n o r m a l so it is the n o r m a l i z a t i o n .

F' is n o n - s i n g u l a r and the a c t i o n of G = Eqo®...@Eqn on F'

extends. Let F = F'/G , V = V'/G and ~ = F ~ V the induced

map. Finally let ~: ~ ~ ~ be the m i n i m a l resolution of the

quotient singularities of F . Then p = ~: ~ ~ ~ is the cano-

nical equivariant resolution of ~ .

2. Since the a c t i o n extends~ ~ has an isolated singularity at

the origin whose resolution is d e t e r m i n e d by the Seifert invari-

ants of K . The topology of V at the o r i g i n is d e t e r m i n e d by

the map 81~: K ~ K . In g e n e r a l K is not a manifold and

may identify orbits of K , some by maps of d i f f e r e n t degrees. One

needs some n o t a t i o n for these objects and an e q u i v a r i a n t classifi-

cation theorem.

3. The central object is o b t a i n i n g the S e i f e r t invariants of

and u n d e r s t a n d i n g the map ~ from the a l g e b r a i c description of V.

 - 65 -

The isotropy groups of orbits in K are easy to read off. The

slice at ~ E K may consist of several disks meeting at ~ The

number of orbits mapping onto the orbit of z

u
is determined by

the number of orbits of the action of ~ in the slice. If k

disks of the slice are mapped into each other by ~a" then there

is one orbit with isotropy group ~/k in K mapping onto the

orbit of ~ by a map of degree k . The action of ~a/k in the

individual slice determines ~ (as an invariant of K ). The

obstruction class b is obtained by the same formula as before.

The genus g(X) of the non-singular curve X = ~ -Q/@* is ob-

tained from the arithmetic genus Pa(X) of the (possibly singular)

curve X = V - Q/@* using the formula

g(~) = pa(X)- z 3x
xEX

where 5x is an invariant of the singular point x E X . The

computations are, of course, harder. They are carried out for

hypersurfaces of C3 in Orlik-~Tagreich E2].

 - 66 -

4. Equivariant Cobordism and the a-Invariant

This chapter is a brief extract from the thesis of 0ssa [I].

First some g e n e r a l n o t a t i o n is introduced then the basic facts

about sl-manifolds are given. Next the fixed point free cobor-

dism group of oriented, closed, smooth 3-dimensional fixed point

free St-manifolds is d i s c u s s e d in detail. It is shown to be free

and g e n e r a t o r s are constructed. An a l g o r i t h m for finding the co-

b o r d i s m class in terms of these g e n e r a t o r s f r o m the Seifert inva-

riants is also obtained.

Using a fixed point theorem in A t i y a h - S i n g e r Ill, an invariant

is defined for fixed point free circle actions. It is a r a t i o n a l

function in Q(t) . This invariant is computed for 3-dimensional

St-manifolds.

4.1. Basic R e s u l t s

All m a n i f o l d s and bundles are assumed smooth and orientable.

G i v e n the v e c t o r bundles ~I ~ XI ' ~2 ~ X2 define ~I ~ ~2 by

the V~hitney sum of the p u l l b a c k s of the p r o j e c t i o n s Pri: X I × X 2

X i , i = 1,2 . as

~I ~ 2 = pr~ ~I ® p r ~ 2

Let G be a compact Lie group, H a closed subgroup and

(H) = [gHg-1 ! g E G ) . A family of subgroups F is called admis-

sible if H E F implies (H) c P . All families of subgroups

will be assumed admissible. Let ~n be a G-manifold and assume

that G is o r i e n t a t i o n preserving. M is called of type (F,F')

if p ~ M then G E F for all p E N and if p E 8M then

P
G E F ~ for all p E ~M . It is called (F,F')-bounding if there
P
 - 67 -

is an (F,F)-manifold W n+1 so that ~,i is an e q u i v a r i a n t sub-

manifold of SW and for every point p E ~W-M , Gp E F' . We

also call W an (F,F')-cobordism for M Two G-manifolds MI

and M2 of type (F,F') are (F,F')-cobordant if the d i s j o i n t

union M 1 + (-M2) is (F,F')-bounding. This is an e q u i v a l e n c e

relation. Denote by ~n(G;F,F') the e q u i v a l e n c e classes of n-

dimensional G-manifolds of type (F,F') and u~.~>(G;F,F') =

® ~(G;~,F,)
n
Let F D P' ~ F" be f a m i l i e s of subgroups of G Then

there is an exact sequence

. . . . ~(G;F, F',) ~ ~ a ~" J ~ _ (~;F',F") . . . .

~h( ;F, ) - %(~;F,~') ~ I
where i and j are induced by i n c l u s i o n mud ~ is r e s t r i c t i o n

to the boundary.

A G-vector bundle of d i m e n s i o n (k,n) is defined as a smooth

G-vector bundle w i t h fiber dimension k over a smooth, closed n-

manifold. Assume that the total space is o r i e n t a b l e and the ac-

tion of G is o r i e n t a t i o n preserving. It w i l l be called of type

(~,H) if

(i) each isotropy group of the zero section contains a subgroup

conjugate to H ,

(it) each i s o t r o p y group of the a s s o c i a t e d sphere bundle is in

- (i) .

A G-vector bundle ~ of type (F,H) bounds if there is a

G-vector bundle ~ with oriented total space over a m a n i f o l d with

boundary so that ~ is e q u i v a r i a n t l y diffeomorphic to the r e s t r i c -

tion of ~ to the b o u n d a r y of its base. Two G-vector bundles

and ~' of type (F,H) are (F~H)-cobordant if the d i s j o i n t

union ~ + (-~,) bounds. Again, (~,H)-bounding is an e q u i v a l e n c e

 - 68 -

relation and the c o l l e c t i o n of equivalence classes ¢~(G;F,H)

forms an abelian group under d i s j o i n t union. Let ,~[(G;F,H) =

¢ (G~F,H) . Note that 12k+1(G;P,H) = 0 follows

~. from the
k~n
o r i e n t a t i o n assumption, e.g. if G is abelian.

Given a G-manifold M" of type (F,F - ( H ) ) the set of

points p E N so that G contains a conjugate of H is a clo-

P
sed G-invariant submanifold of M - ~ . Let ~ be its n o r m a l

bundle in M . Then ~ is a G - v e c t o r bundle of type (F,H) .

It is easily seen that the map M ~ ~ induces an Q. module iso-

morphism
- .> ®
k - -

The inverse map is g i v e n by taking the a s s o c i a t e d disk bundle

of ~ .

4.2. Fixed Point Free SI-Actions

Let Pm be the family of subgroups of S

1 with order ~ m ,
I
Po~ = ~F m and FS all subgroups of S . Note that Nm in Fm

and S1 in FS are m a x i m a l elements. Let us use the s i m p l i f i e d

notation
q'(m) = ~:~n(S1;Fm,~)

1) =

and s i m i l a r l y

Cnk(m) ,k 1
= Wn(S ;Fm,Z m)

¢~(SI) = I,nk(SI[Fs,S 1)

Let M be an St-manifold and H c SI a closed subgroup.

Define I(H) = [p E M I h(p) = p ~ V h E H ] Clearly I(H) is an

invariant submanifold in M . Let N(H) be its n o r m a l bundle.

 - 69 -

We call M an sl-manifold with c qmple~ normal bundles if for

every H the bundle N(H) has the structure of a complex S 1-

veetorbundle satisfying the condition that if H I c H2 then the

bundle N(HI)II(H2) is a complex S1-8ubbundle of N(H 2) . The
corresopnding cobordism groups are denoted by ~n(m) , ~(oo)
and ~(S 1) . Similarly we define complex vector bundles of type
(m) over oriented S1-manifolds where the operation of SI is
compatible with the complex structure to obtain the groups ~ktm~
Vn ~ j
of complex k-dimensional vector bundles of type (m) over n-
manifolds. This yields the exact sequence

....
Q (m-l) - ~--nCm) " ~ } n - 2 k (~') ~ -I
. . . .

Given a complex representation r of Em with no trivial summand

we can form the cobordism group Tn(m,r) of complex St-vector

bundles of type (~m~r) over oriented S1-manifolds. Let Rk(~ m)

denote the set of equivalence classes of complex k-dimensional
representations of ~ with no trivial summand. Clearly
m

~k
,~n(m) = _~ ~nCm, r)
rER-(~ m)

Lemma 1. Let r: E m ~ U(k) be a complex representationo f

m
with no trivial summand. Let
, ,
~(r) be the centralizer of
r(~m) in U(k) Then there is a canonical 0. module isomor-
phism with the singular bordism group of Conn er-Flo~d [I]

~n(m,r) = On_1[B(S1/~m) xB(~(r))] .

Proof. Let ~ E ~n(m,r) and let ~ denote the associated

principal U(k) bundle. Now SI operates on the left on ~ and
 - 70 -

U(k) on the right on ~ . Let

= icE% ~' h e = e r ( h ) ~ vhE~ m } "

Then SI acts on ~ from the left. ~(r) operates as a subgroup

of U(k) on the right on g and hence on q . Define a left

action of ~(r) on ~ by Ge = e c -I . ~his_ gives a left action

of S I x ~(r) on ~ . Define

!
A = [(h,r(h)) ! h E ~m"

a normal subgroup of S I × C(r) . It is easily seen that A is

exactly the isotropy group of every point of q under the action

of S I X ~(r) and q is a principal S 1 x ~(r)/A bundle with

base M/S I defining an element of ~n_1[B(S1x ~(r))/A] and it

follows that

S1 x ~ ( r ) / A ~ $1/~ mx ~ ( r ) .

Conversely, given a principal S1/~m x ~(r) bundle ~ over

~fi/SI, we obtain the principal ~(k) bundle ~ with S I action

over N by n o t i n g that there is a canonical map v ; ~ ×U(k) ~

given by (e,~) ~ e ~ equivariant with respect to the S I action.

It is surjective and v(e1,~1) = v(e2,c2) iff c i ~ I E ~(r) and

-I
e 2 = eIoi~ 2 Thus ~ is the quotient of ~ x U(k) by the
action of ~(r) given by ~(e,s) = (e~-1,~s) .

Let ~n " CPn be the Hopf bundle. Then the 0. algebra

~ (~(k)) is a polynomial algebra generated by the classes

k *
[~n ] , 11 ~ 0 . According to Conner-~loyd [2,(18.1)] one has to

show that if for a k-tuple w = (nl,...,n k) , n I ~ n 2 ~ ...~k~0

we a s s o c i a t e the bundle ~w = ~nl $ ' ' ' ~ k over Pw = C p n l x ' " x c p n k
~vith the classifying map fw' then the classes fm.[Pw] E

H.(BU(k);~) form a ~-basis for H.(BU(k);~) . This is done by

the usual characteristic class argument.
 - 71 -

Recall that every complex representation r : ~ m ~ U(k) is

a sum of linear representations. Denote by rj- ~ m ~ U(1) ,
j = 1,...,m-I the representation that sends the generator

exp(2~i/m) of 2Zm to exp j (2wi/m) . Let krj denote the k-

fold direct sum of rj . Then for some non-negative kl,...,km_ I
with k1+..°+km_ 1 = k the representation r is equivalent to

klr 1 ~ . . . ~ k m _ l r m _ I . Thus ~(r) is isomorphic to U(k I)×...

~, I
x U ( k m _ 1) and since $1/2~m ~ o , we have from Lemma I:

~n(m,r) = Qn_I(BS 1 x B U ( k I) × ... xBU(km_1)) •

Since H,(BU(k);E) has no odd torsion,the KGnneth formula of

singular bordism theory applies~ Cormer-Floyd [2,(44.1)] and one
obtains the following explicit generators. Let S 2q-I denote the
m

(2q-I) sphere [(z I .... ,Zq) E cq ! ~zi~ i = 1} with the ineffictive

SI action t(z I .... ,Zq) = (tmzl, .... tmzq) . Let ~(J)n denote
the Hopf bundle over CP n with SI acting by multiplication by
tj in each fiber.

Theorem 2. ~ (m) = ® ~k
~n(m) is freely generated as an D
n
module by

s2q-1 x (~ (jl) ^ -(Jk)

m in I ®. . . . ~ ~nk )

where q ~ I ; m-1 ~ Jl Z J2 ~'°" Z Jk ~ 1 and n s ~ ns+ I i_~

Js = Js+1

Theorem 3. (a) The canonical module homomorphism

f~rr ,

is injective.
(b) j : ~(m) ~ ® '~k(m) is surjective.
(c) ~.(m) is freely generated as an O.
 - 72 -

module b ~
(Jo)~ (Jl) (Js)
S( ~no - ~nl ~ . • •• @ ~~ n s )

where s > 0 , m > Jo > Jl >''' > Js > I and n o > no+ I if

Ja = Jo+l "

Here S(q) denotes the sphere bundle of the bundle ~ .

Proof. If ~I and ~2 are of type (S I) so that every

isotropy group in S(~1) is ~m and in S(~ 2) of order < m,
then S(D 1 $ ~2 ) is of type (m) and the normal bundle N(~m) of
the fixed set I(Em) is equivariantly equivalent to S(~I) × ~2'
In the exact sequence

.... ~n (m-l) "i ~ ( m ) ~ ® =?n

k ,km)
, ~ ~ n -1 (m_1) ....
n k

~(m) is free on the generators given in Theorem 2. The element

of ~r.(m)
s(~(m) ~ (Jl)~. * (Jk)
~q-1 • gn 1 " ~ gn k )

maps onto the corresponding generator by the remark above so j

is surjeetive and by exactness i is injective. Part (c) follows
from induction on m .
In particular one obtains the following.

Corollary 4. ~.(co) is freely generated as an Q. module

D_Z
(Jo)@ ~ ( J i ) ~ . . @ (is))
~ns
S(~n ° ~n I •

where s > 0 , Jo > Jl > J2 >'''> Js > I and n a _> no+ 1 i_~f
Jo = J~+1 "
 - 73 -

4.3. 3-Manifolds

$I_
The cobordism group of 3-dimensional fixed point free

manifolds is determined as follows.

Theorem I. ~+3(0o) is free abelian with free ~enerators

(Jo) (Jl)
S(~o ~ ~o ) ~ Jo ~ 2JI "

proof. Consider the relations:

(i) [S(5o'(m)~ ~o"(n))l = FS(~(m+n)-

-o ~ ~o"(m))] + [S(~(m+n).o ~ ~n))],
m,n~1

The first is obtained from the S I action on CP 2 given by

tFZo~Zl:Z
2 ] _ = FZo:tmz
~ 1 :tm+nz27 observing that the fixed point

set consists of the three points [I~0~0] , [0:1:0] and [0~0:I]

and the above are their normal sphere bundles. The second follows

by noting that S(~ j)) = S(~ j) ~ ~J)) and letting m = n = j

in (i). Thus it follows from (4.2.4) that the image of

is generated by the above generators. In order to prove that

is an isomorphism we first claim that ~ is onto. This means

that every 3-dimensional orientable fixed point free sl-manifold

has complex normal bundles. This is obvious since these are ori-
ented D2-bundles over S1 . To show that ~ is injective it is

enough to show that the generators given in the theorem are linear-

ly independent in ~3(0o) Here is ~uq outline of this argument.

Using (ii) it suffices to prove that if Y is an oriented 4-di-

mensional fixed point free S1-manifold with boundary

 - 74 -

(Jo (Jl (j)

BY = Jo~2Jl ajo,j I
S(~o )$ ~o ) + j~1 bj S(g I
)

j >1

then the coefficients a and b. are zero. First it is

Jo,Jl J
shown that Y is cobordant to Y~ where Y' is a fixed point

free S I- manifold with complex normal bundles and BY = BY' .

Using (4.2.3a) and a d o ~ w a r d induction on the orders of the

isotropy groups one obtains the announced result.

Next we shall express the cobordism class of an arbitrary

oriented fixed point free S1-manifold

M = [b; (o,g,0~0)~ (ai,~i) ..... (~rg~r)~

in terms of the generators given above. In order to avoid treat-

ing the class b separately we shall think of M in the equiva-

lent presentation

M = [0; (o,g,0,O);(1,b),(~1,~ I) ..... (ar,~r)l .

Remove the interior of an equivariant tube consisting of only prin-

cipal orbits from M and call the r e s u l t i n g manifold-with-boundary

M' . Let V be a tubular neighborhood of an E-orbit with Seifert

invariants (a,S) as described in (1.7)~ a > 0 , (a,~) = I but

B is not n e c e s s a r i l y normalized.

As in (].7) define v and o by

v~ ~ 1 mod a ~ 0 < v <

= ( ~ - 1 ) / ~ .

Choose a cross-section on the boundary torus of M' so that the

action ~&~itten with complex coordinates is

t(zl,z 2) = (zl,tz 2)

t E U(1), !z1! = I , !z2! = 1

 - 75 -

The action in V is d e s c r i b e d by

t(x,z) = (tVx,taz)

!x! < I , !z! = I . Define the e q u i v a r i a n t m a p

by ~ ( Z l , Z 2) = (ZlC'Z~,Z~z2P) ,

Its inverse is the map F given in (1.10). Since ~ has deter-

minant -1 it is o r i e n t a t i o n reversing and it can be used to ob-

tain an oriented m a n i f o l d

r4 = r 4 ( a , B ) = I4, u v .
cp

Let Y = Mx I with M = ~x [0} c Y . Consider the unit ball

in ~2

D~,~ = [ ( z 1 9 z 2) E ~2 11 1,2I + lz212 ~ 1}

with the U(1) action

t(zl,z 2) : (tvzl,taz2)

and let S = ~D denote S3 with the above action.

The map
x(x z) = (,, x z )
~l+xx ~l+x~
defines an o r i e n t a t i o n p r e s e r v i n g equivariant embedding
i
Define Dw•,,,o~ = [ ( z 1 , z2) E :D,J,c~ I Izl ! 2 + lz21 2 <--{]
and
l
Y + = D ~ ( Z - D ~Vg(Z c D Vg(~

Using k sew Y+ and Y_ together along V × [I} c M × [I} to

obtain a 4-manifold with bo~mdary Y = Y U Y w i t h a fixed

- X +
point free S I action.

The b o u n d a r y of Y has three companents 51(a,~) = M × {0] c Y _ ~

 - 76 -

S~ = S( ( a ) ~ { v)) and the result of sewing ~ x [I] and S

v,~. ~0 ~,~
together by k . The latter is obtained by sewing the complement

of V in S a into M' = Z ( a , B ) - V . A careful analysis shows

that

= - - ~o °

In order to emphasize the symmetry of the situation we let ,j=~'

and p = ~ and write the result as:

Lemma 2. With the above notation the fixed point free S 1_

manifold Y has boundary

~Y = M ( ~ , B ) - M ( ~ , ~ ) - S(% o(~)$ %
o(~)) .

Noting that 0 < ~ < a m

the above lemma gives an algorithm

for representing the cobordism class of an arbitrary fixed point

free S l-manifold in terms of the generators of %(co) given in

Theorem 1.

¢°4. The a-invariant

Consider the composition of inclusion maps

Theorem I. The sequence above is exact in the middle.

Corollar 2 2. If M is a £ixe~ point free S1-manifold with

no isotropy group of even order s then M bounds an St-manifold.

Proof. By (4.2.4) im ~ c k e r i . On the other hand we have

the exact sequence of (4.1)

 - 77 -

-~ ~ . ( S I) -~ ~ Jrk.(sI) ~ ~.(oo) i ~($I) ....

so it is sufficient for the converse that keri = im~ c i m ~ .

This follows because an S1-vector bundle of type (S 1) with

fixed point set equal to the zero section has a natural complex

structure inducing the structure of an S1-manifold with complex

normal bundle on the associated sphere bundle.

The next result is stated without proof, 0ssa [I, 2.2.1].

Theorem 3. coker~ is a 2-torsion group.

Thus for every fixed point free S1-manifold ~I, a suitable mul-

tiple 2rM bounds an S1-manifold. This fact will be used to

define an invariant of the St-action on M, a(M) below.

Given an S1-vectorbundle ~ over the compact, oriented

manifold X so that the fixed point set is equal to the zero-

section X c ~, there is a canonical splitting of into a sum

of complex S1-vectorbundles Ok , k _~ I so that t E SI operates

by complex multiplication by tk in the fiber of ~k "

Let nk
c(~ k) = H (1+ xj(k)) , xj(k) of degree 2
j=1
.
be a formal factorization of the total chernclass c(~ k) E H (X;Q).

Let i(X) E H*(X;Q) be the total ~ polynomial of X ,

Hirzebruch [2]. Define a rational function ~(~) E Q(t) by

n k tke2Xj(k)
~(~) : (~(x) ~ ~ +1)[x] ,
k>o j:1 tke2Xj (k)_ I

where IX] is the fundamental class of X , [X] E H.(X;Q) .

Given a closed, oriented S1-manifold l~I with fixed point

set X , its normal bundle O has a canonical complex structure

and therefore it induces an orientation on X from the orienta-

 - 78 -

tion of M . If ~(M) denotes the signature of Mj then a fixed

point theorem in A t i y a h - S i n g e r [I,p.582] implies that

T(M) : ~(~) .

N o w assume that M is an o r i e n t e d fixed point free S1-mani-

fold. For some r we can find an oriented S1-manifold Y so

that ~Y = 2rM . Let ~ denote the n o r m a l bundle of the fixed

point set of Y and define the r a t i o n a l function

~(M) = 2-r(T(Y) - ~ ( ~ ) ) .

To see that ~(M) is i n d e p e n d e n t of the choice of Y one takes

Y' , BY' = 2r'M and c o n s t r u c t s

W = (2r'y) !I (-2ry ~)

to obtain a closed manifold for w h i c h the A t i y a h - S i n g e r theorem

applies. The a d d i t i v i t y of the signature implies the assertion.

Remark. Ossa [I]. a(M) may be e x p r e s s e d as a p o l y n o m i a l

tk+ I
in k > 0 with coefficients in ~[½] .
tk - I '

It turns out that a(M) is d e t e r m i n e d up to an additive con-

stant by the fixed point free cobordism class of M . In order

to compute a(M) for a fixed point free 3-dimensional S1-mani -

fold,we first compute a(M) for the g e n e r a t o r s of %(2) .

L e m m a @. Let ~ = ~(m)
5o ~ c
~(n)
o . Then

t TM + I tn+ I
tm - I tn - I

Let D(~) and S(~) be the a s s o c i a t e d disk and sphere bundles.

Then clearly ~(D(~)) = 0 and we have:

 - 79 -

L e m m a 5.

tm + I tn + I
~(s(~)) : -
tm - I tn - I

Next recall the fixed point free S1-manifold Y = Y(M,a,8) ob-

tained from M in (4.3) with

( ~ ) ~ ~(~))
~Y : M ( ~ , ~ ) - ~ ( ~ , ~ ) - s(~ o ~o "

In o r d e r to f i n d the r e l a t i o n between the a-invariants of M ( a , B )

and M(~,~) it is n e c e s a a r y to c o m p u t e the signature of Y .

Let M = [0;(o,g,O,O); (a1~B1) .... ~(~n_1,~n_1 )] where the (aj,Bj)

are not n e c e s s a r i l y normalized. Direct computation gives:

L e m m a ~.

n-~
where c =
j=1 aj

Given the r e l a t i v e l y prime pair (~B) of p o s i t i v e integers

there is a u n i q u e continued fraction

a/8 = [ao,a I .... ,ak] = ao _ 1

aI - 1
• . 1

ak

with ai ~ 2 , as n o t e d in (2.4). The a u x i l i a r y variables of

the E u c l i d e a n algorithm are defined by P-I = I Po = ao '

Pi+1 = ai+IPi-Pi-1 ' i ~ 0 . Define the r a t i o n a l function

k ~ tPi-1
r(a,S) = Z (1 - t p i + l • + 1) •
i=o t pi - I t pi+l - I

it has the f o l l o w i n g properties

 - 80 -

(i) r(a,-~) = -r(¢,S)

(ii) r(1,o) = o

(iii) if (a,B) and (~,~) are given so that 0 < ~ ~ a and

a ~ - aB = -I as a b o v e , t h e n

r(a,8) = r(~,~) + I - t a + I . t ~ + I
ta - 1 t~- I "

With this n o t a t i o n the ~-invariant of a 3-dimensional closed,

oriented sl-manifold is c o m p u t e d as follows:

TheoremS. Let K = [O;(o,g,O,O);(a1~81),...,(an,~n)} @

T h e n we h a v e
n n ~.

Proof. ~e use induction assuming the statement for all

= ..... with

m <n or

m = n and a' < an or

m

m = n and a~ = an and !p1~ I < I Bn{ .

m

We m a y assume that 8n > 0 for if 8n = 0 then the conclusion

follows trivially and if 8n < 0 t h e n we consider -K = [O,(o,g,~O);

(al,-B1),...,(an,-Bn)} . Let }~ = [O,(o,g,O,O);(a1,81),...

n-1 Bi Bn
.. (an_1,~n_l) } , c = ~ __ and an,Bn,~n, as above. Now
' i=I ai

using the d e f i n i t i o n of a on the fixed point free sl-manifold

Y we h a v e

a(~Y) = T(Y)

t an + I t~ + I
a[M(an, Sn)] - a [ M ( ~ n , B n ) ] + -
an ~n
t -1 t -I
 - 81 -

Using (iii) above and the induction hypothesis, the assertion

follows from the simple identity below:

sign(o+~)(o+~) = t- sign(~+~)+sign(~+~) .

Example 8. Let us compute the a-invariant of the 3-mani-

fold K = [-I;(o,3,0,0);(5,2),(5,2),(5,2),(5~2),(I0,7)} obtained as

the neighborhood boundary of the isolated singularity at O of

the surface V = [z E ¢3 ! z °15 + ~4~ + z 7 z I = 0 ] in (3.10) ~irst we

shall absorb b in the E-orbit (10,7) and write

K = [0;(o,3,0,0);(5,2),(5,2),(5,2),(5,2),(10,-3)] . Next

5 10 I
"2' = 3-½- and -3-= 4--~_ . Hence

t 3 + I t+ I t5+ I t3+ I
r(5,2) = I t~ - I t - I +I t 5 - I 3_ I

r(I0,3)= I t 4 + I t+ 1 + I t7 + I t4+ I t10+1 t7+1

+1
t4- I t- I t 7 - I t4- I t I0 _ I t 7 - I

5 ~i 2 -3 13
and E -- = ~ ~-~ = ]-0 so
i=I ~i 4"~+ 'u

a(K) = 4r(5,2) - r ( I 0 , 3 ) - I .
 - 82 -

5. F u n d a m e n t a l Groups

We n o t e d in chapter I that only some of the Seifert mani-

folds admit St-actions but d e f e r r e d the i n t r o d u c t i o n of the re-

maining ones to this chapter. Using the t e r m i n o l o g y of H o l m a n n

[I] g i v e n in (5.]),the other Seifert m a n i f o l d s are described in

(5.2) and the c l a s s i f i c a t i o n t h e o r e m of Seife~t [I] is proved.

In (5.3) we compute the f u n d a m e n t a l groups and use the method of

0rlik-Vogt-Zieschang [I] to show that if the f t m d a m e n t a l groups

of two Seifert m a n i f o l d s satisfy a c o n d i t i o n (in which case they

will be called "large"),then they are isomorphic only if the m a n i -

folds have the same Seifert invariants (up to orientation). This

gives a h o m e o m o r p h i s m classification of large Seifert manifolds.

In (5.4) we i n v e s t i g a t e "small" Seifert m a n i f o l d s (i.e. whose

fundamental groups are not large) and their h o m e o m o r p h i s m classi-

fication.

5.1 Seifert Bundles

Recall that a bundle % = (X,~,Y) consists of a total space

X , basis Y and continuous onto map ~: X ~ Y . A bundle h o m o -

morphism from ~' = (X',~',Y') is a pair of continuous maps

h: X ~ X' 9
t: Y - Y' making the diagram commutative

X ~ >Y

~t
X' > Y'

It is an i s o m o r p h i s m if h and t are h o m e o m o r p h i s m s .

Following Holmann [I] we define a Seifert product bundle w i t h

 - 83 -

typical fiber F as a triple ~xU)/G ,p', U/G} where U is a

topological space, G a finite group operating on F and U (the

action on U is not assumed effective) and on F xU by g(f,u)

= (gf,gu) and there is a commutative diagram

P
F×U > U

(Fxu)/o , P' > u/o

where p is projection onto the second factor, × and • are

orbit maps of the G actions and p' is the induced map.

We call g = (X,n,Y) a Seifert bundle with typical fiber P

if it is locally isomorphic to a Seifert product bundle with typi-

cal fiber F , i.e. Y has an open cover [Vi, i £ I] so that to

each i we have a Seifert product bundle [(F ×Ui)/Gi,Pi,Ui/Gi]

and a commutative diagram

Pi
F × Ui ,> Ui

Hi =
(F×Ui)/G i l> Ui/G i ti°~i
h.l°Xi

-1 rr
(v i) > vi

where (hi,ti) give a bundle isomorphism in the lower square.

We call G a structure group of the Seifert bundle ~ if

(i) it contains the finite groups G. above,

1

(ii) each non-empty subset of U i , U~ = T?• I (v i n vj) has a fi-

nite (unbranched)cover (Uij,~ij) where Uij = Uji so that

Ti o o . lj = T
j oo ji '
 - 84 -

(iii) for V i o Vj ~ ~ there is a continuous map gij: Uij ~ G

so that by defining fij : (f,u) ~ (gij(u)f,u)
the diagram below is commutative:

f..
1j
× Uij > P × Uij

Sji I i Sij=Xi°(iF×~ij )
h71oh, v
i
> i

If the fiber F equals the structure group G acting on

itself by left translations, we call it a principal Seifert bundle.

The following two results of Holmann [I] will be useful later.

Theorem I. Let ~ = (X,w,Y) be a principal Seifert bundle

with structure ~roup and fiber G . Assume that X, Y and G are

locally compact. Then X is a G-space and the orbits of the

action are the fibers of the Seifert bundle.

Theorem 2. Let a locally compact topological group G act

on a locally compact space X so that each g: X - X is a proper

map and all isotropy groups are finite. Then ~ = (X,w,X/G) is

a pringipal Seifert bundle wit h fiber and structure group G .

Corresponding results hold in the differentiable and complex

analytic cases.

Example (Holmann [I~.) Let ~ = ($3,~,S 2) be the Seifert

bundle with total space S3 and base space S2 given by the or-

bits of the St-action on S3 from (1.5.1)

 - 85 -

t(zl,z 2) = (tnzl,tmz2)

where (m,n) = I and S 3 = [(Zl,Z 2) E C 2 I z1~ 2 + z2z 2 = I] . We

think of the base space S 2 = CP I with homogeneous coordinates

[Xl:X 2] . The orbit map is then given by

(zl,s 2) = •

Consider the open sets in the base space V i = [[Xl:X 2] ECp1!xif0},

i = 1,2 . Let UI and U2 equal the complex numbers with coor-

dinates Yl and Y2' and Gn and Gm the corresponding cyclic

groups of order n and m . Let ~ = exp(2~i/n) operate on UI

by ~(yl) = ~-my I and ~ = exp(2wi/m) operate on U2 by ~(y2 ) =

~-ny 2 . Define the corresponding actions on SI x U i by

~(x,Yl) = (~x,~-myt) and ~(x,Y2) = (-qx,~-ny2)

Define T i: U i ~ V i , H i :
S 1 xU i ~ ~- t (V i) by
n-m

TI(Y 1) = [(1+Y1~1)'--"2-': y~]

m-n

T2(Y 2) = [y~ : (1+Y2~'2) ~'~']

xn xmy I

Hl(X,y I) = (/i+Yi.~i 'I+~1~ )

xny 2 xm
H2(x,y2) = ( i+y32
' I~2~ 2

giving the required Seifert diagrams.

In order to define the action of the structure group we let U12

= U21 equal the complex numbers without the origin and U~ =

1 2
U I - [0} , U 2 = U 2 - {0} and define the covers ~ 1 2 : U 1 2 ~ UI by
I -n in-m
a12(Y ) = ym and ° 2 1 : U 2 1 ~ U2 by o21(Y ) = y !y~ . These
maps satisfy the condition T2oq21 = Tioc12 . Finally let
 - 86 -

g21(Y) = y l y l -1 , g12(Y) = y - l l y [ be maps U12 SI giving rise

to automorphisms f12 and f21 of S I x U12 defined by

f12(x,y) = (y-l!y!x,y) , f21(x,y) = (y!yI-lx,y) satisfying

H2°(islYa21)°f21 = HI °(iSIw°12)

Remark. If we define U12 = 021 as all complex numbers and

extend the maps ~12 and c21 to be branched m-fold and n-fold

covers and consider the locally trivial fiber bundle ~ obtained

from S I x 012 and S I x U21 by identifying S I × U12 and

S I ~ U21 using f12' then we see that ~ is a branched mn-fold

cover of ~ branched along the two E-orbits of ~ . In fact

= (S3,w,S 2) is just the Hopf bundle, and the equivariant branch-

ed ccver is described globally by

n m
zI z2
~(zl,z 2) = ( ).

5.2. Seifert Manifolds

In his classical paper Seifert [I] considered the class of

closed 3-manifolds satisfying the conditions

(i) the manifold decomposes into a collection of simple

closed curves called fibers so that each point lies on a unique

fiber,

(ii) each fiber has a tubular neighborhood V consisting

of fibers so that V is a "standard fibered solid torus". The

latter is the quotient of D 2 × S I by the action of a finite cyc-

lic group as in (1.7).

 - 87 -

The p r o b l e m is to c l a s s i f y all such m a n i f o l d s up to fiber

preserving homeomorphism. In the n o t a t i o n of (5.1) we have

Seifert bundles ~ = (M,~,B) where M is a closed 3-manifold,

the fiber is SI and the s t r u c t u r e group is all h o m e o m o r p h i s m s

of SI . Since this g r o u p retracts onto 0(2) we can r e s t a t e our

problem as follows: Classify all Seifert bundl.es ~ = (M,w,B)

with total space a closed 3-manifold, fiber SI and structure

group 0(2) under bundle equivalence. The first result is a con-

sequence of (5.1.1).

Proposition I. If the structure group reduces to S0(2),

then ~ is a p r i n c i p a l Seifert bundle with typical fiber SI

M admits an S1-action and the c l a s s i f i c a t i o n is g i v e n by

Theorem (1.10).

Considering the g e n e r a l case we may use the a r g u m e n t of (1.9)

to conclude that B is a closed 2-manifold of genus g . Thus

there are only f i n i t e l y many open sets Vi in the cover of B

with Gi ~ 1 . A refinement of the cover enables us to collect

all these in an open set at the base point of B Outside of

this set ~ is a genuine fiber bundle. The structure group 0(2)

contains reflection of the fiber, i.e. along some curve of B

(not h o m o t o p i c to zero) the fiber may r e v e r s e its orientation.

This gives rise to a h o m o m o r p h i s m

: ~I(B) ~ c2

where C2 is the m u l t i p ] i c a t i v e group of order 2 , C 2 = {I,-I]

identified with the a u t o m o r p h i s m group of ~I(S I) = ~ Here

~(x) = I if the fiber preserves its orientation along a curve

representing x and ~(x) = -I otherwise. Select a set of gene-

 - 88 -

rators for w1(B) . The next result is due to 8eifert Eli. We

give the p r o o f of Orlik ~13, see also O r l i k - R a y m o n d E23 for g e n e -

ralizations.

T h e o r e m 2. Up to 8eifert bundle equivalence there are the

following six possibilities:

oi: B is orientable and all g e n e r a t o r s preserve orientation so

M is orientable and is a p r i n c i p a l 8eifert bundle;

o2: B is orientable w i t h g £ I and all ~ e n e r a t o r s reverse ori-

entation so M is n o n - o r i e n t a b l e ;

nl: B is n o n - o r i e n t a b l e and all ~ e n e r a t g r s preserve orientation

s_~o M is n o n - o r i e n t a b l e and ~ is a p r i n c i p a l Seifert bundle;

n2: B is n o n - o r i e n t a b l e and all g e n e r a t o r s reverse orientation

s_£o M is orientable;

n3: B is n o n - o r i e n t a b l e with g £ 2 and one g e n e r a t o r preserves

orientation while all others reverse orientation so M is

non-orientable;

n4 : B is n o n - o r i e n t a b l e with g £ 3 and two g e n e r a t o r s preserve

o r i e n t a t i o n while all others reverse orientation so M is

non-orientable.

Proof. Clearly ~ : ~I(B) - C2 is d e t e r m i n e d by the v a l u e s

on the generators. We shall show that for an a r b i t r a r y h o m o m o r -

p h i s m we can choose n e w g e n e r a t o r s of w1(B) so that the induced

acts on the g e n e r a t o r s according to one of the maps in the

theorem.

If B is orientable and ~ maps all g e n e r a t o r s into +I or

all g e n e r a t o r s into -I, then there is n o t h i n g to show. N o w sup-

 - 89 -

pose ~(ui) = -I and ~(uj) = I . By r e n u m b e r i n g the g e n e r a t o r s

we m a y assume ~(Ul) = I . Let j be the smallest index so that

9(uj) = I . If

(i) j is even: let vj_ I = U j _ l U j ; vj = uj_ 1 and vk = uk

for k / j-l,j .

(ii) j is odd (j~3) and c~(uj+1) = I : let vj -I = u~luj

-lv
-I -1 -I
vj = uj U j _ l U j _ 2 U j _ l U j U j + I ; v j + I = u 3+ j~ u ujT 1j+ 1 and

vk = uk for k / j-l,j,j+1 ;

j is odd (j ~ 3 ) and ~(uj+1) = -I : let vj = ujuj+ I

and vk = uk for k / j .

Repeated application of this procedure defines new generators for

~I(B) so that ~ sends every generator into -I .

A similar argument holds if B is n o n - o r i e n t a b l e . If all

generators are m a p p e d into +I we h a v e a principal bundle, nI .

If all g e n e r a t o r s are m a p p e d into -I we h a v e an o r i e n t a b l e total

space, n2 . Now suppose that some generators preserve orientation

and some reverse it. Let ~(Ul) = -I and ~(u2) = ~(u 3) = ~ ( u 4)

= I . The following change of b a s i s reduces the n u m b e r of o r i e n -

tation preserving generators by two:

-1 -1 -1 -1 -1 -1 - 2 -1
v I = UlU2U 3 ; v 2 = u 3 u 2 u I u 3 u 2 u3u 4 u 3 u 2 u 3 ;
-1 2 -1-1 222
v 3 = u 3 u2u3u 4 , v 4 = u 4 u 3 UlU2U3U 4 ; vi = ui for i > 4 .

Repeated application of this map gives n3 or n4 .

To s h o w that the six b u n d l e equivalence classes are indeed

distinct is t r i v i a l in all eases except for n3 and n4 . Here

we a b e l i a n i z e w1(B) and n o t i c e that the image of UlU2...Ug is

the u n i q u e element of o r d e r 2 in HI(B;Z) . This element com-

mutes in n1(M ) with the h o m o t o p y class of a t y p i c a l fiber for

odd g only for n3 and for even g only for n4 .

 - 90 -

Using the proof of the classifJcation theorem (1.10) for

3-manifolds with St-action and F USE = ~, we obtain the follow-

ing classification theorem of Seifert [I].

Theorem 3. Let ~ = (}~,w,B) be a Seifert bundle with typi-

cal fiber SI 9
structure ~roup 0(2) and total space M a

closed 3-manifold. It is determined up to bundle equivalenqe

(preservin~ the orientation of M or B if they have an~) b$ the

followin~ Seifert invariants:

N = [b;(E,g);(¢l,£1) , .... (ar,5r)] .

Here E is one of Ol,O2,nl,n2,n3,n 4 denotin~ the weighted map

of the 2-manifold B of ~enus g described in Theorem 2; th__~e

(aj,~j) are pairs of relatively prime positive integers

0 ~ Bj ~ aj for E = oi,n2,

0 ~ ~j ~ aj/2 for E = o2,nl,n3,n4;

and b is an integer satisfyin~ the conditions

b E E for E = ol,n 2 and

b E E2 for E = o2,nl,n3,n 4 unless aj = 2 for some j in

which case b = 0 .

Note that M is orientable if E = ol,n 2 and a change of

orientation gives the Seifert invariants

- M = [-b-r;(E,g);(~1,~1-~1),...,(ar,ar-Br)] .

5.3. Fundamental Groups

The fundamental group G = ~I(M) is generated by the "parti-

al cross-section" qo'''''qr and a 1,b 1,...,~g,bg if B is ori-

 - 91 -

entable or Vl,...,Vg if B is n o n - o r i e n t a b l e and the f i b e r h .

The r e l a t i o n s are g i v e n b y : t h e commuting relations of h with

the o t h e r g e n e r a t o r s , the n u l l h o m o t o p i c curves in the E-orbits:

qj3h
3 a - 8. = I , the r e l a t i o n on the " p a r t i a l cross-section" qo~. =I

where ~. = q l . . . q ~ 1 , b l ] . . . [ a g , b g ] if B is o r i e n t a b l e and
2 2
w. = q l . . . q r V l . . . V g if B is n o n - o r i e n t a b l e , and the r e l a t i o n

qo hb = I , w h i c h we e l i m i n a t e by s u b s t i t u t i n g qo = h-b " Thus

for o r i e n t a b l e B we have

G : [a I , bl ' . . ,ag,bg,ql,
. . . . ,qr,hl a i h ~ ~ :h ¢i , bihb~1= h ci , I

qjJh J = I, q l . . . q r [ a l , b l ] . . . [ a g , b g ] = h b }

°I: ¢i = I for all i ,

02: e i = -I for all i ;

and for n o n - o r i e n t a b l e B we h a v e

G = Iv I ..... Vg,ql, .... qr,h i v i h v ~ I = h ci , q j h q ~ 1 = h , q ~J

j h 6J : I ,
ql • "'qrV~ oe • v2g = hb}

n1: ¢i =I for all i ,

n2: c i = -I for all i ,

n3: ~I = 19 ci = -I for i > I ,

n4: Cl = ¢2 = I, e i = -I for i > 2 .

We call M small if it s a t i s f i e s one of the c o n d i t i o n s

below:

(i) 01 , g = 0 , r ,< _2
I I I
(ii) 01 , g = 0 , r = 3 , ~-~+--+~ > I
a2 ~3

(iii) [-2;(Ol,0); (2,1),(2~1),(2,1),(2,1)]

(iv) 01 , g = I , r = 0 ,

(v) o2 , g = I , r = 0 ,
 - 92 -

(vi) n I , g = I , r _< 1 ,

(vii) n 2 , g = 1 , r _< 1 ,

(viii) nI , g = 2 , r = 0 ,

(ix) n2 , g = 2 , r = 0 ,

(x) n 3 , g = 2 , r = 0 ,

otherwise we call M large.

We shall assume in the r e m a i n d e r of this section that M is

large and prove following Orlik-Vogt-Zieschang [1] that the

Seifert invariants of M are determined (up to o r i e n t a t i o n ) by

~I(M) . Small Seifert manifolds will be treated in the n e x t

section.

Lemma 1. The subgroup generated by h is the u n i q u e maximal

cyclic normal subgroup of G and h has infinite order.

Proof. Consider the f o l l o w i n g groups:

Ci = [qi ,h I q i h q ~ 1 = h , qi h = I}

~. ci}
D i : [ai,bi, laihai I : h bihb 1:

The subgroup generated by h is i n f i n i t e cyclic and n o r m a l in

each of these groups. We f o r m the iterated amalgamated free pro-

duct along (h) to c b t a i n G as f o l l o w s :

(i) for orientable B and r > 3 we take

C1 (h)
* C2
 - 93 -

and note that h and qlq2 form a free abelian subgroup of rank
2. Taking

C3(~)C 4 ~)... (~)Cr(~)D1 ~)... ~ ) D g

we find that h and (q3...qr H [ai,bi]h-b)-I also form a free

abelian group of rank 2 so we can amalgamate along these subgrcups.
A similar argument shows the assertion for all classes except for

01 , g = 0, r = 3, -~i
- + ~_ _o + I < I
~--~_ , 01 , g = I, r = I and 02, g =
I, r = 1 , where there are not enough "parts". For these cases
we note that the quotient group G/(h) is a planar discontinuous
group and has no cyclic normal subgroup,
(ii) for non-orientable B the above argument works for all
large Seifert manifolds. This completes the proof.
~e should remark here the following well known fact.

Proposition 2. Let K be a closed 3-maz:ifold. I__f K i__s

0rientable~ let K' = K if not~ let K' e~ual the orientable
double cover of K . Suppose that w1(K') is infinite, not cyc-

lie and not a free product. Then K and K' are aspherical and

Wl(K) has no element of finite order.

From this follows immediately:

Proposition 3. A large Seifert manifold M is a K(G,I)

space.
We shall see later that it follows from Waldhausen [I] that
they are also irreducible 3-manifolds.

Given the planar discontinuous group D defined by {q1''"

..,qr,al,bl,...,~g,bg I qjJ = I, ~i...~r[~1,bl]...[~g,bg]= I} or

,~g _a - -2 . . ~ : 1~
[91 . . . . ,qr,Vl,... I qjJ = 1, ~ l . . . q r V l .
 - 94 -

We define free groups D with generators ~I,...,Qr,A1,BI,...

_ _

..,Ag,Bg or I .... ' r' 71 ..... ~g and words in these groups

5. = QI...Qr[AI,BI]...[Ag,Bg] or ~* = QI"
• - V2
"Qr I""
• • ~2g • Define a h o m o m o r p h i s m D ~ D by mapping capital letters

into lower case letters. Let ~(x) = w(X) = I if we have an ori-

entable fundamental domain and ~(x) = w(X) = ±I according to

whether the ~i (or Vi) occur an even or odd number of times in

(or ~) .

Define the group G as either

[Q1,...,Qr,A1,Bl,...,Ag,Bg,HIAiHA11=T H Cl,BiHB~1 = H I,QjHQjl=H} or

[Q1,...,Qr,V1,...,Vg IViHV~I = HCi,QjHQ~ I = H} where the ci are

the same as in the definition of G . Let H. be as above (with-

out bars) and define the h o m o m o r p h i s m G ~ G by sending capital

letters to lower case letters. The map ~ is defined as above

for G and G , i.e. w(x) = ~(X) = I for x E G and X E

if B is orientable and ~(x) = ~1 (w(X) = ±I) according to the

parity of the number of times vi (V i) occur in x (X).

The next result is due to Zieschang [I]•

Lemma 4• E v e r y autgmorphism A of D is induced by an

automorphism A of D with the property that:

A(Qi ) = M i Qv i
~(~.) : ~ ~ ~-1

where

w(M)~ :
I
C =
•Vr
!I
t is a permutation with
avi
= ~i and w(Mi)~i =

This allows us to prove the following:

Theorem 5. Let M and N' be large Seifert manifolds and

I: G' ~ G an isomorphism. Then we have
 - 95 -

, X. ~i -I
I(qi)= h lmi qvi mi

where is a permutation and w(mi)Ci = p = ~1 . The

1 " Vr --
map I is induced by an isomorphism of the ~roups I: G' ~

where
i(Q.~) = H xi M i Qvi
~i M71

i(~;) = H k N II~ M -I
r
and ~(M)~ = 0 Moreover k = ~ ki+ 2a where o = 0 for
i=I
= 01 o_~r n 2 .

Proof. Since (h) and (h') generate characteristic sub-

groups,the isomorphism I induces a commutative diagram:

0 .... > (h') > G' > D' > I

"~ 111 ~ ! ~ IO

0 > (h) > O , ,> D .> I

Next define an inclusion map ~: D ~ G by Qi " Qi ' ~i " Ai '

Bi " Bi ' ~i ~ Vi and consider the diagram below where Io is

defined to induce I by lemma 4.
o

1!~'
.T lr
/
/
\
/

I G' > D'

-~I o -~ o
t a ~ > ]}
\
\
v / \V
g > 5
.... ^
@
 - 96 -

Considering the solid arrows only this diagram is commutative.

We want to lift the isomorphism I to an isomorphism I of the

"^" groups. Let ~ and q' send capital letters to lower case
^

letters. We can construct generators for G' from generators of

using the composition J = ~Io~'O In order to make the whole

diagram commute (apart from ~), we note that the difference between

I~' and qJ lies in the kernel of ¢ , (h)° Now suppose that X'

is a generator of G' and

hk(X')~ J (X') = Iq'(X') .

Define I by

I(X') = Hk(X')j(X')

I(H') = H5

where I(h') = h6 and 6 = ~1 from 11 above,

This makes the diagram

o > (~') ,>~' >~' > 1

o > (i) > ~ -> 5 > 1

commutative so I is an isomorphism. It follows from lemma 4

-~i --
that Io(Qi ) = Mi Q~i Mil

~o(~) = ~ ~ ~-1

with w(~i)C i = w(~)~ = p .

Letting k i = X(Q~) , k = X(H~) , ~(~i ) = M i , ~(M) = M we have

I(Q!)
z = H ki M i Q~i -
vi Mil

I(H~) = HX N ~ M- 1

It remains to prove the last statement. For orientable B we

 - 97 -

have

H-X~(H$) = J(H$) =J(Q~)...J(Q~)[J(A~),J(B~)]...[J(A~),J(B~)] =

-X(A~):(A~),H-x(B~)i(B~)]...
H-XI~(Q~)...H-Xr~(Q~)[H
-X(AA)^ -x(B~)
[~ ~ !(Q),H ~ ~(B~)] .

If A! and B! commute w i t h i(A') H' and then so do

i(B')
l 1 i i
r
and their c o m m u t a t o r equals [!(A'),I(B:)]I_I ' thus k = Z k i . If
i=I
A'. and B' anticommute with H' then the c o r r e s p o n d i n g commuta-
1 1
tor equals
-2X(AI)-2X(B i) ^ r
H r I(A'.),I(B!)~ so k = Z ÷2o.
I I i= I

For n o n - o r i e n t a b l e B a similar a r g u m e n t works.

This leads us to the f o l l o w i n g h o m e o m o r p h i s m classification

theorem for large Seifert manifolds.

T h e o r e m 6. Let M and M' be large S e i f e r t manifolds.

The f o l l o w i n ~ statements are e~uivalent:

(i) M and M' are e g u i v a l e n t Seifer~ bundles (possibly a f t e r

reversin~ the o r i e n t a t i o n of one),

(ii) M and M' are h o m e o m o r p h i c ,

(iii) N and M' have i s o m o r p h i c fundamental groups.

Proof. Clearly (i) ~ > (ii) ~---> (iii). Assume that we have

an i s o m o r p h i s m I: G' - G . Assume m o r e o v e r that the p e r m u t a t i o n

of theorem 5 is the identity. By l e m m a I we have an induced iso-

morphism Io: G'/(h') - G/(h) between non-euclidean crystallogra-

phic groups. This shows that B' = B , g' = g , r' = r and

a~ = ~i " Also by l e m m a 1 I(h') = h8 with 5 = ~I . Applying

 - 98 -

to the r e l a t i o n q~ aih,Bli = I according to t h e o r e m 5 gives

1

ki C. '5 ai~i I Xi~i+8 '

I = (h m i q i l m ~ 1 ) a i h 8i = miq i m~ h Bi =

-8i~ i lhXiai+8 ' _c(mi)~iSi+Xiai+88~

mih m~ 8i = h

where for x ~ G we let e(x) = ±I according to w h e t h e r x com-

mutes with h or a n t i c o m m u t e s with h . Since h has infinite

order

-e(mi)CiSi+ kiwi+ &gl = 0 .

For 01 and n2 we h a v e e(mi) = m(mi) so c(mi)~ i = ~ ( m i ) { i = p.

Thus

=
Bi o 5 8 i' + plia i

and if 05 = I then the condition 0 < Si < a i implies that

Xi = 0 while if o5 = -1 we get &k i = -I . Substituting these

values we h a v e S i = S'
i or Si = ai - Hi' for all i . For the
!
other classes the condition 0 < ~i j ~i/2 implies that 8i = H i

and k. = 0 for all i .

1
Finally we n e e d a similar computation for b :

I = I( w $ h , - b ') = h k m w . ~ m -I~-5b'
n =

hXmhCbm-lh -6b' = h X + c ( m ) ~ b-Sb'

and since h has infinite order

k +c(m)Cb-Sb' = 0 .
r
For oI and n2 we h a v e c(m) = ~(m) , o(m)~ = 0 and k = E kr
i=I
so r
E X i + o b - 6b' = 0
i=I
if p6 = I then ki = 0 and b = b'; if p8 = -I then 8k i = I
and b = -b'-r as r e q u i r e d .
For the other classes ki = 0 and k = 2~ but b,b' 6 ~2 so
b = b' . This completes the proof.
 - 99 -

5.4 Small Seifert Manifolds

This section is based on 0rlik-Raymond [27.

(i) The manifolds oi, g = 0, r ~ 2 (lens spaces).

Since these manifolds all admit S1-aetions we can use the equi-

variant method of chapter 2 to identify them. The manifold

L(b,1) = [b;(o1,0)] was discussed there. The standard orienta-

tion gives S 3 = L(-I,0) = L(1,1) aud we note that L(0,1) =

S2 x SI .

The ma&ifold [b;(Ol,0);(~,£)] is identified similarly.

By lemma (2.2.3) it is the boundary of the linear plumbing accord-

ing to the graph

-b-1, - b1, . - .b 2 .. .... . . . . . . ,s

-b

where ~ - 3 = Ebl'''''bs] ° According to lemma (2.2.1) the result

of this linear plumbing is L(p,q) where

2q = r1b +-l , b , . . . ,b s ] = b +1 a.1 _ &(b+l)~-(¢-6) = b~+~

~-6

so we see that {b;(Ol,0);(~,~)] = L(b~+~,~) .

For r = 2 we apply the same argument: [b;(Ol,0);(a1,B1),(a2,~2)]

is the boundary of the equivariant linear plumbing

- b, l ' S l -~._
b l , s1__ -_
1 ,
-bl
........ '
1 - b -, 2 _ _ _-+b_2~ 1. . ... ... . .. b2,s2

~1 ~2
where ~1 - ~-----~
= [b1'1'''''b1 'Sl ] and -~ 2 --~ 2 = [b2'I'''" ,b2,s2 ].

It is L(p,q) with

q = [bl,sl,...,b 1,1,b+2,b2,1,...,b2,s2] .

First we note that the result of a reverse plumbing

-b -b I
 - 100 -

is d e t e r m i n e d from the p r o d u c t of m a t r i c e s

(-lbl :)~10 :),,, i: 10)(bl~)=s t~-PS-lps-P's,1),,s

and by i n d u c t i o n

PS= PS ' P's= Ps-1 ' Ps-1 = Ps ' = PS-1

we
Thus have for the d e t e r m i n a t i o n of L(p,q) using (2.2.3):

1 0~-P
~lt~ 01/¢~1-0c1\ -~111 =
ala 2 + a i B 2 + a2~ I
.)
me 2 - n~ 2

where m = -by I - v I - Pl ' n = -V I satisfy the c o n d i t i o n

ma I - n ( b ~ l + ~ 1 ) = I .

The m a n i f o l d is L(p,q) with p = b~la 2 + ~ i ~ 2 + a2B I and

q = mm 2 - n ~ 2 •
The m u t u a l h o m e o m o r p h i s m classification of these m a n i f o l d s

is g i v e n by the w e l l - k n o w n classification of lens spaces: L(p,q)

and L(p',q') are h o m e o m o r p h i c if and only if !P! = !P'I and

q ± q, m 0 mod p or q.q, ~ ~I (mod p) . The fact that they are

not h o m e o m o r p h i c to any other Seifert m a n i f o l d will f o l l o w once

we have proved that they are the only ones w i t h finite cyclic fun-

d a m e n t a l groups.

(ii) The m a n i f o l d s oi, g = 0 , r = 3 , 11 + ~ 2 + ~ 3- - > I .

There are only four possible sets of ~i satisfying these condi-

tions called the "platonic triples": (2,2,a3), (2,3,3), (2,3,4)

and (2,3,5) . They have finite, non-abelian fundamental groups

 - 101 -

and will be d i s c u s s e d in detail in the next chapter where we

shaLl also show that those w i t h (2,2,~3) called "prism m a n i f o l d s "

are h o m e o m o r p h i c to m a n i f o l d s n2, g = 1 , r ~ I . Note that (h)

is in the center of NI(M) and

a1 a2 a3
~i (M)/(h) = {ql,q2,q3!qlq2q3=ql =a2 =q3 =I]
has no center so (h) is the whole center and the ~j are in-

variants of w1(M ) . The order of HI(M;E)

P = Ibala2a 3 + ~i~2a 3 + a 1 ~ 2 a 3 + m l a 2 ~ 3 1

is sufficient to d i s t i n g u i s h the m a n i f o l d s with given (al,a2,a3)

up to orientation. Since we shall see that the only other Seifert

manifolds w i t h finite f u n d a m e n t a l groups are the lens spaces and

the p r i s m manifolds, their h o m e o m o r p h i s m classification is com-

pleted.

(iii) The m a n i f o l d ~ = {-2;oi,0); (2,1),(2,1),(2,1),(2,1))

is h o m e o m o r p h i c to M' = [0; (n2,2)~ . This is seen by n o t i n g

that the orientable S I bundle over the I~Ioebius band is h o m e o m o r -

phic to the m a n i f o l d obtained by sewing two E-orbits of type

(2,1) into a fibered solid torus. Doubling the former by an ori-

entation reversing homeomorphism gives M' . D o u b l i n g the latter

by an o r i e n t a t i o n r e v e r s i n g h o m e o m o r p h i s m gives

[0; ( O l , 0 ) , ( 2 , 1 ) , ( 2 , 1 ) , ( 2 , - I ) , ( 2 , - 1 ) ] = M . We shall see in chap-

ter 7 that M fibers over SI w i t h fiber the torus and the self-

homeomorphism of the fiber is of order 2 . It turns out that I~

is a flat R i e m a n n i a n m a n i f o l d doubly covered by SI x S1 x S I and

the c o v e r i n g can be made e q u i v a r i a n t with respect to the S I ac-

tion on M , see chapter 8.

The other small Seifert m a n i f o l d s are easily seen not to be

 - 102 -

homeomorphic to each other or any of the large ones with the ex-

ceptions m e n t i o n e d below, compare Orlik-Raymond [2q. We shall

briefly mention their special p r o p e r t i e s and r e t u r n to them in

chapter 7.

(iv) The m a n i f o l d s [b; (o1,1)) are torus bundles over SI .

(v) The m a n i f o l d s [b; (o2,1)) are K l e i n bottle bundles

over SI .

(vi) The m a n i f o l d s nl, g = I , r ~ I give rise to the dif-

ferent S I actions on p2xS I and N , the n o n - o r i e n t a b l e S 2-

bundle over SI .

(vii) The m a n i f o l d s n2, g = I , r ~ I . Here M = [0;(n2,1)]

is seen as the result of taking S2 x I fibered by intervals p ×I

and c o l l a p s i n g each b o u n d a r y component by the a n t i p o d a l map. The

sphere S 2 x [½] decomposes M into a connected sum of two real

projective spaces, M =]RP 3 # ~ p 3 . The other m a n i f o l d s are h o m e o -

morphic to the p r i s m m a n i f o l d s of (ii) and will be treated in de-

tail in the n e x t chapter as orbit spaces of finite groups acting

freely on S3

(viii) The m a n i f o l d s {b; (ni,2)] are the same two K l e i n

bottle b u n d l e s as u n d e r (v).

(ix) The m a n i f o l d s [b; (n2,2)] are torus bundles over SI

distinct from (iv).

(x) The m a n i f o l d s [b; (n3,2)~ are the "other two" K l e i n

1
bottle bundles over S not obtained in (v) and (viii).
 - 103 -

6. Free Actions of Finite Groups on S3

There has b e e n no s i g n i f i c a n t progress in the p r o b l e m of

finding all 3-manifolds w i t h finite fundamental group since the

results of H. Hopf [I] and Seifert and T h r e l f a l l [I] d e t e r m i n i n g

orthogonal actions on S3 . These articles are somewhat d i f f i c u l t

to read and the object of this chapter is to p r e s e n t old k n o w l e d g e

w i t h n e w terminology. The basic theorem of section I is that if

G is a finite subgroup of S0(4) acting freely on S 3, then

there is an a c t i o n of SI on S3 commuting w i t h G so that the

orbit space S3/G is again an S1-manifold. Thus the orbit spaces

of o r t h o g o n a l actions are S1-manifolds with finite f u n d a m e n t a l

groups. These are d i s c u s s e d in section 2. In section 3 we list

following Milnor r2] the groups that satisfy the algebraic condi-

tions for an action but do not act orthogonally.

The i n t r i g u i n g fact remains that if one could find a 3-mani-

fold with finite fundamental g r o u p not h o m e o m o r p h i c to one listed

above, then either it w o u l d be the orbit space of a n o n - o r t h o g o n a l

action on S3 or its u n i v e r s a l cover w o u l d provide a counterex-

ample to the 3-dimensional Poinear@ conjecture.

6.1. Orthogonal Actions on S3

In order to u n d e r s t a n d the structure of finite subgroups of

S0(4) that can act freely on S 3, we shall d e c o m p o s e S0(4) .

It is u s e f u l to think of S0(4) both as a g r o u p of orthogonal

transformations of R4 and as a m a t r i x group of 4 ×4 real or-

thonormal matrices. It is clear that the m a x i m a l torus of $0(4)

is T 2 = S0(2) × $0(2) and the center is g e n e r a t e d by the i d e n t i t y

map e and the antipodal map a = -e . Let C = [e,a] denote
the center of S0(4) .
 - 104 -

Lemma I. The f o l l o w i n g sequence is exact:

i P

Proof. F r o m Lie group theory we have that Spin(4)/center =

S0(4)/C = Ad Spin(4) = Ad(Spin(3) × Spin(3)) = Spin(3)/center ×

Spin(3)/center = S0(3) × S0(3) .

In order to g a i n g e o m e t r i c insight we shall n o w give a direct

proof. Consider the m a x i m a l torus T2 g i v e n by the m a t r i c e s

os ~ -sin ~ 0 0

The

called
l
subgroup

right
0

generated

rotations,
cos

sin ~

by all

R
~

The
-sin

cos

l-dimensional

subgroup
0

0
<

<~

circles

generated
r# <

<2rr
2st

~ = ~

by
.

~ ~ -~
is

mod 2w is called left r o t a t i o n R , L . Note that ROL = C and

abstractly R~L ~ S3 . Every element g E S0(4) is d e c o m p o s e d

into a r i g h t and left r o t a t i o n but this d e c o m p o s i t i o n is only de-

fined m o d u l o a . Moreover, every right rotation commutes with

every left r o t a t i o n and v i c a versa. Specifically, if we choose

coordinates so that g is g i v e n by the m a t r i x above, then for

some r i g h t rotation by Xr and left r o t a t i o n by XI we have

= Xr+Xl +2kw

= Xr- X I + 2k'~
and hence

X r = ~(~+¢) + (k+k')~

~-I = ½(~-~) + (k-~,)~

 - 105 -

are the possible choices of angles for right and left rotations.

Thus g can be d e c o m p o s e d into two pairs (Mr,X1) and

(Xr+W,Xl+W) d i f f e r i n g by the a n t i p o d a l map. In order to elimi-

nate this i n d e t e r m i n a c y we construct double covers

Pr: R ~ S0(3) and Pl: L ~ S0(3) as follows:

Given a vector v in R4 and a right r o t a t i o n r by the angle

M r , there is a unique plane through v rotated in itself by r .

There is also a unique left r o t a t i o n 1 rotating the same plane

by X 1 =-Mr so that the r o t a t i o n rl leaves v fixed. It ro-

tates the R 3 perpendicular to v by an angle Mr Xr-Xl 2Mr"

The same c o n s t r u c t i o n applies for left rotations.

Thus if g 6 S0(4) is d e t e r m i n e d in a suitable coordinate

system by the angles (~,@), then its image in S0(3) × SO(3) may

be i d e n t i f i e d by two R 3 rotations (X~,Xi) fixing a g i v e n v e c t o r

where
, ~ ~ + , ×~ ~ ~-@ (mod 2w) .
Xr

L e m m a 2. If

M r' ~ XI' ~ W (mod 2w)

then both g and ag have fixed points on S3 . If

' ~ + ' (mod 2w)

Xr -X I

then either g o_rr ag has f i x e d points on S3 . If n e i t h e r

e qngruence h o l d s then b o t h g ~ud ag are free on S3 .

Proof. Recall that ~ ~ Mr+M1 (mod 2w) and ~ ~ Mr-M1

(mod 2w) so g has fixed points on S3 if Bud only if at least

one of these angles is zero so 7 r ~ X1 ~ 0 (mod 2~) . F r o m the

 - 106 -

relations M r' ~+- 2M r , X~ ~ 2M1 (mod 2w) we obtain the required

formuli. The converse is a similar computation.

Let G c SO(4) be a finite subgroup acting freely on S3 o

Let H = p(G) and H 1 = PrlHCSO(3) , H 2 = Pr2HcSO(3) . Then

clearly H c H I ×H 2 but H itself is not n e c e s s a r i l y a direct

product of subgroups.

The finite subgroups of SO(3) were first found by F. Klein.

They are the

cyclic group C n of order n , C n : Ix Ix n : 1} ;

dihedral group D2n of order 2n , the group of space symme-

tries of a regular plane n-gon generated by rotations and a re-

flection
D2n = [x,y I x2 = yn = (~#)2 = I~ ;

tetrahedral group T of order 12, the group of symmetries of

a regular tetrahedron,

T = ~x,y!x 2 = (xy)3 = y3= I~ ;

octahedral ~roup O of order 24, the group of symmetries of

a regular octahedron or , equivalently the cube

o = ~ x,ylx 2 = (xy) 3 = y 4 = I~ ;

icosahedral ~roup I of order 60, the group of symmetries of

a regular icosahedron or , equivalently the dodecahedron

I = ~x,yfx 2 = (xy) 3 = y5 = I~ .

Lemma 3. Every finite subgroup of S0(3) is one of the

above.
 - 107 -

Proof. (Wolf [1]) If G is a f i n i t e subgroup of S0(3)

and g E G g # I, then g is a r o t a t i o n by an angle 8g about

a line Lg through the origin. Let ~g be the intersection of

Lg with the u n i t sphere S2 consisting of the two "poles" Pg =

[pg,p~] which are the only fixed points of g on S2 . We call

two p o i n t s x,y E S2 G-equivalent if gx = y for some g E G .

Let [CI,...,Cq} be the equivalence classes of p o l e s for all n o n -

trivial elements of G . If p is a pole • let Gp be the sub-

group preserving p : Gp = 1 U [g E G - 11 P E Pg} . Let p belong

to the class Ci and enumerate Ci as [glp,g2p,... griP] with

gl = I and the gi a system of r e p r e s e n t a t i v e s of the cosets of

-1
Gp in G . In p a r t i c u l a r Ggip = giGpgi exhaust all the con-

jugates of Gp in G and the Ggip all h a v e the same order ni °

If N is the order of G then N = rin i .

Note that G has N - 1 non-trivial elements and each one

has 2 poles. Since exactly ni - I non-trivial elements of G

preserve a pole p E Ci we h a v e the identity

q
2(N-1) = i~iri(ni-1)

SO

2(I-~) = qE ( 1 - - ~ t ) .
i=1 ni

Since N ~ ni ~ 2 we see that q is 2 or 3 and one of the f o l -

lowing must hold:

(i) q = 2 , n I = n 2 = N>I

(ii) q = 3 , 2 = nl ~ n2 ~ 3 n2 < n3 with the p o s s i b i l i t i e s

a) n I = n 2 = 2 , N = 2n 3 _> ¢ ,
b) n I = 2 , n 2 = n 3 = 3 , N = 12 ,
c) n I = 2 , n2 = 3 , n 3 = 4 , N = 24 ,
~) nI = 2 , n2 = 3 , n 3 = 5 , N = 6 0 .
 - 108 -

It is n o w a simple geometric argument to show that these cases

indeed correspond to the a l r e a d y listed groups.

~e can n o w combine lemmas 2 and 3 noting that D 2 n , T, 0

and I have elements of even order and go t h r o u g h the p o s s i b l e

subgroups of HI × H2 to obtain:

Lemma 4. At least one of HI and H2 is cyclic.

This enables us to prove the m a i n theorem of this section

due to S e i f e r t and T h r e l f a l l [I].

Theorem 5. Let G be a f i n i t e s u b g r o u p of S0(4) acting

freely on S3 . Then there is an S1-action on S3 so that the

action of G is e q u i v a r i a n t and the orbit space s3/o

an St-manifold.

Proof. We may assume that HI is cyclic. Since R ~ S 3,

its p r e i m a g e G r = p~1(H I ) is cyclic and we can embed it in a

circle subgroup ~ of R . Note that this is not true of every

cyclic subgroup of SO(4) . Since every element of G decomposes

into a left and a right rotation and the left r o t a t i o n s commute

with E while the right rotations are contained in Z we see

that G is e q u i v a r i a n t with respact to E .

It is easy to see by direct argument that the converse is

also true, i.e. every St-manifold w i t h finite fundamental group

is the orbit space of a free orthogonal action of a finite group

on S3 . We shall list the g r o u p s and the orbit spaces in the

next section.
 - 109 -

6.2 Groups and Orbit Spaces

We proved in (6.1) that if G is a finite subgroup of S0(4)

acting freely on S3 and H I c S0(3) , H 2 c S0(3) are the pro-

jections of G , then either H1 or H2 is cyclic. Assume that

HI is cyclic of order m . Before we list the possible groups

note that if G has even o r d e r , t h e n a E G and G/C ~ H so G

is a C2 central extension of H . Writing H = [e,hl,...,h k]

we have G = [~e,±hl,...,±hk~ . On the other hand if G has odd

order then G ~ H .

The double cover S 3 - S0(3) gives rise to finite subgroups

of S3 doubly c o v e r i n g those of SO(5) . Corresponding to D2n

we have D~n of order 4n

D*4n = [x,y I x 2 = (xy) 2 = yn]

and c o r r e s p o n d i n g to T, 0, I we have the b i n a r y tetrahedral

group T* of order 24, the b i n a r y octahedral group 0* of order

48 and the b i n a r y i c o s a h e d r a l group I* of order 120 p r e s e n t e d by

[x,y I x 2 = (xy) 3 = y n x % = I] for n = 3,4,5.

It can be shown that these are in fact the only finite sub-

groups of S3 . Thus if HI = e then G is one of these groups.

Also, if HI is a cyclic group of r e l a t i v e l y prime order to one

of the above g r o u p s s t h e n the direct product will act freely.

It r e m a i n s to investigate the n o n - t r i v i a l possibilities.

Pirst note that if H is a s u b g r o u p of HI × H2 then the elements

of the f o r m (hl,e) E H f o r m a subgroup H~ c H I and similarly

Hi c H2 so that H' = H1.

2 ..
'H , c H is an invariant subgroup. The

q u o t i e n t groups

H/H' . H 1 / H ~ ~ H2/H ~ ~
 - 1 1 0 -

are i s o m o r p h i c so H consists of elements (hl,h2) with the pro-

perty that the coset of hI in H1/H ~ corresponds to the coset

of h2 in H2/H ~ under the i s o m o r p h i s m with F .

We again assume that H 1 = Cm is cyclic.

If H 2 = Cn is also cyclic~then we assert that H is also

cyclic. This is clear if (n,m) = I . Otherwise suppose that F

is of order f so H~ has order m' = m / f and Hi has order

n' = n/f . Clearly they are also cyclic. We shall prove that if

G acts freely on S 3, then H must also be cyclic. If a gene-

rates HI and b generates H2 then H~ consists of all powers

of af and H' of b f Given an element of F, the elements of

2
HI c o r r e s p o n d i n g to it in the coset d e c o m p o s i t i o n mod H' are
I
those of the f o r m a kf+p for fixed p and all p o s s i b l e k . If

it c o r r e s p o n d s to a g e n e r a t o r of F~ then its order is f and

(f,p) = I . Let k equal the p r o d u c t of all primes in m not

in f •p (or k = I if no such prime exists). Then (kf+p,m) = I

and u = a kf+0 has order m and therefore generates HI . We

can find a s i m i l a r generator v for H2 . It r e m a i n s to show

that (u,v) generates H ° Since at least one of the p r e i m a g e s

of (u,v) in S0(4) is fixed point free, it f o l l o w s from (6.1.2)

that (m',n') = 1 . Find p,q so that pm' + q n ' = I . Then

clearly pm ~ f (mod n) and qn ~ f (Inod m) so u qn = u f and

v pm = v f . F r o m this we get for a r b i t r a r y k,l,0 that

(ukf+P, v lf+o) = (u,v)kqn+lpm+p

proving the a s s e r t i o n that H is cyclic.

Assuming that H2 is one of the other groups D2m , T,0,I

and using similar arguments it can be shown that only two more

types of groups occur.

 - 111 -

If H I = 02k_i , H 2 = D2(2n+1 ) , H'I = C2k-2 ' Hi = C2n+1

and HI/H~ ~ H2/H ~ *~ C 2 then we obtain a group H with double
cover in S0(4) equal to

D'k(2 = [x,y Ix2k=l, y2n+l=1, xy-l=y-lx~l k>2, n > l .

2n+1 ) -- --

Note that D~(2n+1 ) = D~(2n+1 ) .

If H I = C3k , H 2 = T , H'I = C3k_I , H' = C 2 X C 2

2
and

HI/H ~ ~ H2/H ~ ~ C 3 then we obtain a group H with double cover

in S0(4) equal to

T'8.3k = [x,y,z I x 2 = ( x y ) 2 --~ 2 , zxz-l=y, zyz -S = xy, z3k=l },k>1 .

Note that T~4 = T*

24 "
Thus we have the following conclusion, see H. Hopf rl],
Seifert-Threlfall ~I] and Milnor F2].

Theorem I. The following is a list of all finite subgroups of

S0(4) that can act freel 2 on S3 :

Cm, D* D' , T*, T' 0", I* and the direct product of

4m' 2k(2n+1) 8.3 k'
any of these groups with a cyclic group of relatively prime order.

Orbit spaces of finite groups acting freely and orthogonally

on a sphere are called spherical Clifford-Klein manifolds. The

3-dimensional ones correspond to $eifert manifolds with finite

fundamental group by (6.1.5) and are listed as follows, see

Seifert-Threlfall FI].

Theorem 2. The Seifert manifolds with finite fundamental

group are:
 - 1 1 2 -

(i) M = {b;(o1,0);(a1,~1),(a2,82)} , here we allow ~ = 1,

= 0 , are lens spaces (see 5.4) with ~1(~) = Cp where p =

Ib~la2 +a182 + ~1a21 ;

(ii) M = [b;(o1,0);(2,1),(2,1),(a3,83)] are called prism

manifolds. Let m = (b+I)63 + 83 ; i_~f (m,263) = 1 then ~i(~)

= CmxD~63 , and if m = 2m' then n e c c e s s a r i l y m' is even and

(m,,63) = 1 and letting m' = 2km '' we have ~I(M) =Cm. X D ~ k + 2 3;

(iii) M = [b; (01,0);(2,1),(3,82),(3,~3)) , let m = 6b+3+

2(~2+~3) , i_~f (m,12) = 1 then ~i(~[) = C m x T * , and

i_~f m = 3km ', (m,,12) = I then ~I(M) = Cm, × T ~ . 3 k ;

(iv) M = {b;(oi#);(2,1),(3,~2),(4,~3) } , let m = 12b+6 +

4# 2 + 3 8 3 , it follows that (m,24) = I and w1(M) = C m x 0 * ;

(v) M = [b;(o1,0);(2,1),(3,~2),(5,~3) ] , let m = 30b + 15 +

1082 + 6 8 3 , it follows that (m,60) = I and w1(M) = C m x I * ;

(vi) M = [b;(n2,1);(a1,~1)] with n = Iba 1+ 811 / 0 are

homeomorphic to prism manifolds so that

if 61 is odd then ~I(M) = C61 >~ D*4n and

i_~f 61 = 2k6~ , (6~,2) = 1 then ~I(M) = C D~k+2 n •

Proof. Except for (vi) the proof consists of verifying the

group isomorphisms. It remains to prove that every prism maul-

fold also admits a Seifert bundle structure of type n2 over the

projective plane. If G is the group, acting on S3 with cyclic

HI and H 2 = D2n the dihedral group then we consider the maximal

cyclic subgroup Cn of D2n and the cyclic group C~n c G map-

ping onto Cn . Since C*2n consists of left r o t a t i o n s ,

 - 113-

C*2n c L ~ 8 3 , it can be e x t e n d e d to a circle group F c L . If

5 is a left r o t a t i o n of order 4 in the g r o u p

D* whose image
4n
is the r e ~ l e c t i o n of D 2 n , then for every element y E F we have
6y8-1 = y -1 . Thus 8 maps the orbits of the circle action in-

duced by £ into each other revers in~ the o r i e n t a t i o n and S3/G

admits a S e i f e r t f i b r a t i o n of class n2 . Since Wl(~) is finite

the orbit space is p2 and r < I .

M

Remark. It can be shown d i r e c t l y that apart from the lens

spaces whose h o m e o m o r p h i s m c l a s s i f i c a t i o n was g i v e n in (5.4) two

3-dimensional spherical Clifford-Klein manifolds are h o m e o m o r p h i c

if and only if their f u n d a m e n t a l groups are isomorphic. Note al-

so that u n d e r (vi) n = Iba I + ~ i I = 0 if and only if M =

[0;(n2,1)! =mP3~P 3 , see (5.4).

6.3. Non-orthogonal Actions

It is not k n o w n w h e t h e r there exists a smooth free action

of any group G on 83 not conjugate to one of the orthogonal

actions above. Since every such action has as orbit space a

closed, orientable 3-manifold M with fundamental group G , it

follows that G must have cohomology of period 4. We see from

(6.1.2) that G can have at m o s t one element of order 2. All

finite groups not a p p e a r i n g in (6.2.1) satisfying these conditions

are listed by ~ i l n o r [2] as follows:

(i) Q(8n,k,1) = [x,y,z I x 2 = (xy) 2 = y 2 n , z k l = I, xzx-1=zr, y z y - ~ 1)

where 8n,k,1 are pairwise relatively prime integers so that if

n is o d d 3 t h e n n > k > 1 ~ 1 and if n is e v e n ~ t h e n n ~ 2 ,

k>l>1. m
 - 114 -

(ii) 0'48.3k k _> I is the extension I ~ C3k - O'48.3k - O* ~ I

with the property that its 3-Sylow subgroup is cyclic and the

action of O* on C3k is given as follows~ The commutator sub-

group T* c O* acts triviall~ while the remaining elements of

O* carry each element of C3k into its inverse.

(iii) the product of any of these groups with a cyclic group of

relatively prime order.

The smallest group on this list is Q(16,3,1) of order 48 that

may or may not be the fundamental group of a 3-manifold.

 -115-

7. Fibering over S1

In this chapter we shall find the Seifert manifolds that

admit a locally trivial fibration with base S1 and fiber a 2-

manifold. This was originally done by Orlik-Vogt-Zieschang [1]

for almost all cases and completed by Orlik-Raymond [21. These

results are recalled in section 2. In the meantime, however, a

beautiful theory of injective toral actions has been developed by

Conner-Raymond [I] and we shall discuss these general considera-

tions first. Tollefson [I] and Jaco [I] noted independently that

the product bundles M = [0;(ol,g)] fiber over S1 in infinitely

many distinct ways, i.e. with infinitely many mutually non-homeo-

morphic fibers. An outline of this argument is given in Section 3.

7.1. Injective Toral Actions

This section consist of results of Conner-Raymond [I].

Let X be paracompact, pathconnected, locally pathconnected

and have the homotopy type of a CW complex. In the applications

we shall assume that X is a manifold. An action of the torus

group T k = S I ~ s I ×... >~S I (k times) on X is called injective

if the map

i ~l(Tk, 1) ~ ~I(X, x)

defined by f~(t) = tx is a monomorphism for all x

In this case we have a central extension

0 . ~k . NI(X ) - F ~ I

and only finite isotropy groups occur.

 -116 -

Theorem I. Let (Tk,x) be an action and HI(X;~) be fini-

tel 2 generated. Then (Tk,x) fibers equivariantl2 over Tk if

and only if the induced map

x
f. : HI(Tk,1) - HI(X,x )

is a m o n o m o r p h i s m .

Note that if f.x is a m o n o m o r p h i s m then so is x

f~ and the

a c t i o n is injective. For the p r o o f we start w i t h an injective

action and c o n s i d e r subgroups of ~1(X,x) containing im ~ .

Let BH be the covering space a s s o c i a t e d with H and bo E BH

be a base point c o r r e s p o n d i n g to the constant path at x . The

action of Tk may be lifted to BH

T k × B H ..... > B H

Tk x X > X

since in the c o r r e s p o n d i n g diagram of f u n d a m e n t a l groups imf~ mH.

T h e o r e m 2. If im fx# c H and H is normal then the action

(Tk,BH) is e q u i v a r i a n t l y homgomorphic to (Tk,Tk~<Y) , where the

Tk a c t i o n is ~ust left t r a n s l a t i o n on the first factor.

The most important case is w h e n ~ = id: w 1(X,x) ~ ~1(X'x)

and H = im(f x) Note that in this case ~I(BH) = H = ~.k so Y

is simply connected.

The proof of theorem 2 consists of first showing that there

is a n a t u r a l splitting H _~ 2Zk , < k e r ~ . This follows because

h E ~1(X,x) lies in H if and only if there is t E ~,k so that

x
~of (t) = ~(h) E L and since f# is a m o n o m o r p h i s m t is unique.

Define an e p i m o r p h i s m p : H -2~ k by p(h) = t in the above for-

 -117-

mula. We have p(f~(t)) = t and ker ~ = ker p . Define q: H

ker ~ by q(h) = h. fx~

~ P t ~h-1 )) . Clearly x c ker q
im f~ and

since it is a central subgroup it is the whole kernel. Note that

if h E ker ~ then q(h) = h and h = fXp(h).q(h)

# proving the

splitting of groups. Next we use i n d u c t i o n on k . For k = I

let ~
be the g e n e r a t o r of ~I($1,1) r e p r e s e n t e d by exp(2~it) ,
b
0 ~ t ~ I . Then f~o(w) = e x p ( 2 w i t ) b ° r e p r e s e n t s the g e n e r a t o r

of the ~ f a c t o r in W1~H ) = H and by the n a t u r a l i t y of the

splitting b° must have trivial i s o t r o p y group, i.e. if

e x p ( 2 ~ i t / n ) b o , 0 ~ t ~ I , is a closed loop then n e c e s s a r i l y

n = I . A similar a r g u m e n t applies for arbitrary b E BH showing

that the S~action is free. Induction on K proves that (~,BH)

is free. The fact that the p r i n c i p a l Tk-bundle over BH is tri-

vial is obtained u s i n g the L e r a y - H i r s c h theorem and the splitting

H ~k × ker ~ .

F r o m the g r o u p of c o v e r i n g transformations N = ~I(X,x)/H

and the p r o j e c t i o n in the s p l i t t i n g onto Y we obtain an N - a c t i o n

on Y w h i c h turns out to be p r o p e r l y discontinuous (all isotropy

groups are finite and the slice t h e o r e m holds).

The n e x t step in the p r o o f of theorem I is to classify ac-

tions of N on Tk x Y w i t h the p r o p e r t y that

(i) Tk acts on the first f a c t o r by left translations,

(ii) the a c t i o n of N commutes with this T k action and is equi-

v a r i a n t with a g i v e n p r o p e r l y discontinuous action (N,Y) by the

p r o j e c t i o n map.

Such actions are in one-to-one correspondence w i t h elements of

HI(N;Maps(Y,Tk)) where the N-module structure on the abelian

group W a p s ( Y , T k) is g i v e n by (af)y = f(Y~) for f E Maps(Y,Tk),

E N . Thus the action is g i v e n by a map m : Tk × Y × N ~ T k so

 - 118-

that for t E Tk , y E Y , a E N we have (t,y)a = (m(t,y,~),ya).

Now m(t,y,a) = tm(1,y,~) by the left action of Tk so it is

sufficient to consider maps m: Y >~ N ~ Tk satisfying m ( y , a 8 ) =

m(y,a)m(ya,p) . The c o r r e s p o n d i n g action is (t,y)a = (tm(y,a),ya).

Consider these maps as Z1(N;~aps(Y,Tk)) , the l-dimensional co-

cycles. Two such maps m1(Y,~ ) and m2(Y,a ) are cohomologous

if they give rise to e q u i v a r i a n t actions. Then there is a map

g :Y ~ Tk so that we have an e q u i v a r i a n t h o m e o m o r p h i s m

~: (Tk,T k × Y , N ) I - (Tk,T k × Y , N ) 2

defined by F(t,y) = (tg(y),y) in w h i c h case

m2(Y,a) = m 1 ( Y , ~ ) g ( y ) g ( y a ) -I .

If the c o h o m o l o g y class of m is of finite order, say n , then

there is a map g: Y - Tk for w h i c h

(.) g(y)g(y~)-1 m m(y,~)n for all ~ E N .

In p a r t i c u l a r if N is a finite group of order n , then every

element of HI(N;Maps(Y,Tk)) has finite order d i v i d i n g n .

The last step in the p r o o f of theorem I is to show that

g i v e n the map g satisfying (*),the space X fibers over Tk

w i t h structure group (~n)k, where we think of (Zn)k c T k as the

p r o d u c t of n-th roots of unity. Let C = [(~,y)l~ng(y)= 1] c

Tk x Y . It admits an action of (En)k since if k 6 (~n)k and

(~,y) E C then (kT,y) 6 C . Also, C is an invariant subset of

the action (T k x Y , N ) because by (*) if (T,y) 6 C then

nm(y,~)ng(ya) = ~ng(y) = I showing that (Tm(y,a),ya) 6 C .

Thus there are actions ((~n)k,c,N) . Let W = C/N with the in-

duced (~n)k action, let [~,Y] E W be the e q u i v a l e n c e class of

(~,y) under the action of N on C and w: T k × Y - X the N
 - 119 -

orbit map. Define a new Tk-equivariant map G: T k x W - X by

G(t,[~,y]) = w(t~,y) = tTn(1,y) . The fact that G is well de-

fined follows f r o m w(tfm(y,~),ya) = tf~(1,y) . If G(t,[T,y]) =

G(to,[~o,Yo]) ~henforsome mEN y a = y o and t~m(y,a) = tot o . Now

tn tnTn m, ,n , , tn tnTn t . tnTn ,
= ~y,a) g~ya) and o = o og~Yo ) = o o g~ya) so it
follows that tn = tn and therefore there is a X 6 (~n)k such
o
that kt o = t , kTm(y,a) = T° and (tk-1,[k~,y]) = (to,[~o,Yo])

showing that if (En)k acts on Tk x W by k(t,[v,y]) =

(tk-1,[k~,y]) then G induces a Tk-equivariant homeomorphism

of (T k x W ) / ( E n ) k with X . The f i b r a t i o n over Tk is g i v e n by

the map (t,[~,y]) - tn with fiber W and structure group (~)k

x
The p r o o f is completed by n o t i n g that if f . : HI(Tk,1) -

- HI(X,x) is a m o n o m o r p h i s m , t h e n provided HI(X,x ) is f i n i t e l y

generated, we have a direct summand L of rank k with

im f~ c L and an e p i m o r p h i s m ~: w1(X,x) - L . The g r o u p N =

L/~(imf~) is therefore finite.

Observe that the c o n s t r u c t i o n depends on the choice of the

map g: Y - T k . Different choices may even give fibers of dif-

ferent h o m o t o p y type as we shall show in s e c t i o n 3.

For X a closed 3-manifold and k = 1 we obtain the follow-

ing statement.

Corollary 3. A Seifert m a n i f o l d ~ of class oI o~r nI

admits an e q u i v a r i a n t fibration over SI if and only if the order

of the principal orbit h i_~n HI(M;E) is infinite.

Note that if there is a fibration, then the c h a r a c t e r i s t i c

map of the fiber (3.11) is of finite order. We shall see in the

n e x t section that large Seifert m a n i f o l d s of the other classes do

 - 120 -

not admit a fibration over SI 9 v~ile some small Seifert mani-

folds admit non-equivariant fibrations over SI so that h has

finite order in HI(M;~) and the characteristic map is of infi-

nite order.

7.2, Fibering Seifert Manifolds over S

A 3-manifolds is called irreducible if every tamely embedded

2-sphere bounds a 5-cell. The following result is due to

Waldhausen [I], see (8.1).

Theorem I. Large Seifert manifolds are irreducible.

The basic result on fibering 3-manifolds over SI is due to

Stallings [12.

Theorem 2. Let M be an irreducible compact 3-manifold.

If ~i(~) has a finitely generated normal subgroup N / [1],Z 2 ,

with quotient NI(M)/N ~ E then M fibers over SI with fiber

a compact 2-manifold T and NI(T) ~ N .

These manifolds were classified by Neuwirth [I~. In particular

for closed manifolds we have:

Theorem 3. Let M2 be any closed irreducible 3-manifold

and MI a closed manifold satisfyin~ the conditions of t h e o r e m 2 ,.

Then MI is homeomorphic to M2 if and only if w1(~ is iso-

morphic to ~I(M2) .

The next result is from Orlik-Vogt-Zieschang [I].

Theorem 4. Let G be the fundamenta I ~roup of a large

Seifert manifold and H the maximal cycli0 normal subgroup gene-

 - 121 -

rated by h . There is a f i n i t e l y g e n e r a t e d n o r m a l ~ u b g r o u p

N c G with G/N ~ ~ if and only if [G,G] N H = [I}

Proof. If [G,G~ 0 H = [I] then H injects into G/~G,G] :

HI(M;Z ) and since it is an infinite cyclic subgroup of G its

image is c o n t a i n e d in an infinite summand of G/rG,G] . We can

construct a homomorphism ~: G - ~ so that ker~0 H = {I} .

Then we have the commutative diagram

0
T
v
H

1 >N >G ~,>~ >0

I > N' >G/H ~>~/~H--> 0

where N' is the kernel of the induced map G/H ~ ~ / ~ H . Since

ker ~ 0 H = [I} we see that ~ is an isomorphism. But G/H is

finitely generated and Z/oH is finite so N' and hence N is

f i n i t e l y generated. Note that this argument has elements of the

p r o o f of (7.1.1).

Conversely, if N is a f i n i t e l y g e n e r a t e d n o r m a l subgroup

with G/N ~ ~ then it follows from the fact that M is large

and from the above theorem of Stallings that N is the f u n d a m e n -

tal group of a closed 2-manifold. If N 0H ~ ~1] then N con-

tains an infinite cyclic normal subgroup. This is only possible

for the torus and the K l e i n bottle~ Let N' = N for the torus

and let N' be the free a b e l i a n subgroup of rank 2 in N for

the K l e i n bottle. Clearly, N' 0 H ~ I and N'/N' 0 H must be a

cyclic group since in G/H (M large~) two elements commute if

and only if they are the powers of some other element. On the
 - 122 -

other hand N'/N' A H would be a cyclic normal subgroup of G/H

and this is a contradiction. Thus N D H = [I] and clearly

Corollary 5. Let M be a large Seifcrt manifold. It fibers

over SI if and only if the order of the fiber h i_~n HI(M;E)

is infinite.

Since for classes other than oI and nI we have the homo-

logy r e l a t i o n 2h = 0 , this corollary gives the same condition

as (?.I.3).
L o o k i n g at the h o m o l o g y relations one can see i m m e d i a t e l y (3.11)

that

(i) for oI the order of h is infinite in HI(M;~) if

and only if

p = bal...~r+B1a2...~r+...+al...ar_iBr = 0

(ii) for nI the order of h is always infinite in HI(M;E).

For a m a n i f o l d M let A(M) denote its h o m e o t o p y group, the

group of isotopy classes of s e l f - h o m e o m o r p h i s m s divided by the

subgroup of those isotopic to the identity. For a group G we

denote by Aut(G) the full group of a u t o m o r p h i s m s of G and by

In(G) the subgroup of inner automorphisms°

If M is a B-bundle over $I~ then it is d e t e r m i n e d by the

characteristic map ~: B - B . If B ~ S2,p 2 then theorem 3

says that M is d e t e r m i n e d by its f u n d a m e n t a l group. Now a well-

k n o w n theorem of N i e l s e n states that

A(B) = Aut(~IB)/In(~iB)

so the isotopy class of ~ is d e t e r m i n e d by the induced automor-

 - 123 -

phism ~: NI(B) - NI(B) up to inner automorphisms.

Given an automorphism of w1(B) we call the manifold, obtained

as a fiber bundle over SI with characteristic map some ~ whose

induced map agrees with ~ up to an inner automorphism, M . From

the previous discussion it follows that M is well defined. We

let

~I(B) = (Xl,...,Xnl %)

where ~. = [Xl,X2],...,[Xn_l,Xn] if B is orientable and w.

= x ,..., x2n if B is non-orientable. A presentation of Wl(M )

is then given by

w1(M ) = (x1,...,Xn,Xl~.,xxix-1 =$(xi) , i = I .... ,n).

Now consider the small Seifert manifolds, see Orlik-Raymond

[2]. The two fibers we shall encounter are the torus T and the

Klein-bottle, K . Recall that A(T) is isomorphic to the multi-

plicative group of unimodular 2 x 2 integer entry matrices. It

can be generated by

~1 =
(:;) (:i)(:I
' ~2 = ~ ~3 =
0

and a presentation is given by

2 3 )2 )2
A(T) = (~1,~2,~31~I = ~ = ~ =~1~2= (~1~ 3 = (~2~ 3 = 1).

The orientation preserving automorphisms (matrices with determi-

nant +I) form a subgroup of index 2

A+(T) = (~1,~21~I=M~=~M~= 1)

isomorphic to the free product of C4 and C6 amalgamated along

the subgroups isomorphic to C2 . This shows that the only ele-

 - 124 -

ments of finite order in Fk+(T) are powers of ~I and ~02 and

their conjugates.

It is known that A(K) = ~ 2 +2~2 and generators may be given

2 2
as the following automorphisms of w1(K) = (Xl,X21XlX2=1):

$1(xI ) = x2 ' ~1(x2) = Xl ; ~2(xI ) = Xl I ' ~2(x2 ) = x21 "

Now let us consider the small Seifert manifolds.

(i) oi, g = 0 , r < 2 are either lens spaces or S2 x SI ,

the latter if and only if p = bala 2 + Sla 2 + a l s 2 = 0 . From this

equation we conclude that ~2 = al and ~2 = -(b~I+~I) so b=-I
and 82 = ¢1 -81 • Thus the complete set of S1-actions on S2xS I

is given by the collection [-1;(o1,0),(a1,81),(¢1,al-81)] . The

order of h is infinite in HI(S 2 × $ I ; ~ ) .

I I I
(ii) oi, g = 0 , r = 3 9 ~-~+a-~+~-~ > I have finite HI(M;~)
and cannot fiber over S1 .

(iii) M : {-2;(o1,0);(2,1),(2,1),(2,1)~,1)}sa~sfies the con-

dition for an injective action and it is easily seen that h has

infinite order in HI(M;E) In fact there is an equivariant fi-

bration of M over SI with fiber T and ~ = ~ 6 ~(T), see

(ix) below.

(iv) M = [b;(oi,1 )] are T-bundles over SI . Specifically,

nI(M) = (al,bl,hl[al,bl]h-b,[al,h],[b1~h]) and the map f(a 1) = X l ,

f(bl) = x , f(h) = x 2 defines an isomorphism with M for ~ =

(~2)-b 6A+(T)whose matrix is (~-~) Note in particular

that for b # 0 ~ has infinite order in A+(T) and h has

finite order in HI(M;E) . Of course, for b = 0 we have M =

S 1 × $ 1 × S1 .

(v) M = [b;(o2,1)) are two of the four K-bundles over S I.

 - 125 -

With the n o t a t i o n above we have

{0;(o2,1)] = Mid = K x S 1 and [1;(o2,1)] = M~1¢2

by f ( a 1) = x I , f ( b 1) = x~lx , f ( h ) = XlX 2 .

(vi) nl, g = I , r ~ 1 give the possible S I actions on

p2xS I and N and both fiber over S1 .

(vii) n2, g = 1 , r ~ 1 are the prism manifolds with finite

fundamental groups and [0;(n2,1)} =~p3 #iRp3 so they do not

1
fiber over S .

(viii) M = [b;(nl,2)} are the same two K-bundles over S1

as under (v),

I
[0;(nl,2)] = Mid = K x S and [1;(nl,2)] = M~1@2

The first is obvious. The second is given by f(vl) = xl, f(v2)=x ,

f(h) = x~2x 2 .

(ix) M = [b;(n2,2)] are T-bundles over SI Specifically,

Wl(M) 2 2.-b , VlhV ~ Ih, v2hv ~ lh)

= (Vl,V2,hlVlV2n and the map f(v I ) = x ,
f(v2)=Xl x-1 9 f(h) = x 2 defines an isomorphism with M for

(~2)b E A+(T) whose m a t r i x is -I . For b / 0 the

order of ~ is infinite and Wl(~) is centerless. For b = 0

the manifold [0;(n2,2)] is homeomorphic to [-2;(oi,0);(2,1),

(2,1),(2,1),(2,1)] as noted in (5.4). Thus the latter is also

a T-bundle over SI with characteristic map of order 2 and

matrix (-: _~) .

(x) M = {b;(n3,2)} are the other two K-bundles over S1 ,

{0;(n3,2)] = M$2 and [1;(n3,2)] = M:I

The first is given by f(v I) = Xl x-1 , f(v2) = x 9 f(h) = x~lx~ I 9

the second by f(vl) = x , f(v2) = x -I x I , f(h) = x2x 1 •

 - 126 -

7.3. Non-uniqueness of the Fiber

The choice of the map g: Y - T k in the proof of (7.1.1)

determines the fiber. The n o n - u n i q u e n e s s is c l e a r l y seen by the

following example of T o l l e f s o n [I].

Let T(m) denote a closed orientable 2-manifold of genus

m = k(g-1) + I where g > I and arrange T(m) in ]R 3 with k

arms each of genus (g-l) about one hole at the origin, see pic-

ture b e l o w for k = 3 , g = 3 .

/ ', / ',, / j

Let ~: T(m) ~ T(m) generate a free Ek action by r o t a t i n g

through the angle 2wi/k and consider the 3-manifold M that

is a T(m)-bundle over SI with characteristic map ~ . It ad-

mits an obvious free S1-action as follows: If Ix,t] ET(m) xI~x,O)

= (~(x),1) is the e q u i v a l e n c e class of a point and s E S I =~/~

then define

[s](rx,t]) = [x,t+~s] .

The a c t i o n is e q u i v a r i a n t with respect to the ~k action and its

orbit space is T(g) . Thus M = [b;(ol,g)] and since it fibers

 - 127 -

over S I, it follows from (7.2.4) that b = 0 , hence M = T ( g ) × S 1.

Thus for m = k(g-S) + I we can embed T(m) in T(g) x S I as

a non-separating surface with complement T(m) x I so that the

projection map p: T(g) x S 1 ~ T(g) restricted to T(m) is a

covering. A m u c h stronger statement about incompressible surfaces

in St-bundles due to W a l d h a u s e n [1S may be found in (8.1.3).

 - 1 2 8 -

8. Further Topics

The important results of Waldhausen [1,2,3] occupy a central

position in the theory of 3-manifolds in general and Seifert mani-

folds in particular. It would carry us too far afield to give a

detailed account of his work so we have to restrict ourselves in

section I to a description of the most relevant results. In his

book Wolf [I] determines all closed 3-dimensional flat riemarmian

manifolds. There are six orientable and four non-orientable such

manifolds and in section 2 we identify them as Seifert manifolds.

Section 3 lists Seifert manifolds with solvable fundamental groups

as determined by L. Moser [I]. We consider finite groups acting

on Seifert manifolds in section 4. Some remarks on foliations in

section 5 and on flows in section 6 conclude the notes.

8.1. Waldhausen's Results

Waldhausen [1,2,3] works in the pie cewise linear category

so manifolds have combinatorial triangulations, submanifolds are

subcomplexes and maps are piecewise linear. Manifolds are always

orientable compact 3-manifolds and may have boundaries. Regular

neighborhoods of submanifolds are also compact and chosen suffici-

ently small with respect to the already given submanifolds of the

manifold in question. In general the embedding of a surface F

in a manifold M is proper, F 0 ~M = ~F and F is orientable,

hence 2-sided. A s2stem of surfaces has a finite number of dis-

joint components. Homeomorphisms are assumed to be surjective.

An isotopy of X is a level preserving map h: X × I - X×I so

 - 129 -

that at each l e v e l h l x × t = ht: X ~ X is a h o m e o m o r p h i s m . We

shall assume that h = id and call an isotopic deformation sim-

o
ply a deformation. Two s u b s p a c e s of X , YI and Y2 are isoto~ic

if there is an a m b i e n t isotopy of X so that hi(Y1) = Y2 ° Two

surfaces ~ and G in M or bM with ~NG = bF = bG are

called parallel if there is a surface H and e m b e d d i n g f: H × I

~ so that f(H×O) = F and f(H×] U bH×I) = G . A surface

F in M is called B-parallel (boundary-parallel) if there is

a surface F in 8M parallel to F . For curves in sufaces we

define parallel and b-parallel similarly.

The f o l l o w i n g construction is often repeated. Given a system

of surfaces F in ~ a new (not n e c e s s a r i l y connected) manifold

is o b t a i n e d by cuttin~ up M along F , i.e. let U(F) be a

regular neighborhood of F in M and let ~ = ~ - U(F-~ . We can

thus v i e w M as a s u b m a n i f o l d of M . Note that the c o n s t r u c t i o n

is w e l l defined up to an i s o t o p y of F . Given another system of

surfaces G in M in g e n e r a l position w.r.t. F , the n e w s y s t e m

= G D~ , however, depends on p r i o r deformations of F .

A system of surfaces F in M or bM is c o m p r e s s i b l e if

one of the f o l l o w i n g holds:

(i) there is a simple closed curve k in F that does n o t

bound a 2-cell in F and an e m b e d d i n g of a 2-cell D in M

so that D c M and D0F = k ;

(ii) there is an e m b e d d i n g of a 3-cell E in ~ so that

E D F = bE .

The n e g a t i o n of c o m p r e s s i b l e is d e n o t e d incompressible. Thus

M is i r r e d u c i b l e if it contains no i m c o m p r e s s i b l e 2-sphere.

Here are some of the m a i n r e s u l t s of W a l d h a u s e n [1]:

 - 130 -

Theorem I. Let be an i n c o m p r e s s i b l e s y s t e m of surfaces

in ~ .~,d ~ = M-U(F) . is i r r e d u c i b l e if and only if

is irreducible.

Let B be a compact, not n e c e s s a r i l y orientable 2-manifold

and p: M - B an St-bundle over B w i t h orientable total space.

Thus if M is closed it is a Seifert m a n i f o l d of class o1 or

n2 . A subspace X c M is v e r t i c a l if X = p-1(p(X)) and h o r i -

sontal if pIX is an embedding.

Lemma 2. Let p: M - B be an St-bundle. If B is not S2

or p2 then M is irreducible.

Note that the St-bundles over S2 are lens spaces and k n o v ~

to be irreducible or S2 × S I while the sl-bundles over p2 are

prism manifolds and irreducible or [0;(n2,1)] =~p3 ~Rp3 . If a

m a n i f o l d has irreducible orientable double cover, then it is itself

irreducible so the above lemma proves the irreducibility of all S l-

b u n d l e s with the noted exceptions~ p2 × S I and N .

Theorem 3. Let p: M - B be ~ S1-bundle where B is not

S2 o_~r p2 . Let G be a system . of. i.n c.o m p.r e.s s i b l e surfaces in N

so that no b o u n d e d component of G i_~s ~-parallel. Then there

is an ambient isotopy s O that the r e s u l t is either that

(i) G is v e r t i c a l so each component of G is an annulus

or a torus; o r

(ii) p!G is a covering map.

The basic r e s u l t on the h o m e o m o r p h i s m s of St-bundles is the

following:

T h e o r e m 4. Let p~ M - B and p': M ~ - B' b_~e S1-bundles.

 - 131 -

Assume that n e i t h e r B nor B' is S 2, p2, D 2 or SI × I and

if B or B' is the torus or K l e i n bottle then the bundle has

no c r o s s - s e c t i o n . Let ~: N ~ M' be a h o m e o m o r p h i s m . There

exists a h o m e o m o r p h i s m ~: M ~ M' so that

(i) $ is isotopic to ~ ,

(ii) there is a map p(~): B ~ B' making (@,p(@)

bundle isomorphism.

Given a manifold M , a system of tori T = TIU...UT n , n ~ 0 in

the interior of M with regular neighborhood U(T) is called a

g r a p h structure ("Graphenstruktur") on M if M - int U(T) is an

St-bundle. M is then called a g r a p h m a n i f o l d ("Graphenmannig-

faltigkeit"). In order to define a simple graph structure let T I

be a component of T and U(TI) its r e g u l a r n e i g h b o r h o o d homeo-

morphic to torus x interval w i t h b o u n d a r y components T' and

T" . Let MI be the component of M - i n t U(T) meeting T' and

M2 meeting T" . The n a t u r a l isomorphisms

HI(T' ) <--> HI(U(TI)) <--> HI(T" )

a l l o w us to talk about i n t e r s e c t i o n s of h o m o l o g y classes of curves

on T' and T" A g r a p h structure is simple (and the g r a p h mani-

fold is simple) if it is not one of the following:

(i) MI is not identical to ~'~2 and MI is the bundle

over the annulus,

(ii) the f i b e r of NI is h o m o l o g o u s to the fiber of M2 !

(iii) MI is a solid torus and a m e r i d i a n curve has inter-

section n u m b e r I w i t h a f i b e r of ~2 '

(iv) M1 is a solid torus and a m e r i d i a n curve is homologous

to a fiber of M2 ,

(v) MI is the $1-bundle over the ~ o e b i u s band and we

 - 132 -

think of it e m b e d d e d as a c r o s s - s e c t i o n in NI so that its boun-

dary is h o m o l o g o u s to the fiber in M2 ,

(vi) both MI and M2 are St-bundles over the M o e b i u s

band with embedded cross-sections whose boundaries are h o m o l o g o u s ,

(vii) M-intU(T1) has two components, one called Q is

obtained by sewing two orbits of type (2,1) into D 2× S I and the

other is n o t a solid torus,

(viii) MI and M2 are i d e n t i c a l and i s o m o r p h i c to torus x

interval and the c o m p o s i t i o n of n a t u r a l isomorphisms

H I ( T ' ) - H I ( U ( T 1 ) ) - H I ( T " ) - HI(M1) - HI(T' )

maps an e l e m e n t onto itself or its inverse,

(ix) MI and M2 are solid tori,

(x) T = ~ and M is a bundle over S2 or p2 .

Waldhausen Eli gives a complete classification of g r a p h m a n i f o l d s

up to h o m e o m o r p h i s m and shows that Seifert manifolds are special

cases of g r a p h manifolds. Here are the m a i n results.

Theorem 5. A simple graph manifol~ is irreducible.

Theorem 6. Let M and N be simple graph manifolds with

graph structures T = T I ,i. ..UT m and T' = T~ U. . .UT'n ' Assume

that the p a i r (M,N) is not one of the e x c e p t i o n s below. Then

5ivcn a homeomorphism ~: M ~ N there exists an isotopic homeo-

morphism @: M - N so that $(T) = T' .

Exceptions:

(i) M = M - i n t U(T) is a bundle over the m-holed 2-sphere

and m solid tori w i t h m < 3 ; or M is a b u n d l e over the m-

holed projective plane and m solid tori w i t h m ~ I . The same

for N = N-intU(T') .
 - 1 3 3 -

(ii) = M - i n t U(T) is torus x interval and N = N -intU(T')

is a bundle over the n-holed 2-sphere and n solid tori w i t h

n < 3
m
- or v i c a versa.

(iii) M is the m a n i f o l d Q above and N is the St-bundle

over the M o e b i u s band - or v i e a versa.

(iv) M = {-2;(o1,0);(2,1),(2,1),(2,1),(2,1)] , N = {0;(n2,2)}

- or v i c a versa.

We shall call an orientable Seifert m a n i f o l d sufficiently large

if it is not on the list below.

(i) Ol, g = O, r ~ 2
(ii) O1, g = O, r = 3

(iii) n2, g = 1, r ~ I
(iv) S1 × s l x S1

(v) {0;(n2,2)]

(vi) [-2;(o1,0);(2,1),(2,1),(2,1),(2,1)]
(vii) [-1;(n2,1);(2,I),(2,1)]

A corollary of theorem 6 is the f o l l o w i n g result.

T h e o r e m 7. Let M and N be s u f f i c i e n t l y large orientable

Seifert manifolds. Given a homeomoryhism ~: M - N there exists

an isotopic h o m e o m o r p h i s m 4: M ~ N so that ¢ induces a Seifert

bundle isomorphism.

The proof consists of showing that if we take a simple closed

curve about each component of E* in M* (and N*) and c o n s i d e r

their inverse images, then this c o l l e c t i o n of tori gives rise to

a simplegra~hstructure on M (and N). In p a r t i c u l a r this proves

the i r r e d u c i b i l i t y of these m a n i f o l d s up to a few exceptions as

claimed in (7.2.1).

This is c o n s i d e r a b l y stronger than (5.3.6) where we showed

only the existence of some Seifert bundle isomorphism. M u c h more
 - 134 -

is true, however. According to Waldhausen [2] two irreducible,

sufficiently large closed orientable 3-manifolds are homeomorphic

if their fundamental groups are isomorphic. The notion of "suffi-

cently large" means that M is not a ball and contains an incom-

pressible surface. Equivalently, an irreducible closed manifold

N is sufficiently large if and only if HI(M) is infinite or

~I(M) is a non-trivial free product with amalgamation. For ori-

entable Seifert manifolds the notion coincides with the definition

above. As a corollary to this result of Waldhausen [2] we may

state:

Theorem 8. Let ~ be a sufficiently large orientable

Seifert manifold and N an irreducible ' closed~ orientable 3-

manifold. If there exists an isomor2hism ~: ~i M - ~IN then

there exists a homeomorphism ~: M ~ N inducing ~ .

Waldhausen [2] also makes some comments about the homeotopy

group A(N) of M . The following Nielsen-type theorem holds for

sufficiently large manifolds but will be stated here only for

Seifert manifolds.

Theorem 9. Let M be a sufficiently large Seifert man~fold.

Then there is a natural is qmorphism

A(M) ~ Aut(wiM)/In(~1~ ) .

Letting F(M) denote the group of fiber preserving homeo-

morphisms of M modulo those that are isotopic to the identity

by fiber preserving isotopies, Waldhausen [2] shows that the natu-

ral map

F(M) - A(M)

is an isomorphism for sufficiently large Seifert manifolds.

 - 135 -

Surjectivity follows from theorem 7 and injectivity from the

methods developed in Waldhausen [2]. It requires deforming an

isotopy into a fiber preserving isotopy. Not much is known about

the structure of £(M) .

Recall that if the orientable Seifert manifold M admits an

S1-action, then h is in the center of Wl(M) . The following

remarkable conversion of this fact is obtained in Waldhausen [3].

Theorem 10. Let M be .an irreducible I closed I orientable,

sufficiently lar6e 3-manifold. If NI(N) has a non-trivial

center then M is homeomorphig to a Seifert manifold of class o I

and therefore admits an sl-action.

Several of these results may be extended to non-orientable

Seifert manifolds by lifting to the orientable double cover. Let

N = [b;(c,g);(~l,~1),...,(~r,Pr )] be a non-orientable Seifert

manifold. According to Seifert [I~ its orientable double cover is

= [-r;($,&);(a1,81),...,(~r,Gr),(a1,a1-P1),...,(ar,ar-~r))

~ere

c °2 I nl n3 n¢

01 1 01 n2 n2
2gli g-1 2g-2 2g-2

8.2. Flat Riemannian M~uifolds

In this section we shall identity as Seifert manifolds the

closed flat riemannian 3-manifolds found by Wolf [13. Let E(n)

denote the group of rigid motions of Rn ° Every rigid motion

consists of a translation, ta by a vector a followed by a ro-
 - 136 -

tation A . Write the m o t i o n (A,t a) . Clearly A is an element

of 0(n) and a is an a r b i t r a r y v e c t o r in Rn . Thus the eucli-

dean group E(n) is the s e m i - d i r e c t product of O(n) and Rn

satisfying the f o l l o w i n g product rule:

(A, ta)(B,tb) = (AB,tAb+a) •

~e write E(n) = 0(n).R n . Obviously E(n) is a Lie group acting

on Rn and R n = E(n)/0(n) as coset space.

A flat compact, connected r i e m a n n i a n m a n i f o l d Mn is the

orbit space of Rn by the free p r o p e r l y d i s c o n t i n u o u s action of

a discrete subgroup r c E(n) , M n = Rn/F . It admits a c o v e r i n g

by the torus Tn . The group ~ has an a b e l i a n n o r m a l subgroup

F* of rank n and finite index. As a group F* = F 0 R n . It

follows also that r has no n o n - t r i v i a l element of finite order.

The g r o u p of deck t r a n s f o r m a t i o n s Y in the c o v e r i n g T n ~ M n is

called the h o l o n o m y group of M n , ~ = r/r* .

The f o l l o w i n g result is from Wolf [I,p.117].

Theorem I. There are just 6 affine d i f f e o m o r p h i s m classes

of compact connected orientable flat 3-dimensional riemannian

manifolds. They are r e p r e s e n t e d by the m a n i f o l d s R3/F where r

is one of the six groups ~i g i v e n below. Here A is the trans-

lation lattice, [al,a2,a3] are its g e n e r a t o r s , t i = tai , and

= D/r* is the holonomy.

~ I' ~ = [I~ and F is g e n e r a t e d by the trans!ations

[tl,t2,t 3) with [a i] linearly independent.

~ 2" ~ = Z2 and F is g e n e r a t e d by [a,tl,t2,t 3] where

a 2 = t I , at2 a-1 = t~ I and at3~-1 = t~1;~lis orthogonal to a2

and a3 while a = (A,tal/2) with A(al) = a I , A(a 2) = -a 2 ,

A(a3) = -a 3 .
 - 137 -

a3
~3 " ~ = ~3
= tl , at2 a-1 =
and
t3 and
r is generated
at3 a-1 = t~It~ I ; a I
by [a,t1,t2,t31 where
is orthogonal

to a2 an__~d a 3 , !!a21! = !!a31! and [a2,a 3] is a hexagonal plane

lattice, and a = (A,tal/3) with A(al) = al, A(a2) = a3, A(a3) =

= -a2-a 3 .

~4 " ~ = ~4
~4 = t I , at2 ~-I = t 3
and F
and
is generated
at3a-1
by
= t~ I ; [ai)
[a,tl,t2,t 3]
are mutually
where

orthogonal with ia2!l = !la3', while a = (A,tal/4) with A(al)=al,

A(a 2) = a 3 , A(a 3) = -a 2

~5" Y = ~6 and r is generated by [a,tl,t2,t 3] where

6 = t I , at2~ -I = t 3 , ~t3~ -I = t~It3 ; aI .is . ortho~onal
. . . to a 2

and a3, iIa2!! = l!a311 and [a2,a3] is a hexagonal plane lattice,

and a = (A,tal/6) with A(al) = a I , A(a2) = a 3 , A(a3) = a3-a 2.

where
~6 "

Y~a = tlt 3
= ~'2 x Z 2
and
and r is ~enerated by [a,0,Y;tl,t2,t3]

2
a = t I , at2 m-1 = t21 , at3 ~-I = t31

@tl ~-I = tT I , B2 = t 2 , Bt3~-1 = t31

-I
YtIY = t71 , Yt2Y -I = t 21 , y 2 = t3

The [a i] are mutually orthogonal and

a = (A,tal/2) with A(al) = a I , A(a2) = -a 2 , A(a3) = -a3;

= (B,t(a2+a3)/2) with B(al)=-al, B(a2) = a 2 , B(a3) = -a 3 ;

y = (C,t(a1+a2+a3)/2) W ith C(al)=-a I, C(a2)=-a 2, C(a 3) = a 3 •

Theorem 2. The six compact, connected orientable flat

riemannian 3-manifolds of theorem I are the Seifert manifolds:
 - 138 -

M1 {0;(o I 1)} S 1 >" 81 x 81

M 2 = [-2;(oi,O);(2,1),(2,1),(2,1),(2,1 )} i_~s the. T 2 bundle
over SI with matrix of the characteristic map (-~ _~I of order 2;

M 3 = [-I;(oi,0);(3,1),(3,1),(3,1)} is the T 2 buntle over S 1

with matrix of the characteristic m a p ( ~_ -11) of order 3 ;

M 4 = {-I;(oi,0);(2,1),(4,1),(4,1)} is the T 2 bundle over S I

with matrix of the characteristic map (~_ O1) of order 4 ;

M 5 = [-I;(oi,0);(2,1),(3,1),(6,1)] is the T 2 bundle over S I

with matrix of the characteristic ma~ (_~ iJ of order 6 ;

M 6 = [-I;(n2,1);(2,1),(2,1)} is the manifold obtained from

takin~ the two Seifert fibrations of Q , one as a solid torus
with two orbits of type (2,1) and the other as the circle bundle
over the Moebius band with orientable total space, and sewin~
them to~ether by a fiber preservin6 homeomprphism. It is also
the orbit space of the orientation preservin~ free involution on
the Seifert bundle over S2 with total space M2 which identi-
fies fibers over antipodal ~oints of the base.space by an orien-
tation reversin~ homeomorphism.

Proof. Let G i = w1(Mi) . It suffices to show that

i ~ Gi for i = I,...,6 . It will be clear from the isomor-
phisms in the first five cases that there is an S I action on
$I~$I× S I making the action of the holonomy group equivariant
and the fibration over SI will also be equivariant. M6 admits
no St-action.
 - 139 -

2 _~ G 2 by ~(~) = ql ' T(t2) = q~ l q l ' ~(t3) = q2q31

~ 3 ~ G3 by T(a) = q ~ l , ~ ( t 2 ) = q~lq2

~ 4 ~ G4 by T(¢) = q2 ' m(t2) = qlq~ 1

~ 5 ~ G5 by T(a) = ql ' m(t2) = q~2q2

-1
~ 6 ~ G6 by ~(a) = ql ' ~(¥) = Vl

For these isomorphisms the groups are reduced by Tietze transfor-

mations to have only the given generators. The isomorphism for

G5 was found by A. Strsm. It is interesting to note that the G i

are all solvable groups, see (8.3).
The next result is again due to Wolf [I,p.120].

Theorem 3. There are just 4 affine diffeomorphism classes

of compact connected non-orientable flat 3-dimensional riemannian
manifolds. They are represente d by the manifolds R3/r where F
is one of the 4 groups ~i ~iven below. Her___~e A is the trans-

lation lattice, [al,a2,a 3} ~re its ~enerators, ti = ta , Y =r/r*

i
is the holonomy, and F o = r o SO(3).R 3 so that R3/r ° - R3/F i_~s

the 2-sheeted orientable riemannian covering.

~I' ~ = ~2 and F is generated by [¢,tl,t2,t3] where

¢ 2 = tl ' et2c - I = t2 ' ¢t3c -1 = t ~ l ; a I and a2 are ortho~onal

t_~o a 3 while c = (E,tal/2) with E(al) = al, E(a2) = a 2 and

E(a 3) = -a 3 r° is ~enerated by [t1~t2,t 3] .

~2" Y = ~2 and F is ~enerated by [¢,tl,t2,t3] where

2 -1 -1 1
c = tl, ~tlc = t2,ct3¢ = tlt2t ~ ; the orthogonal projection
of a3 on the (al,a2)-plane is (a1+a2)/2 ; ~ = (E,tal/2) with
E(al) = al, E(a2) = a2, E(a3) = a1+a2-a 3 . r° is generated by

[tl,t2,t 3 ]
 - 140 -

~3" Y = ~2 x ~ 2 and £ is generated by [¢,a,tl,t2,t3]

where a 2 . tl ' .c2 t2
. ' ea¢-1
. t2a, at2a-1 t~1, at3a-1 = t3-I,
ctl e-1 = t I and ¢t3¢-I = t; I ; th__~e a i are mutually orthogonal
and

= (A, t a l / 2 ) with A(al) = al, A(a2) = -a2, A(a3) = -a3,

e = (E,ta2/2) with E(al) : al, E(a2) = a 2 , E(a3) = -a3.

Fo is generated by [a,tl,t2,t 3] •

N4"2 = E2 x Z 2 and F is generated by [¢,a,tl,t2,t 3]

where = tl ' 2 = t2 ' ¢~c-I = t2t3a, at2a-1 = t~ 1, ~t3~-I= t~ I
-I
ctle = tl, ct3~-I = t~ I ; the a i are m u t u a l l ~ o r t h o g o n a l and

a = (A, tal/2 ) with A(al) = al, A(a2) = -a2, A(a3) = -a3,

c = (E,t(a2+a3)/2) with E(al) = al, E(a2) = a2, E(a3) =-a3.

Fo is generated by [a,tl,t2,t3} .

Theorem 4. The four compact conn.ected, non-orientable flat

3-dimensional riemannian manifolds are the four Klein-bottle
bundles over S1 . Let 2 2 . Then
w1(K) = (Xl,X2[XlX2)
N 1 = [0;(nl,2)] = K×S 1 ,

N2 = [1;(nl,2)} is the K-bundle over S1 with character-

istic ma,p ,~(Xl) : x21, ,~(x2) = X l 1,
N 3 = {0;(n3,2)] is the K-bundle over S1 with characteris-
tic map ~(Xl) = Xl I 9 ~(x2) = x21
1
N 4 = [I;(n3,2)] is the K-bundle over S with characteris-
tic map ~!i(xi) = x2, ¢(x2) = x I

Proof. Again we let B i = ~1(Ni) and show that ~i ~ Bi'

Note that N1 and N2 admit S1-actions while N3 and N4 do
not.
 - 141 -

~ B1 by • (¢) : Vl, T(t2) : h, ~(t3) : VlV 2 ;

~ B2 by ~(¢) = v 1, T(t 3) : VlV 2 ;

~83 ~ ~3 by • (c) VlV 2 T(a) v~ 1 T ( t 3) h -1

~34 ~ B 4 by • (e) = VlV2, ~(~) = V l V 2 V I .

The groups are again reduced by Tietze transformations to have

only the g i v e n generators. The isomorphisms for B3 and B4

were found by A. Strum. The orientable double cover is M1 for

N1 and N2 and M2 for N3 and N4 . Clearly the Bi are

also solvable groups, (8.3).

8.3. Solvable Fundamental Groups

Let G be a group and G (I) = [G,G] be its c o m m u t a t o r sub-

group. Define inductively G (m) = [G(m-I),G (m-l)] and call G

solvable if the series terminates, i.e.

G D O (I) ~ . . . D G (m) = 1

for some m . Typical example is an abelian group. A well-known

example of a n o n - s o l v a b l e group is the b i n a r y i c o s a h e d r a l group I*,

since [I*,I*] = I* . The subgroups and factor groups of solvable

groups are sovable and the e x t e n s i o n of a solvable group by a solv-

able group is solvable. An e q u i v a l e n t definition is that G has

a finite series of normal subgroups

G D G 1 •... oG n : 1

each Gi normal in Gi_ I so that Gi_I/G i is abelian for all i.

If Gi_I/G i is in the center of G/G i for all i , then G is

called nilpotent.

If G is the f u n d a m e n t a l group of a Seifert manifold, then

 - 142 -

G is solvable if and only if the planar discontinuous group

G/(h) is solvable. These considerations give the following re-

sult essentially due to Moser [I].

Theorem 1. The Seifert manifolds with solvable fundamental

groups are:

(i) M = [b;(oi,1)] , T2-bundles over SI ; G is a n i l p o t e n t

extension of ~x~ by E ;

(ii) M = {b;(o1,0);(2,1),(2,1),(2,1),(2,1)} , for b : -2

M is a T 2 bundle over S I , otherwise ~ is the orbit space of

a free ~2-action on one of the manifolds of (i), G is an exten-

sion of a n i l p o t e n t group by ~2 ;

I I I
(iii) oi, g = O, r = 3, ~11 + a-~ + aU ~ I except for

(al,a2,a 3) = (2,3,5) where I* is a direct summand of G ; for

(3,3,3), (2,4,4) and (2,3,6) M either fibers over S 1 , see

(8.2.2) or it is the orbit space of one of the finite groups E3'

~4 o__rr ~ 6 acting freely on one of the manifolds of (i) s_£o G i_~s

a single or double c~clic extension of a nilpotent group; for

(2,2,n), (2,3,3) and (2,3,¢) G is finite, see (6.2.2);

(iv) oi, g = O, r ~ 2 are lens spaces or S2×S I s__£o G i_.£s

finite or infinite cyclic;

(v) M : [b;(n2,2)] are T-bundles over SI so G is an

extension of ~×~ by ~ ;

(vi) n2, g = I, r ~ I , here [0;(n2,1)] =~3 ~p3 with

G = ~2 * ~2 which is an extension of E b_~v E2 while the other

manifolds have finite fundamental groups, see (6.2.2);

(vii) M = [b;(n2,1);(2,1),(2,1)] are orbit spaces of the

free orientation preservin~ E2 actions on manifolds of (ii) that
 - 143 -

induce the antipodal map in the orbit space of the S1-action; G

is the double extension of a n i l p o t e n t group by cyclic groups;

(viii) M = {b;(02,1 ) } are K-bundles over S I , s_qo G i_£s

an extension of a solvable 6TOUp by E2 ;

(ix) M = [b;(nl,2)} same as (viii);

(x) M = {b;(n3,2)] are the other two K-bundles over $1;

(xi) nl, g = I, r ~ I are the manifolds p2 x S I and N

s_oo G is ~ ×~2 o_~r ~ ;

(xii) M = [b;(n1,1);(2,1),(2,1)] are orbit spaces of the

free orientation reversing E2 actions on manifolds of (ii) that

induce the antipodal map in the orbit space of the sl-action; G

is the double extension of a n i l ~ o t e n t group b2 cyclic groups.

8.4. Finite Grou~ Actions

If M = {b;(¢,g);(al,G1),...,(ar,Pr)] admits an S1-action,

so c = oI or n I , then every finite s u b g r o u p Z kcS I acts on

I~ with orbit space a Seifert manifold M' whose invariants were

computed by Seifert E1,p.218]:

M, = { b ' ; ( c , g ) ; ( ~ l , ~ , 1)
, .... ,(~,~'it)}
where
b' = kb , aj' = aj/(aj,k) , 8[
3 = kSj/(aj,k) •

These Seifert invariants may need normalization. The action of

~k is free on M if and only if (aj,k) = I for j =1,...,r .

Note that the h o m e o m o r p h i s m s of the action are isotopic to the iden-

tity.

The example of M6 in (8.2.2) shows that not every finite

 - 144 -

group acts as a subgroup of the circle. Tollefson ~2~ investi-

gates when a free Ek action on a 3-manifold M embeds in an

S1-action. It is c l e a r l y necessary that a homeomorphism genera-

ting the a c t i o n be h o m o t o p i c to the identity. Such an a c t i o n is

called proper. Let M' be the orbit space and ~: M ~ M' the

orbit map. The action is called E-classified if there is a com-

mutative diagram

M > S1

~ v
M' > S1

where p: S 1 - S I is the u s u a l k-sheeted covering of the circle.

In p a r t i c u l a r such maps exist if HI(~';~) has no k-torsion.

Two Zk-actions U,v: ~ k ×14 ~ M are called weakl Z e q u i v a l e n t if

there is a g r o u p automorphism A: ~ k ~ Ek and a h o m e o m o r p h i s m

H: M ~ M so that u(g) = H-Iv(A(g)) H for all g E Zk . The

m a i n result of T o l l e f s o n [2] is:

Theorem I. Let M be a closed~ orientable~ irreducible

3-manifold. A
-- Z-classified free E p -action on M (p --
> 2 prime)

is proper if and only if it is w e a k l y equivalent to some ~p-actio_.____&n

embedded in an effective St-action on M .

In the course of the proof it is shov~ that M fibers over

I
S and the E
- a c t i o n is e q u i v a r i a n t w i t h r e s p e c t to the fibration.
P
Notice that in some cases a S e i f e r t - m a n i f o l d may cover itself,

e.g. it follows from the o p e n i n g remarks of this section that

M = [-1;(ol,g);(a,1),(~,a-i))

is a p r o p e r k-sheeted covering of itself for every k ~ I mod a .

For g = 0 M = S2× S I but for g > 0 M is i r r e d u c i b l e and a

non-trivial 2-manifold bundle over SI . Tollefson ~3] proves

 - 145 -

that if M is a closed, connected 3-manifold that is a n o n - t r i -

vial connected sum and covers itself, then ~ =~p3 #IRP 3 . It is

the k-fold cover of i t s e l f for e v e r y k but none of these free

q-actions are p r o p e r in the above sense. If the c o v e r i n g action

is proper, then T o l l e f s o n [3~ shows that the m a n i f o l d M is irre-

ducible and if HI(~;E) h a s no e l e m e n t of o r d e r k , then ~ fibers

over S1 .

8.5. Foliations

Let M be a s m o o t h m a n i f o l d with tangent bundle TM . A

k-plane field on M is a k-dimensional subbundle ~ of TM .

If L is an i n j e c t i v e l y immersed, smooth submanifold of M so

that TL x = o x C T M x for all x E L j then L is called an i n t e g r a l

submanifold of ~ . A k-plane field c is c a l l e d completely

integrable if the f o l l o w i n g three equivalent conditions are satis-

fied:

A. M is c o v e r e d by open sets U with local coordinates

Xl,...,x m so that the submanifolds defined by Xk+ I = constant,

...,x m = c o n s t a n t are i n t e g r a l submanifolds of o .

B. ~ is s m o o t h and t h r o u g h every point x E M there is an

integral submanifold L of a .

C. ~ is s m o o t h and if X and Y are v e c t o r fields on M

with X x , Y x E ~x for all x E M then the b r a c k e t [X,Y~ x E ~x "

An i n t e g r a b l e k-plane field is called a foliation and the

maximal connected integral submanifolds are c a l l e d leaves. The

leaves of a f o l i a t i o n partition the m a n i f o l d . The f o l l o w i n g re-

sult is due i n d e p e n d e n t l y to L i c k o r i s h , Novikov and Z i e s c h a n g .

 - 146 -

Theorem I. Every closed T orientable 3-manifold admits a

codimension one foliation.

The proof goes roughly as follows. The Reeb foliation on D 2 x S I

is obtained by considering a function with graph below

f
..............................4,............................................................... ~ X
\

and all its translates along the x-axis. Rotate to obtain a

foliation of D 2 xR and identify integral translates to obtain

the Reeb foliation on D2 x~ . It has one compact leaf, 8D 2 x S I

and all other leaves are homeomorphic to R2 . The union of two

Reeb foliations foliates S3 . Every orientable closed 3-mani-

fold is obtained from S3 by a finite number of (1,1)-surgeries

according to Wallace. Remove the necessary number of solid tori

from S3 and alter the foliation of S3 at the boundary tori by

the procedure of "dropping off leaves"

to foliate the resulting manifold. Now sew in the required copies

of D2 x SI with Reeb foliations to obtain the manifold in question.

 - 147 -

Wood [I] showed that n o n - o r i e n t a b l e closed 3-manifolds also

admit codimension one foliations. A celebrated theorem of N o v i k o v

proves that every codimension one f o l i a t i o n of S3 has a compact

leaf.

The r a n k of a d i f f e r e n t i a b l e manifold M is the m a x i m u m

number of l i n e a r l y independent C 2 vector fields on M which

commute pairwise. If M is a closed manifold, then the rank of

M is the l a r g e s t integer k so that there exists a non-singular

action of Rk on M with all orbits of d i m e n s i o n k . This

action defines a foliation of M . The f o l l o w i n g was proved by

Rosenberg-Roussaire-~eil [I].

Theorem 2. Closed orientable 3-manifolds have the follow-

ing rank:

(i) S I ×SIx SI has r a n k 3 ;

(ii) N h a s rank 2 if and onl 2 if it is a n o n - t r i v i a l

torus bundle over $I;

(iii) all others h a v e rank 1 •

The proof is o u t l i n e d in the p a p e r as follows. If % is a non-

singular action of R2 on the closed, orientable manifold V •

then the orbits are R 2 9 RxS 1 or T2 • It is k n o w n that if all

orbits are R2~ then V is T3 . If V has r a n k 2 ,then there

must be orbits homeomorphic to R × S I or T2 . If all orbits are

homeomorphie to R × SI ,

then ¢ is m o d i f i e d to a C°-close action 91 w h i c h has a com-

pact orbit. It is k n o w n that not every compact orbit of @ can

separate V into two c o n n e c t e d components• One can find k com-

pact orbits T1,...,T k which do n o t separate V but have the

 - 148 -

property that for every other compact orbit T the u n i o n

T U T 1U... UT k separates V . Let W be the m a n i f o l d obtained

by cutting V along the T i , i = 1,...,k • Then ~W consists

of 2k tori and every torus orbit in the i n t e r i o r separates W

into connected components. By a t r a n s f i n i t e argument it is ob-

tained that ~ has no compact orbits in the i n t e r i o r of W • The

crucial step is to show that W ~ T 2 x~0,1] so V is obtained as

a T 2 bundle over SI .

An e x p l i c i t action of R2 on a T 2 bundle over SI is de-

fined as follows: Let f: T 2 ~ T 2 be the orientation preserving

characteristic map of the bundle and V = T 2 x I/f . As n o t e d

earlier f is isotopic to a l i n e a r map F E A+(T 2) = GL+(2,~)

and V is d i f f e o m o r p h i c to T 2 x I/F . Since the group GL+(2~R)

is c o n n e c t e d there is an i s o t o p y F L with ~ = id F I = F -1
b -0 ~ •

Choose it so that Ft = F° for t < c and Ft = FI for I-¢<t~I

for some small ¢ > 0 . Any two constant vector fields on F2

which are l i n e a r l y independent define two l i n e a r l y independent

commuting vector fields on T2 . For t C [0,1] let X(t) =Ft(1,0)

~d Y(t) = Ft(0,1 ) . Then X(t) and Y(t) are two l i n e a r l y

independent vector fields on T 2 xt . Moreover, d~1(X(1)) = (0,1)

= X(0) and dFI(Y(1)) = (0,1) = Y(0) , hence X(t) and Y(t)

define two l i n e a r l y independent vector fields on V .

It is i n t e r e s t i n g to note that if V has no compact orbits,

then F = (~ t), so V is the Seifert manifold [-a;(oi,1)} o

8.6. Plows

A Cr flow on a Cr manifold M is a Cr action ~:MxR ~ M

of the additive reals on M . Such actions arise n a t u r a l l y from

the i n t e g r a t i o n of a Cr vector field on M . Conversely, differ-
 - 149 -

entiation of a C r+1 f l o w gives rise to a C r v e c t o r field on M.

The f o l l o w i n g is an example of a flo~ on S 3 = [(Zl,Z 2) E

C2 I z1~1+z2~ 2 = 1) . Let (p,q) be r e l a t i v e l y prime integers and

define

a(zl,zy,t) = (zle2~ipt,z2 ey~iqt) .

This is clearly the R action o b t a i n e d from l i f t i n g the correspon-

ding S 1 action to the u n i v e r s a l cever of SI For p = q = I

this is called the Hopf flow on S3 . These flows have only c l u e d

orbits. The f o l l o w i n g recent r e s u l t of E p s t e i n [I] proves that if

all orbits are closed on a 3-manifold,then this is the most gene-

ral situation.

Theorem I. Let a: M x R
•
~ ~ be a C r action (I ~ r Zoo) o f

the additive ~roup of real n u m b e r s on ZI , with every orbit a cir-

cle. Let M be a cgmpact 3 - m a n i f o l d ~ o s s i b l y w i t h boundary.

Then there is a C r action ~': M x S I ~ M w i t h the same orbits

as ~ .

If n o n - c o m p a c t orbits are p r e s e n t j t h e n the structure of flows

is still unknown. The f o l l o w i n g result is due to Seifert [2]. Let

C be the v e c t o r field of C l i f f o r d - p a r a l l e l v e c t o r s whose integral

curves, the C l i f f o r d circles, give the Hopf f l o w and let ~ be a

continuous v e c t o r field on S3 w h i c h differs sufficiently little

from C , that is, the angle b e t w e e n a vector of C and that of

is at every point of S3 smaller than a s u f f i c i e n t l y small a.

T h e o r e m 2. A continuous v e c t o r field on the 3-sphere which

differs sufficiently little from the field of C l i f f o r d - p a r a l l e l s

and w h i c h sends through every point exactly one integral curve

has at least one closed intesral curve.

 - 15o -

The q u e s t i o n posed by S e i f e r t [2] w h e t h e r this is true for

all flows on S3 is still open and is n o w r e f e r r e d to as the

Seifert Conjecture.

Added in proof: Paul Schweitzer has obtained a counterexample to

this conjecture.
 - 151 -

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