A Brief Introduction to Homotopy

Theory
Mohammad Hadi Hedayatzadeh
February 2, 2004

Ecole Polytechnique Federale de Lausanne

In this article, we study the elementary and basic notions of homotopy
theory such as cobrations, brations, weak equivalences etc. and some
of their properties. For technical reasons and to simplify the arguments,
we suppose that all spaces are compactly generated. The results may hold
in more general conditions, however we do not intend to introduce them
with minimal hypothesis. We would omit the proof of certain theorems
and proposition, either because they are straightforward or because they
demand more information and technics of algebraic topology which do not
lie within the scope of this article. We would also suppose some familiarity
of elementary topology. A more detailed and further considerations can be
found in the bibliography and the end of this article.
1
Contents
1 Cobrations 3
2 Fibrations 8
3 Homotopy Exact Sequences 14
4 Homotopy Groups 24
5 CW-Complexes 35
6 The Homotopy Excision And Suspension Theorems 47
References 53
2
1 Cobrations
In this and the following two chapters, we elaborate the fundamental tools
and denitions of our study of homotopy theory, cobrations, brations and
exact homotopy sequences.
Denition 1.1. A map i : A X is a cobration if it satises the
homotopy extension property (HEP), i.e. , given a map f : X Y and a
homotopy h : AI Y whose restriction to A0 is f i, there exists
an extension of h to X I.
This situation is expressed schematically as follows:
A
i
0

AI
iId

h
.x
x
x
x
x
x
x
x
x
Y
X
i
0

~
~
~
~
~
~
~
~
X I
H
F
F
F
F
F
where i
0
(u) = (u, 0)
We may write this property in an other equivalent way. i : A X is a
cobration, if there exists a lifting H in the following diagram
A
h

Y
I
p
0

X
f

|
|
|
|
Y
where p
0
() = (0).
Remark 1.2. We do not require H to be unique, and it is usually not the
case.
Remark 1.3. Some authors suppose that i : A X is an inclusion with
closed image. We will show that this can be derived from the denition,
however we need some denitions rst.
3
Denition 1.4. The mapping cylinder of f : X Y is dened to be the
pushout of the maps f : X Y and i
0
: X X I, and we note it M
f
,
so M
f
Y
f
(X I).
We often construct new spaces and new maps from the given spaces and
maps, and one way of such a construction is to take the pushout of two
maps. The following proposition states that the class of cobration is closed
under taking pushouts. Thus we may take pushout of two maps without
any restriction.
Proposition 1.5. Suppose that i : A X is a cobration and g : A B
is any map, then the induced map B B
g
X is a cobration.
There is another proposition that is useful to making new cobration from
ones given.
Proposition 1.6. If i : A X and j : B Y are cobrations then so
is i j : AB X Y .
In the denition of a cobration we asked too much, and in fact it suces
to have the HEP for the following diagram.
A
i
0

AI

i
.w
w
w
w
w
w
w
w
iId

M
i
X

i
0

}
}
}
}
}
}
}
}
i
0

X I
r
G
G
G
G
If there exists such a map, and if we are given a map f : X Y and a
homotopy h : A I Y , the universal property of pushout implies that
there is a map M
i
Y whose composition with r is a map that extends
h and its restriction to X is f. Thus we have the following proposition.
Proposition 1.7. A map i : A X is a cobration if and only if there
exists a ller in the last diagram.
4
Example 1.8. The inclusion i
s
: Y Y I given by i
s
(y) = (y, s) is a
cobration.
Now we are able to prove the following proposition.
Proposition 1.9. If i : A X is a cobration then it is an inclusion with
closed image.
Proof. Consider the previous diagram. Observe that

i is injective on
A(0, 1] since M
i
can be regarded as X0

AI/i(a) = (a, 0). Now

if i(a) = i(b) then r(i(a), 1) = r(i(b), 1), so

i(a, 1) =

i(b, 1), thus a = b. This

shows that i is injective. For the second part, we rst show that M
i
is a
closed subspace of X I. Dene : M
i
X I, to be identity on X
I and (a, t) = (i(a), t) for all a A, t I clearly it is a well dened map.
Now the composition r is a map that lls the pushout diagram
A
i

i
0

AI

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
X

i
0

i
0

U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U M
i
r

F
F
F
F
F
F
F
F
M
i
Uniqueness of such a map implies that it is the identity map, so is injective
and we can identify M
i
with its image. Since r is surjective, we have M
i
=
r(X I) = z X I [ r(z) = z =: , in order to show that is a
closed subspace of X I, we have to show that its intersection with each
compact subset of X I is closed. So suppose that C is compact in X I,
and y C C, so r(y) is dierent from y, C is Hausdor, there exist
open subsets V
1
and V
2
such that,
y C V
1
, r(y) C V
2
, C V
1
V
2
= , r(y) C V
2
implies that y r
1
(V
2
), now put U := V
1
r
1
(V
2
) C it is an open
subset of C that contains y and we have U = , for if z U we
have r(z) = z U so r(z) V
1
V
2
C = which is a contradiction. So
5
C C is open in C, and so C is a closed subset of C. Now consider
the inclusion X 1

M
i
,
1
(M
i
) is closed in X 1 and we have

1
(M
i
) = A1, thus A is closed in X.
An important property of cobrations is that every map is a cobration
up to homotopy equivalence, more precisely if f : X Y is a map we can
write it as the composite X
j
M
f
r
Y where j(x) = (x, 1) and r(y) = y
on Y and r(x, s) = f(x) on X I.
Proposition 1.10. In the above notation j is a cobration and r is a ho-
motopy equivalence.
There is a characterization theorem for cobrations, but we need the fol-
lowing denition before.
Denition 1.11. A pair (X, A) with A X is an NDR-pair (NDR for
Neighbourhood Deformation Retract ) if there is a map u : X I such that
u
1
(I) = A and a homotopy h : X I X such that h
0
= Id, h(a, t) = a
for all a A and t I and h
1
(x) A if u(x) < 1. (X, A) is a DR-pair if
u(x) < 1 for all x X in which case A is a deformatioin retract of X.
Proposition 1.12. If (X, A) and (Y, B) are NDR-pairs then (X
Y, X B A Y ) is an NDR-pair. If (X, A) or (Y, B) is a DR-pair then
so is (X Y, X B AY ).
Now we have the characterization theorem
Theorem 1.13. Let A be a closed subset of X. The the following statements
are equivalent:
(i) (X, A) is an NDR-pair
(ii) (X I, X 0 AI) is a DR-pair
(iii) X 0 AI is a retract of X I
(iv) The inclusion i : A X is a cobration
Sometimes it is important to work in the category of spaces under a given
space,so dene this category as follows.
6
Denition 1.14. A space under A is a map i : A X. A map of spaces
under A is a commutative diagram
A
i
.~
~
~
~
~
~
~
j

@
@
@
@
@
@
@
X
f

Y
a homotopy between maps f and f

under A is a homotopy h that at each

time t is map under A, i.e., h(i(a), t) = j(t) for all a A and t I. We
write h : f f

rel A. It gives rise to a notion of a homotopy equivalence

under A, called a cober homotopy equivalence.
Proposition 1.15. Assume given a commutative diagram
A
d

B
j

X
f

Y
in which i and j are cobrations and d and f are homotopy equivalences.
Then (f, d) : (X, A) (Y, B) is a homotopy equivalence of pairs, i.e.,
there are homotopy inverses e of d and g of f such that g j = i e together
with homotopies H : g f Id and K : f g Id, that extend homotopies
h : e d Id and k : d e Id. In particular if f is homotopy equivalence
between two spaces under A, then it is a cober homotopy equivalence.
7
2 Fibrations
In this chapter we dualize the notions and theory of the last chapter.
Denition 2.1. A surjective map p : E B is a bration if it satises the
covering homotopy property, i.e., given a map f : Y E and a homotopy
h : Y I B there exists a lifting of h to E,whose restriction to Y 0
is f.
Remark 2.2. As we have seen for cobration there is an equivalent deni-
tion of a bration in which we can better see the dualization.
E
p

E
I
p
0

p
I

Y
f
?
?
?
?
?
?
?
?
h

A
A
A
A
A
A
A
A
H

}
}
}
}
B
B
I
p
0

where p
0
() = (0)
Remark 2.3. Again we do not require the uniqueness of such a lifting.
The class of brations is closed under the base extensions, i.e.,
Proposition 2.4. Pullbacks of brations are brations.
Denition 2.5. Let p : E B be a map. Its mapping path space is a
pullback of p and p
0
: B
I
B and we note it N
p
E
p
B
I
= (e, ) [
(0) = p(e).
The mapping path space is the dual of mapping cylinder and we will see
that it plays the same role for the brations.
Denition 2.6. Let p : E B be a map and let N
p
be its mapping path
space. A map s : N
p
E
I
such that s(e, )(0) = e and p s(e, ) = is
called path lifting function.
We have seen that for a map i : A X to be a cobration it suces to
admit a homotopy extension for its mapping cylinder. It turns out that for
8
a map p : E B to be bration it suces to have a homotopy lifting for
its mapping path space, or equivalently we have
Proposition 2.7. A map p : E B is a bration if and only if it admits
a path lifting function.
Proof. Replace Y by N
p
in the test diagram of the equivalent denition
of brations; necessity is then clear. So suppose that we have a path lifting
function s : N
p
E
I
and maps f : Y E and h : Y B
I
. There is
an induced map g : Y N
p
, since N
p
is a pullback. The composite s g
gives the required homotopy lifting.
As an application of this proposition, we have the following example
Example 2.8. If p : E B is a covering, then p is a bration with a
unique path lifting function.
Example 2.9. The evaluation map p
s
: B
I
B given by p
s
() = (s) is
a bration.
The relation between brations and cobrations is stated in the following
proposition
Proposition 2.10. If i : A X is a cobration and B is a space then
the induced map p = B
i
: B
X
B
A
is a bration.
Proof. It is an easy task to show that we have the following homeomor-
phisms
B
M
i
= B
X{0}AI

= B
X

p
(B
A
)
I
= N
p
If r : X I M
i
is a retraction given by Theorem 1.13 then
B
r
: N
p

= B
M
i
B
XI

= (B
X
)
I
is a path lifting function, for if (e, ) N
p
the homeomorphism sends (e, )
to B
M
i
, the function that is e on X and

on A I, where

(a, t) =
(t)(a), so B
r
(e, )(x, 0) = (e, )(r(x, 0)) = (e, )(x) = e(x), so we have
B
r
(e, )(0) = e through the identication B
XI
= (B
X
)
I
. We have also
pB
r
(e, )(t)(a) = B
r
(e, )(i(a), t) = (e, )(r(i(a), t)) = (e, )(a, t) = (t)(a)
hence B
r
is a bration.
9
We have seen that every map can be factored as cobration followed by a
homotopy equivalence. We can dualize this property and get the following
proposition,which will be of great use later.
Proposition 2.11. Any map can be factored as a homotopy equivalence
followed by a bration.
Proof. Given a mapf : X Y , dene : X N
f
, by (x) =
(x, c
f(x)
) where c
f(x)
is the constant path at f(x) and dene, : N
f
Y ,
by (x, ) = (1). We have f = . Now let : N
f
X be the
projection, then =id and id via the homotopy h : N
f
I N
f
by setting
h(x, )(t) = (x,
t
), where
t
(s) = ((1 t)s)
Now suppose given a test diagram
A
i
0

N
f

AI

x
x
x
x
h

Y
Write g(a) = (g
1
(a), g
2
(a)) and set

h(a, t) = (g
1
(a), j(a, t)) where
j(a, t) =

g
2
(a)(s + st) if 0 < s < 1/(1 + t)
h(a, s + st 1) if 1/(1 + t) s 1
So satises the covering homotopy property,and thus it is a bration.
In Example 2.8 we have seen that a covering is a bration ,so we may
think of brations as a generalization of coverings. This idea leads to a local
criterion that allows us to recognize brations when we see them. But we
need a denition before stating it.
Denition 2.12. A numerable open cover of a space B is an open cover O
such that for each U O there are continuous maps
U
: B I such that

1
U
(0, 1] = U and that the cover is locally nite, i. e, each b B has an
open neighbourhood that intersects nitely many U O.
Theorem 2.13. Let p : E B be a map and let O be a numerable open
cover of B. Then p is a bration if and only if p : p
1
(U) U is a
bration for every U O.
10
Now we give the denition dual to that of cober homotopy equivalence.
Denition 2.14. A space over B is a map p : E B. A map of spaces
over B is a commutative diagram
D
q

@
@
@
@
@
@
@
f

E
p
.~
~
~
~
~
~
~
B
A homotopy between maps of spaces over B is a homotopy that at each time
t is a map over B. A homotopy equivalence over B is called ber homotopy
equivalence.
Proposition 2.15. Assume given a commutative diagram
D
q

E
p

A
d

B
in which p and q are brations and d and f are homotopy equivalences.
Then (f, d) : q p is a homotopy equivalence of brations. In particular
if f : D E is a homotopy equivalence and we have a commutative
diagram
D
q

@
@
@
@
@
@
@
f

E
p
.~
~
~
~
~
~
~
B
then f is a ber homotopy equivalence.
The statement means that there are homotopy inverses e of d and g of f
such that qg = ep together with homotopies H : gf id and K : f g
id that cover homotopies h : e d id and k : d e id.
Using the covering homotopy property of brations, p : E B induces a
translation of bers along path classes and we will see that this change of
ber is in fact a homotopy equivalence. Let F
b
be the ber over b B, i.e. ,
the set F
1
(b), let i
b
: F
b
E be the inclusion and let B
I
be a path
joining b to b

. There exists a lift

in the following diagram
11
F
b
i
0

i
b

E
p

F
b
I

y
y
y
y
y
y
y
y
y

B
since p is a bration. We have p

(e, t) = (t) so at time t we have a map,

t
: F
b
F
(t)
, and in particular we have

1
: F
b
F
b
, and we note
[] [

1
] : F
b
F
b

which is called the translation along the path class []. The following propo-
sition conrms that [] is in fact well dened.
Proposition 2.16. Using the above notations, if

then

and
in particular

1
. Therefore the homotopy class of

1
which is noted by
[] is independent of the choice of in its path class.
One can easily see that
[c
b
] = [id] and [ ] = [] [] with (1) = (0)
so [] has the inverse [
1
] and we have
Proposition 2.17. If B
I
is a path starting at b and ending at b

then
F
b
and F
b
are homotopic equivalent. Therefore if B is path connected any
two bers of B are homotopic equivalent.
These translations are functorial in the sense of the following proposition
Proposition 2.18. Let p and q be brations in the following commutative
diagram
D
q

E
p

A
f

B
and let : I A be a path from a to a

. Then the following diagram is

commutative up to homotopy equivalence
12
F
a
[]

F
f(a)
[f]

F
a

g
F
f(a

)
if further h : f f

and H : g g

in the commutative diagram

D I
qid

E
p

AI
h

B
then the following diagram commutes up to homotopy equivalence
F
a
g
z
z
z
z
z
z
z
z
g

E
E
E
E
E
E
E
E
F
f(a)
[h()]

F
f

(a)
where h(a)(t) = h(a, t).
13
3 Homotopy Exact Sequences
In this section we associate two exact sequences of spaces to a given map.
These sequences will play an important role in homotopy theory, and we
will see that they are very useful for calculating higher homotopy groups.
All spaces in this section are based spaces, and we shall write generically
for the basepoints.
Denition 3.1. For spaces X and Y , [X, Y ] denotes the set of based ho-
motopy classes of based maps X Y . This set has a natural basepoint,
namely the homotopy class of the constant map from X to the basepoint of
Y . F(X, Y ) denotes the subspace of Y
X
consisting of the based maps,
with the constant base map as basepoint.
Denition 3.2. The wedge product of X and Y is the pushout of two
inclusions X and Y and is noted by XY . Explicitly, XY =
(x, y) X Y [ x = or y = . The smash product of X and Y is
dened by
X Y = X Y/X Y
.
Proposition 3.3. Let X, Y, Z be three spaces, we have:
(i) We have a natural homeomorphism
F(X Y, Z)

= F(X, F(Y, Z)).
(ii) We can identify [X, Y ] with
0
(F(X, Y )),where
0
(X) denotes the set
of path components of X.
Denition 3.4. Let X be a space. The cone or reduced cone on X is
CX = X I, where the basepoint of I is 1. More explicitly we have
CX = X I/( I X 1).
The unreduced cone on X which is again noted by CX is dened by
CX = X I/X 1.
14
We can view S
1
as I/I and denote its basepoint by 1. We can now dene
some important objects of our study.
Denition 3.5. The suspension or reduced suspension of a space X is
X = X S
1
= X S
1
/( S
1
X 1) = X I/( I X I).
The unreduced suspension of X is dened to be the quotient of XI obtained
by identifying the set X 1 to a point and the set X 0 to another
point.
We have the dual constructions and denitions as follows
Denition 3.6. The path space of a space X is the based space PX =
F(X, I), where I is given the basepoint 0. Thus its elements are paths in X
starting from the basepoint.
Denition 3.7. The loop space of a space X is the based space X =
F(S
1
, X). Its points are loops at the basepoint of X.
To see how these constructions are related one to another we have the
following proposition
Proposition 3.8. There are natural homeomorphisms
F(X, Y )

= F(X, Y )
[X, Y ]

= [X, Y ].
Proof. To see the rst statement dene the function

: F(X, Y ) F(X, Y )
that corresponds to each map f : X Y the map f

: X Y dened
by f

(x)(s) = f(x s). The other statement follows from Proposition 3.3
by taking
0
of both side of the last homeomorphism.
We can endow [X, Y ] with a multiplication as follows. For f, g : X
Y dene
([g] + [f])(x t) = (g

(x) f

(x))(t) =

f(x 2t) if 0 t 1/2

g(x (2t 1)) if 1/2 t 1
It is easy to see that this multiplication is well dened.
15
Proposition 3.9. [X, Y ] is a group and [
2
X, Y ] is an Abelian group.
Denition 3.10. If in the denition of a cobration we require that all
maps and spaces be based then we have a based cobration which we call
cobration throughout this section. We call a space X nondegenerately based
or well pointed if the inclusion X is a cobration in the unbased sense.
Remark 3.11. A based map i : A X that is a cobration in the unbased
sense is a cobration in the based sense as well since, the basepoint of X
lies in A.
Denition 3.12. We write Y
+
for the union of a space Y and a disjoint
basepoint. The space X I
+
is called the reduced cylinder on X. We write
in this section M
f
for the based mapping cylinder Y
f
(X I
+
) of a based
map f : X Y .
Remark 3.13. Observe that we can identify X I
+
with X Y/ Y .
Remark 3.14. Observe also that a based homotopy X I Y is the
same thing as a based map X I
+
Y . As we have seen for the unbased
case in Proposition 1.7 a map i : A X is a cobration if and only if M
i
is a retract of X I
+
.
Denition 3.15. For a based map f : X Y dene the homotopy cober
to be
Cf = Y
f
CX = M
f
/j(X)
where j : X M
f
sends x to (x, 1).
Remark 3.16. Quotienting out Y we get X from C
f
, more precisely we
have C
f
/Y

= X.
Proposition 3.17. The inclusions X CX that sends x to (x, 0) and
Y C
f
are cobrations.
Proof. For the rst inclusion consider a test diagram
X
i
0

X I

h
.v
v
v
v
v
v
v
v
v
Z
CX
i
0

{
{
{
{
{
{
{
{
CX I
H
H
H
H
H
H
16
dene a homotopy H : CX I Z by
H((x, t), s) =

g(x, t + s(1 + t)) if 0 s (1 t)/(1 + t)

h(x,
1
2
(s(1 + t) + t 1)) if (1 t)/(1 + t) s 1
For the second inclusion, observe that it is the pushout of f and the latter
cobration.
Denition 3.18. A sequence S
f
S
g
S

of pointed sets is exact if

g(s) = if and only if f(s

) = s for some s

.
Denition 3.19. Let f : X Y be a based map and i : Y C
f
be the
cobration cited above. Let
: C
f
C
f
/Y

= X
be the quotient map and
f : X Y
be the map dened by
(f)(x t) = f(x) (1 t).
Then the cober sequence generated by f is
X
f
Y
i
C
f

X
f
Y
i
C
f

2
X

2
f

2
Y
Remark 3.20. It is not very hard to show that this sequence is an exact
sequence of pointed sets.
Proposition 3.21. For a based map f : X Y we have

n
C
f

= C

n
f
n > 1.
Proof. We show the case n = 1 and the general case follows by induction
on n. Now consider the map
C
f
= (Y
f
CX) C
f
= Y
f
CX
(c t)

c t if c Y
(x t, s) if c = (x, s) CX
One can easily show that this map is a homeomorphism.
17
Proposition 3.22. If i : A X ia cobration, then quotient map

i
: C
i
C
i
/CA

= X/A
is a based homotopy equivalence.
We shall note the homotopic inverse of
i
, by
i
. There is yet another
important proposition that we use to show one of the main results of this
section.
Proposition 3.23. In the following diagram, the left triangle commutes and
the right triangle commutes up to homotopy equivalence
X
f

Y
i(f)

C
f
(f)

i(i(f))

D
D
D
D
D
D
D
D
X
f

Y
. . .
C
i(f)

i(f)

(i(f))

z
z
z
z
z
z
z
z
z
Theorem 3.24. Suppose that X, Y, Z are three based spaces. The cober
sequence generated by the map f : X Y , induces the sequence
[C
f
, Z] [Y, Z] [X, Z] [C
f
, Z] [Y, Z] [X, Z]
which is an exact sequence of pointed sets, or of groups to the left of [X, Z],
or of Abelian groups to the left of [
2
X, Z].
Proof. According to Remark 3.20 we have only to show that at each
stage the inverse image of the basepoint is contained in the image of pre-
vious map. We now show a key point of the proof, for each pair of maps
f : X Y, i(f) : Y C
f
where i is the inclusion of the codomain of
f into the homotopy cober of it, and each map g : Y Z whose com-
position with f is null homotopic, there exists a map g : C
f
Z whose
restriction to Y is g, that is the sequence in question is exact at rst stage,
but we shall show that each pair of consecutive maps in the sequence is of
this form up to homotopy. To see this, consider the diagram
X
f

Y
i(f)

C
f
g=gh
.~
~
~
~
Z
where h : g f c

is a based homotopy, since h is constant on the set

18
X 1 I we may consider it as a map CX Z, using the fact
that C
f
= Y
f
CX, h and g induce the required map g. We can show by
induction on k that there is a commutative diagram up to homotopy

k
X

k
f

k
Y

k
i(f)

k
C
f

k
(f)

k+1
X

k
X

k
f

k
Y
i(
k
f)

C(
k
f)
(
k
f)

k+1
X.
Thus it would suce to show the exactness for the two pairs of maps
(i(f), (f)) and ((f), f). So let g [C
f
, Z] such that g i(f) c

,
from the rst argument we have that the sequence
Y
i(f)
C
f
i(i(f))
C
i(f)
induces the exact sequence
[C
i(f)
, Z] [C
f
, Z] [Y, Z]
so there exists h : C
i
(f) Z such that h i(i(f)) g. Applying Propo-
sition 3.22 and Remark 3.16 we have h
i(f)
: X Z and applying
the commutativity of the rst triangle of the diagram of Proposition 3.23
h
i(f)
(f) g since
i(f)
i(i(f)) = (f) implies that
i(i(f)) = id
X
i(i(f))
i(f)

i(f)
i(i(f)) =
i(f)
(f).
Thus the sequence
[X, Z] [C
f
, Z] [Y, Z]
is exact. To show the exactness of the sequence
[Y, Z] [X, Z] [C
f
, Z]
suppose given g : Y Z with g(f) c

so we have g
i(f)
: C
i(f)

Z Using the result of the rst pair for the pair (i(i(f)), (f)) , there exists
h : Y Z such that g
i(f)
h (i(f)) so
g = g id
X
g
i(f)

i(f)
h (i(f))
i(f)
and using the commutativity up to homotopy of the second triangle of the
diagram of Proposition 3.23 we have (i(f))
i(f)
f since
i(f)
and

i(f)
are homotopic inverses of each other and therefore g h (f).
19
Now we turn to the case of brations and the homotopy exact sequences
associated to them.
Denition 3.25. A based bration is bration where all maps and spaces
in the denition of bration are required to be based. We call them simply
bration in this section.
Remark 3.26. A based bration is necessarily a bration in the unbased
denition, to see this we restrict to spaces of the form Y
+
in the test diagrams
and observe that Y
+
I
+

= (Y I)
+
. It is less obvious that if a map
p : E B is a bration in the unbased sense then it satises the covering
homotopy property for test diagrams in which Y is nondegenerately based.
Remark 3.27. A based homotopy XI
+
Y is the same thing as based
map X F(I
+
, Y ) where F(I
+
, Y ) is the space Y
I
with the basepoint
c

: I Y ,the constant path at the basepoint of Y .

Denition 3.28. In this section the mapping path space Nf of a map f :
X Y is the subspace of X Y
I
, dened by Nf = (x, ) [ (1) = f(x)
given the basepoint (, c

).
Denition 3.29. The homotopy ber of a map f : X Y , Ff is the
pullback of the following diagram
Ff

PY
p
1

X
f

Y.
Explicitly we have
Ff = X
f
PY = (x, ) [ (1) = f(x) X PY.
Remark 3.30. Since : Ff X is a pullback of the bration p
1
:
PY Y , it is a bration.
Denition 3.31. Let : Y Ff be the inclusion () = (, ). The
following sequence is called the ber sequence generated by f

2
X

2
f

2
Y

Ff

X
f
Y

Ff

X
f
Y
where
(f)()(t) = (f )(1 t) for X.
20
Dual to the long exact sequences of pointed set induced from the cober
sequences, we have the following theorem whose proof is dual to that of The-
orem 3.24. But likely to the proof of Theorem 3.24 we need two propositions
before.
Proposition 3.32. If p : E B is a bration, then the inclusion :
p
1
() Fp dened by (e) = (e, c

) is a based homotopy equivalence.

Proposition 3.33. In the following diagram, the right triangle commutes
and the left triangle commutes up to homotopy


X
f

((f))

G
G
G
G
G
G
G
G
G
Y
(f)

Ff
(f)

X
f

Y
F(f)
((f))

x
x
x
x
x
x
x
x
Theorem 3.34. Suppose that X, Y and Z are three based spaces. The ber
sequnece generated by the map f : X Y induces the sequnece
[Z, Ff] [Z, X] [Z, Y ] [Z, Ff] [Z, X] [Z, Y ]
which is an exact sequence of pointed sets, or of groups to the left of [Z, Y ],
or of Abelian groups to the left of [Z,
2
Y ].
Remark 3.35. The group structure of the space [Z, Y ] is imposed by
Proposition 3.8 and Proposition 3.9 .
We have seen that for any space X and any map f : X Y between
spaces we can dene two other spaces and maps, namely X, X and f :
X Y and f : X Y these constructures preserve identities
and composition of maps thus they are functors from the category of pointed
topological spaces to itself. Proposition 3.8 suggests that the pair (, )
forms an adjoint pair. In fact it is the case and naturality is straghtforward
and there exisit unit and counit natural trasformation for these functors,that
are dene in the follwoing denition.
Denition 3.36. For a based space X and a based map f : X Y dene
: X X and : X X
21
by (x)(t) = xt and ( t) = (t) for x X, X and t S
1
. And dene
: Ff C
f
and : Ff C
f
by
(x, )(t) = (x, , t) =

(2t) if t 1/2
(x, 2t 1) if t 1/2
for (x, ) Ff.
There is a connection between cober sequences and ber sequences which
is useful in the calculation of homotopy groups of some spaces as we will see
later.
Proposition 3.37. Let f : X Y be a map between spaces. The fol-
lowing diagram is homotopy commutative where the top row is obtained by
Applying the functor to a part of the ber sequence generated by f and the
bottom row is obtained by applying the functor to a part of the cober
sequence generated by f:
Ff
p

X
f

Ff
p

X
Y

Ff
p

X
f

Y
i

C
f

X
Y
i

C
f

X
f

Y
i

C
f
.
Proof. If we number the squares of this diagram from the top left to the
right bottom from 1 to 8 the commutativity of squares 1, 2, 7, 8 follows from
the naturality of and . The commutativity of square 3 and 4 follows from
the commutativity of squares 6 and 5 respectivly, since the functors and
are adjoint to eachother. Thus it suces to see that squares 5 and 6 are
homotopy commutative, which is a machinery task.
Proposition 3.38. Let f : X Y be a map of based spaces. Then the
following diagram is homotopy commutative, where j : X M
f
is the in-
clusion and r : M
f
Y is the retraction and is induced by the quotient
map M
f
C
f
22
Fj = X
j
PM
f
Fr=idPr

O
O
O
O
O
O
O
O
O
O
O
O
X
f
PY = Ff

p
p
p
p
p
p
p
p
p
p
p
p
C
f
Proof. We write PM
f
as (
1
,
2
) and dene a homotopy
H : X
j
PM
f
I C
f
as follows
H(x, , t)(s) =

1
(
2s
2t
) (1 t)
2
(
2s
2t
) if 0 s
2t
2
x (2s 1) if
2t
2
s 1
now it is easy to verify that this map is indeed the required homotopy.
23
4 Homotopy Groups
In this section we dene and describe one of the most important objects of
algebraic topology, homotopy group of a topological space, these construc-
tions make it possible to translate topological properties of a space in an
algebraic language,in a homotopy-invariant fashion, indeed we want to study
the homotopy type of a space so that these homotopy group functors must
factor through the homootpy category of topological spaces. These invari-
ants supply us with means to do algebraic manipulations which correspond
to homotopy-invariant constructions in the category of topological spaces.
Throughout this section we suppose some some familiarity we elementary
algebraic topology such as fundamental groups and covering spaces.
Proposition 4.1. We have the following canonical homeomorphisms
S
n

= S
1
S
1

= S
n1

=
n
S
1

= I
n
/I
n
for n 1
Denition 4.2. For n 0 and a based space X dene the n-th homotopy
group of X at the basepoint to be the set of homotopy classes of based
maps S
n
X, i.e.

n
(X) =
n
(X, ) = [S
n
, X].
Remark 4.3. In view of Proposition 3.9
n
(X) is a group for n 1 and is
an Abelian group for n 2.
Proposition 4.4. For n 0 we have

n
(X) =
n1
(X) = =
0
(
n
X).
Proof. We show by induction on n that
n
X = F(S
n
, X), indeed we have
X = F(S
1
, X) by denition and from Proposition 3.8 and Proposition 4.1
we draw that

n
(X) =
n1
(X) = F(S
n1
, X) = F(S
n1
, X) = F(S
n
, X)
applying
0
to both side of the equality, we have that
n
(X) =
0
(
n
X).
Now replacing X by
k
X and n by nk, one shows the required equalities.
24
Proposition 4.5. For A X, the homotopy ber of the inclusion
i : A X, Fi may be identied with the space of paths in X that begin at
the basepoint and end in A. We note this space byP(X; , A).
Denition 4.6. For n 1 dene

n
(X, A) =
n
(X, A, ) =
n
(P(X; , A)).
These are called relative homotopy groups.
Remark 4.7. Again we have that
n
(X, A) is a group for n 2 and an
Abelian group for n 3 and
n
(X, A) =
0
(
n1
P(X; , A)).
Proposition 4.8. For n 1 we have

n
(X, A, ) = [(I
n
, I
n
, J
n
), (X, A, )]
where
J
n
= I
n1
I I
n1
0 I
n
and J
1
= 0 I
and we consider the homotopy classes of maps of triples,i.e. the image of
each component is included in the corresponding component.
In the identication Fi = P(X, , A), the projection on A from Fi corre-
sponds to the endpoint projection of P(X, , A) on A ,i.e., the endpoints of
the paths in P(X, , A), which we note p
1
: Fi A. Thus we have the
ber sequence generated by the inclusion i : A X as follows

2
A
2
X Fi A X

Fi
p
1
A
i
X.
Applying
0
() = [S
0
, ] to this sequence and using Proposition 4.4, we
obtain the long exact sequence associated to i which is

n
(A)
n
(X)
n
(X, A)


n1
(A)
0
(A)
0
(X).
Remark 4.9. We can give explicitly the denition of :
n
(X, A)

n1
(A) as follows. By Proposition 4.8 we know that an element f of

n
(X, A) is a map (I
n
, I
n
, J
n
) (X, A, ), thus restricting it to (I
n1

1, I
n1
1) and using the fact that S
n1
= I
n1
/I
n1
we have a
based map S
n1
A this map is the image of f under . The maps

n
(A)
n
(X) and
n
(X)
n
(X, A) are induced by the inclusions
(A, ) (X, ) and (X, , ) (X, A, ).
25
Proposition 4.10. Let p : E B be a bration with B path connected.
Fix a basepoint B and let F be the bre over and x a basepoint
F E. Let : F Fp be the based homotopy equivalence dened by
Proposition 3.32 . Then the following diagram is homotopy commutative:



2
E
id

Fi
p

p
1

E
id

Fi
p

p
1

E
id

2
E

2
p

2
B

Fp

E
p

Fp

E
where Fi = P(E, , F) and p() = p B for Fi .
Proof. The most right square and the third square are commutative which
can be shown by writing down the maps explicitly. The next to last square
is homotopy commutative, for dene a homotopy h : (p) p
1
by
h(, t) = ((t), p([1, t])) where [1, t](s) = (1 s + st)
i.e., [1, t] is the path going from (1) to (t). The other squares are ob-
tained from these three squares by applying the functor ,hence they are
commutative as well.
Proposition 4.11. With same hypothesis and notations of last proposition
we have that p

:
n
(E, F)
n
(B) is an isomorphism for n 1.
Proof. Passing to long exact sequences of homotopy groups described
above ,using the fact that homotopy equivalences induce isomorphisms on
the homotopy groups and using the ve lemma, we achieve the proof.
Remark 4.12. This result could also be derived directly from the covering
homotopy property of p.
Remark 4.13. Using

to identify

(F) with

(Fp) where

stands for
generic homotopy groups,we may rewrite the long exact sequence obtained
from the bottom row of the diagram as

n
(F)
n
(E)
n
(B)


n1
(F)
0
(E) .
A little path lifting argument shows that
0
(F)
0
(E) is a surjection.
26
Now we see some examples of homotopy groups.
Example 4.14. If X is contractible, then
n
(X) = 0 for all n 0. Since
X is homotopic to singleton whose homotopy groups are trivial.
Example 4.15. If X is discrete then
n
(X) = 0 for all n 1. Since xing a
basepoint in X, and using the fact that S
n
is connected for n 1 we observe
that any based and continuous map S
n
X has image in thus it is
trivial.
Example 4.16. If p : E B is a covering, then p

(E)

(B)
is an isomorphism for all n 2. To see this, consider the exact sequence
of Remark 4.13 and observe that for a covering the preimage of a point is
a discrete set, thus
n1
(F) = 0 for n 1 from the previous example so

n
(E)

=
n
(B).
Example 4.17.
1
(S
1
) = Z and
n
(S
1
) = 0 for n 2. We already know
the rst statement . To see the second statement observe that the map
p : R S
1
,t e
it
is a covering, so we have
n
(S
1
)

=
n
(R) which is
trivial since R is a contractible space.
Example 4.18. If i 2, then
1
(RP
i
) = Z
2
and
n
(RP
i
)

=
n
(S
i
) for
n 2. Again the rst statement is a covering space argument,for the second
one observe that the projection
: S
i
RP
i
= S
i
/Z
2
is a covering, and use Example 4.16 .
Proposition 4.19. For all spaces X and Y and all n, we have

n
(X Y )

=
n
(X)
n
(Y ).
Proof. Projections p
1
: X Y X and p
2
: X Y Y dene a
canonic function that sends a given map f : S
n
X Y
n
(X Y )
to the pair (p
1
f, p
2
f)
n
(X)
n
(Y ). Conversely given a pair of
maps (f
1
, f
2
)
n
(X)
n
(Y ) we can dene a map
f : S
n
X Y
n
(X Y ) , s (f
1
(s), f
2
(s))
. It is now easy to verify that these functions are homomorphisms, one
inverse of the other.
27
Proposition 4.20. If i < n, then
i
(S
n
) = 0.
Proposition 4.21.
2
(S
2
)

= Z and
n
(S
3
)

=
n
(S
2
) for all n 3.
Proposition 4.22. If X is the colimit of a sequence of inclusions X
i

X
i+1
of based spaces, then the natural map
colimit
i

n
(X
i
)
n
(X)
is an isomorphism for each n.
So far we have xed a basepoint and considered homotopy groups with
respect to it,now we want to know the relation between homotopy groups
when we change the basepoint. Since the inclusion of the basepoint in S
n
is a cobration, we conclude from Proposition 2.10 that the evaluation at
the basepoint p : X
S
n
X is a bration. We can identify
n
(X, x) with

0
(F
x
),where F
x
is the ber over x X which is in fact the loop space
n
X
with respect to the base point x,since two dierent points of F
x
lie in the
same path component if and only if there is a based homotopy h : S
n
I
X, and so homotopic maps S
n
X
n
(X, x) correspond to points in
the same path components of
0
(F
x
). We have seen in Proposition 2.16
that a path class [] : I X from x to x

induces a homotopy equivalence

[] : F
x
F
x
, we continue to write [] for the induced bijection
[] :
n
(X, x)
n
(X, x

).
Denition 4.23. For any space X, the folding map : X X X is
the unique map X X X which restricts to the identity map X X
on each wedge summand.
Denition 4.24. The pinch map p : S
n
S
n
S
n
is the map that is
obtained by collapsing an equator to the basepoint.
Proposition 4.25. Given two maps f, g : S
n
X, then [f] + [g] is the
homotopy class of the composite
S
n
p
S
n
S
n
fg
X X

X.
Remark 4.26. There is another way to see the addition in
n
(X, x). The
cobration S
n
induces the following pushout diagram

S
n

S
n
S
n
S
n
28
thus the inclusion S
n
S
n
S
n
and its composition with S
n
are
cobrations, by Proposition 1.5. Again from Proposition 2.10 the induced
map X
S
n
S
n
X is a bration and we have a diagram
X
S
n
S
n
p
p

X
S
n
p

X X.
the ber over x in left hand bration is the product F
x
F
x
,where F
x
is the
ber over x in the right hand bration, since from the last pushout diagram
any map S
n
S
n
X is a pair of maps S
n
X which coincide on the
basepoint of S
n
. Thus by identication F
x
=
n
X, the ber over x in the
left hand bration is
n
X
n
X. Now given two elements f, g X,we
see the pair (f, g) as an element of the ber in X
S
n
S
n
which is sent into
the ber F
x
=
n
X by the map p
p
,but this element is exactly the composite
S
n
p
S
n
S
n
fg
X X

X
by denition. Thus the addition in
n
(X, x) is obtained by restricting the
map p
p
to the ber over x and then applying
0
.
Proposition 4.27. The bijection [] :
n
(X, x)
n
(X, x

) is an iso-
morphism.
Proof. Using Proposition 2.18 which states the naturality of translations
of bers with respect to maps of brations, and with a little extra argument
we can show that the following diagram is homotopy commutative
F
x
F
x

[][]

F
x
[]

F
x
F
x

F
x
.
Addition in homotopy groups are induced by the horizontal maps on passage
to
0
, hence these translations of bers are compatible with additions in
homotopy groups, and they are homomorphisms.
There is a similar result for relative homotopy groups,the idea is the same,
for a A we identify the group
n
(X, A, a) with the homotopy class of
29
maps of triples [(CS
n1
, S
n1
, ), (X, A, )], via the homotopy equivalence
(I
n
, I
n
, J
n
)

= (CS
n1
, S
n1
, ) obtained by quotienting out J
n
. We have
two cobrations S
n1
and S
n1
CS
n1
which give rise to the
bration
p : (X, A)
(CS
n1
,S
n1
)
A
that to each function (CS
n1
, S
n1
) (X, A) associate its evaluation
at the basepoint. Again we identify
n
(X, A) with
0
(F
a
). A path class
[] : I A from a to a

induces a homotopy equivalence [] : F

aa
,and
again we write [] for the induced isomorphism
[] :
n
(X, A, a)
n
(X, A, a

).
From the naturality stated in Proposition 2.18 we deduce the following the-
orem which shows how the homotopy groups behave when we change the
basepoint. The proof uses essentially Proposition 2.18 .
Theorem 4.28. If f : (X, A) (Y, B) is map of pairs and : I A is
a path from a to a

, then the following diagram commutes

n
(X, A, a)
f

[]

n
(Y, B, f(a))
[f]

n
(X, A, a

)
f

n
(Y, B, f(a

)).
If h : f f

is a homotopy of maps of pairs and h(a)(t) = h(a, t), then the

following diagram commutes

n
(X, A, a)
f

n
n
n
n
n
n
n
n
n
n
n
n
f

Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q

n
(Y, B, f(a))
[h(a)]

n
(Y, B, f

(a)).
As a consequence we have the following proposition.
Proposition 4.29. If f : X Y is map between topological spaces, and
: I A a path from a to a

then the following diagram commutes

30

n
(X, a)
f

[]

n
(Y, f(a))
[f]

n
(X, a

)
f

n
(Y, f(a

)).
If h : f f

is a homotopy and h(a)(t) = h(a, t), then the following diagram

commutes

n
(X, a)
f

.p
p
p
p
p
p
p
p
p
p
p
f

O
O
O
O
O
O
O
O
O
O
O

n
(Y, f(a))
[h(a)]

n
(Y, f

(a)).
Proposition 4.30. A homotopy equivalence of spaces or of pairs of spaces
induces an isomorphism on all homotopy groups.
Proof. We show only the claim for the relative case, other one is the same.
So, suppose that we have a pair of maps of pairs f : (X, A) (Y, B) and
g : (Y, B) (X, A) and a pair of homotopy of pairs h : f g id
Y
and
k : g f id
X
, we show that
f

:
n
(X, A, a)
n
(Y, B, f(a))
is an isomorphism. We now that the induced homomorphism on homo-
topy groups by the identity map is the identity homomorphism,and that the
composition of induced homomorphisms of two maps is the induced homo-
morphism of the composition of them, now using the second diagram of the
last theorem we have

n
(X, A, a)
id
X
o
o
o
o
o
o
o
o
o
o
o
gf

=g

R
R
R
R
R
R
R
R
R
R
R
R
R

n
(X, A, a)
[k(a)]

n
(X, A, g f(a)).
Since [k(a)] is an isomorphism, it follows that
f

:
n
(X, A, a)
n
(Y, B, f(a))
31
is a monomorphism and that
g

:
n
(Y, B, f(a))
n
(X, A, g f(a))
is an epimorphism. Similarly, we have
g

:
n
(Y, B, f(a))
n
(X, A, g f(a))
is a monomorphism and that
f

:
n
(X, A, g f(a))
n
(Y, B, f g f(a))
is an epimorphism. Now the statement of the Theorem 4.28 says that for
the path k
a
= k(a, ) : I X we have a commutative diagram

n
(X, A, a)
f

[k
a
]

n
(Y, B, f(a))
[fk
a
]

n
(X, A, g f(a))
f

n
(Y, B, f g f(a)).
[k
a
] and [f k
a
] are isomorphisms,
f

:
n
(X, A, g f(a))
n
(Y, B, f g f(a))
is surjective, thus the monomorphism
f

:
n
(X, A, a)
n
(Y, B, f(a))
is an isomorphism.
Now we dene one of the most important classes of maps in the category of
topological spaces. The homotopy theory is in part the study of these maps.
As we will see in the next section they are isomorphisms in the homotopy
category of CW-complexes.
Denition 4.31. A map e : Y Z is an n-equivalence if for all y Y ,
the induced map
e

:
q
(Y, y)
q
(Z, e(y))
is an isomorphism for q < n and a surjection for q = n. e is a weak
equivalence if it is an n-equivalence for all n or equivalently if the induced
homomorphisms on all homotopy groups are isomorphisms.
32
Remark 4.32. If in the denition the space Y is path-connected, then it
suces to verify the induced homomorphisms for a xed basepoint.
Proposition 4.33. Any homotopy equivalence is a weak equivalence.
Proof. This is clear from proposition 4.30 .
There is a powerful proposition which we shall use many times in the
following sections. We take CX to be the unreduced cone. Again we view

n+1
(X, x) as the set of relative homotopy classes of maps
(CS
n
, S
n
) (X, x).
Proposition 4.34. The following conditions on a map e : Y Z are
equivalent.
(i) For any y Y, e

:
q
(Y, y)
q
(Z, e(y)) is an injection for q = n
and a surjection for q = n + 1.
(ii) Given maps f : CS
n
Z, g : S
n
Y , and h : S
n
I Z such
that f [ S
n
= h i
0
and e g = h i
1
in the following diagram, there
are maps g and

h that make the entire diagram commute
S
n
i
0

S
n
I

h
.u
u
u
u
u
u
u
u
u
S
n
i
1

g
.y
y
y
y
y
y
y
y
Z Y
e

CS
n
i
0

z
z
z
z
z
z
z
z
CS
n
I

h
I
I
I
I
I
CS
n
.
i
1

g
E
E
E
E
(iii) The conclusion of (ii) holds when f [ S
n
= e g and h is the constant
homotopy at this map, i.e., h(s, t) = f(s) = e g(s) for all s S
n
and
t I.
Proof. (ii)= (iii). This is trivial.
(iii)=(i). For n = 0 observe that a map f : CS
0
Z is the same thing
as a path in Z from in f(s
0
, 0) to f(s
1
, 0). Thus if there is a path in Z
from e(y
o
) to e(y
1
) we can see it as a map f : CS
0
Z, the existence of
33
g : CS
0
Y which makes the diagram commute, implies the existence of
a path in Y from y
0
and y
1
. Now suppose that n > 0, and that e

([g]) = 0
that is e g c
e(y)
, where c
e(y)
is the constant map at e(y). So there is a
homotopy f : S
n
I Z with f
0
= e g and f
1
= c
e(y)
,we can see this
homotopy as a map f : CS
n
Z since it is constant on the set X 1.
Thus we have a map f : CS
n
Z whose restriction to S
n
is egAgain
the lifting g gives a homotopy g c
y
thus we have [g] = 0 in
n
(Y, y),this
shows the injectivity. Now we show that
n+1
(e) is surjective. Given a map
f
n+1
(Z, e(y)), as we said before we have f : (CS
n
, S
n
) (Z, e(y)),now
let g be the constant map at y, so we have a map g : (CS
n
, S
n
) (Y, y)
and a homotopy

h : f e g relative to S
n
, so g
n+1
(Y, y) is mapped to
f. Hence the surjectivity of
n+1
(e).
(i)=(ii). We give a sketch of the proof. The idea is to use (i) to show that
the nth homotopy group of the ber F
y
over y is trivial, and to use the given
part of the diagram to construct g and

h.
34
5 CW-Complexes
We dene a large class of spaces, called CW-complexes, which play a very
important role in homotopy theory and are the cornerstones of cellular ho-
mology theory. As we said before and we will see in this section any weak
equivalence between them is a homotopy equivalence,and we will see that
any space is weakly equivalent to a CW-complex.
Denition 5.1. The unit n-disk D
n+1
is the set x R
n+1
[ |x| 1. The
n-sphere S
n
D
n+1
is the set x R
n+1
[ |x| = 1 = D
n+1
.
Denition 5.2. A CW-complex X is the colimit of the successive inclu-
sions X
n
X
n+1
of subspaces X
n
,that are constructed by the following
inductive procedure. X
0
is a discrete set of points (called vertices),suppose
that we have constructed X
k
for k n, and suppose that A
n+1
is an index-
ing set and for any A
n+1
we have an attaching map j

: S
n

X
n
,
then X
n+1
is the pushout of the following diagram

A
n+1
S
n

X
n

A
n+1
D
n+1

X
n+1
.
That is, X
n+1
is constructed from X
n
by attaching (n+1)-disks along attach-
ing maps j

. More explicitly X
n+1
is the quotient space of X
n
(

D
n+1
)
by the identication j

(s) = s, for all A

n+1
and s S
n
. Each resulting
map i

: D
n+1

X
n+1
or by abuse of language the image of such maps, is
called an (n+1)-cell or simply a cell. The space X
n
is called the n-skeleton.
If the 0-skeleton X
0
is the disjoint union of a space A with a (possibly empty)
discrete set,then the resulting space is the relative CW-complex (X, A). In
both relative or absolute case, we say that the CW-complex has dimension
n if X = X
n
. A CW-complex is nite if it has nitely many cells. A sub-
complex Y of a CW-complex X is a subspace and a CW-complex such that for
each n 0 its indexing set B
n
is contained in A
n
and the maps D
n

X
n
35
are composition of the inclusion Y X and the maps D
n

Y
n
for all
B
n
. In other word, Y is the union of some of the cells of X. A map
of pairs f : (X, A) (Y, B) between relative CW-complexes is said to be
cellular if f(X
n
) Y
n
for all n.
Remark 5.3. The maps D
n
X
n
are sometimes called characteristic
maps.
Remark 5.4. When A is a subcomplex of a CW-complex X we can view
(X, A) as a relative CW-complex.
Of course the topology of the colimit is the weak topology, i.e.,a subspace
is closed if and only if its intersections with skeleta are closed.
Proposition 5.5. If X is a CW-complex, then each cell is contained in a
nitely many X
n
.
Example 5.6. For any n , the n-sphere S
n
is a CW-complex with one ver-
tex and one n-cell. The attaching map is the obvious map S
n1
,
then the characteristic map D
n
S
n
is the projection D
n
D
n
/S
n1
=
S
n
. If m < n then the only cellular map S
m
S
n
is the trivial map and
if m n then every map S
m
S
n
is cellular.
Example 5.7. RP
n
is a CW-complex with m-skeleton RP
m
and with one
m-cell for each m n. The attaching map is the projection j : S
n1

S
n1
/Z
2

= RP
n1
, thus RP
n
is homeomorphic to RP
n1

j
D
n
. Explicitly
write x = [x
1
, ..., x
n+1
],

x
2
i
= 1, for a typical point of RP
n
. Then x is in
RP
n1
if and only if x
n+1
= 0, the required homeomorphism is obtained by
identifying D
n
and its boundary sphere with the upper hemisphere
(x
1
, ..., x
n+1
) [

x
2
i
= 1 and x
n+1
0
and its boundary.
The category of CW-complexes is closed under many constructions of
topological spaces, so we can make complexes from ones given. The follow-
ing propositions are not very hard to prove, they follow directly from the
denitions.
Proposition 5.8. If (X, A) is relative CW-complex, then the quotient space
X/A is a CW-complex with a vertex corresponding to A and one n-cell for
each relative n-cell of (X, A).
36
Proposition 5.9. For CW-complexes X
i
with basepoints that are vertices,
the wedge product

i
X
i
, is a CW-complex which contains each X
i
as a
subcomplex. More over we have (

i
X
i
)
n
=

i
X
n
i
.
Denition 5.10. The Euler characteristic (X) of a nite CW-complex is
the alternating sum

(1)
n

n
(X), where
n
(X) is the number of n-cells of
X.
Proposition 5.11. Let A be a subcomplex of a CW-complex X, let Y be CW-
complex, and let f : A Y be a cellular map, then the pushout Y
f
X is a
CW-complex with n-skeleton Y
n

f
X
n
, and with Y as a subcomplex and has
one cell for each cell of X that is not in A. The quotient complex (Y
f
X)/Y
is isomorphic to X/A. Further more if X and Y are nite we have the
following formula relating the Euler characteristics (A), (X), (Y ) and
(Y
f
X)
(Y
f
X) = (Y ) + (X) (A).
Proof. Only the last statement about the Euler characteristics needs a
verication, the others follow immediately from the denition. In fact we
have
n
(Y
f
X) =
n
(Y ) +
n
(X)
n
(A), since the map f is cellular and
therefore the n-cells of A are sent to Y
n
, from the rst part of the proposition
we have that (Y
f
X)
n
= Y
n

f
X
n
thus the n-cells of A are have double
contribution in the sum
n
(Y ) +
n
(X) once as n-cells of Y and once as n-
cells of X, but this is superuous, and we have to subtract once the number
of n-cells of A,
n
(A) indeed we have
n
(Y
f
X) =
n
(Y ) +
n
(X)
n
(A),
which shows the desired result on Euler characteristic.
Proposition 5.12. The colimit of a sequence of inclusions of subcomplexes
into CW-complexes, X
i
X
i+1
, is a CW-complex that contains each of
the X
i
as a subcomplex.
Proposition 5.13. For p, q with p+q = n, there is a canonical homeomor-
phism
(D
n
, S
n1
)

= (D
p
D
q
, D
p
S
q1
, S
p1
D
q
).
Proof. This follows immediately from the homeomorphisms
D
n

= I
n
and S
n

= I
n
/I
n
and I
n
= I
p
I
q
.
37
This proposition allows us to endow the product of CW-complexes with
a CW-structure. Explicitly we have the following result
Proposition 5.14. The product X Y of CW-complexes X and Y is a
CW-complex with an n-cell for each pair consisting of a p-cell of X and a
q-cell of Y , with p+q = n. In particular For a CW-complex X, the cylinder
X I is a CW-complex that contains X partialI as a subcomplex and in
addition, has one (n + 1)-cell for each n-cell of X.
Proof. One can easy show by induction on n that the n-skeleton of XY ,
is the space
(X Y )
n
=

p+q=n
(X
p
Y
q
)
note also that the indexing set for the product complex in dimension n is

p+q=n
A
p
B
q
, where A
p
and B
q
are respectively the indexing set of X
and Y in dimensions p and q. For the last statement we use the fact that the
unit interval I is a CW-complex with two vertices and one 1-cell attached to
them in the obvious way, and the result follows from the general case.
We can now produce many useful spaces from a given CW-complex, and
to lie still in the category of CW-complexes.
Proposition 5.15. Let X and Y be CW-complexes, X

and Y

be based
CW-complexes with basepoints that are vertices, and let f : X Y be a
cellular map, then the following spaces are all CW-complexes
(i) The smash product X

Y

.
(ii) The suspension X

.
(iii) The cone CX

.
(iv) The mapping cylinder M
f
.
(v) The mapping cone C
f
.
Proof. To see that X

Y

is a CW-complex, recall from Proposition 5.9
that the wedge product X

Y

is a CW-complex, and from Proposition
5.14 that the product X

Y

is a CW-complex, now the space X

Y

=
X

Y

/X

Y

is a CW-complex by Proposition 5.8, since the wedge X

Y

38
is a subcomplex of the product X

Y

, this shows (i). (ii) follows from (i),
since X

= X

S
1
and S
1
is a CW-complex from Example 5.6 . (iii) is
clear from Proposition 5.14 and the denition of the cone CX

. To see (iv),
we apply Proposition 5.11 to the relative CW-complex (X I, X) and the
map f . A same argument shows (v).
In view of Proposition 5.14 we can dene an appropriate homotopy in the
category of CW-complexes,
Denition 5.16. A cellular homotopy h : f f

between cellular maps

X Y of CW-complexes is a homotopy that is itself a cellular map h :
X I Y
The last proposition in the last section, has a powerful consequence and
generalization which allows to show many results about CW-complexes.
This generalization is known as HELP(=homotopy extension and lifting
property).
Theorem 5.17 (HELP). Let (X, A) be a relative CW-complex of dimen-
sion n and let e : Y Z be an n-equivalence. Then given maps
f : X Z, g : A Y and h : A I Z such that f [ A = h i
0
and
e g = h i
1
in the following diagram, there are maps g and

h that makes
the entire diagram commute
A
i
0

AI

h
.x
x
x
x
x
x
x
x
x
A
i
1

g
.}
}
}
}
}
}
}
}
Z Y
e

X
i
0

~
~
~
~
~
~
~
~
X I

h
F
F
F
F
F
X.
i
1

g
A
A
A
A
Proof. It is enough by induction on i to consider the case that A = X
i
and X = X
i+1
. Then working one cell at a time it reduces to the case where
A = S
i
and X = D
i+1
which follows directly from Proposition 4.34 at the
end of last section.
39
Remark 5.18. This theorem says in part that if e is the identity map of
Y , then the inclusion A X has the homotopy extension property, and
therefore is a cobration.
Remark 5.19. By passage to colimits, n can be in the theorem.
Theorem 5.20 (Whitehead). If X is CW-complex and e : Y Z is an
n-equivalence, then e

: [X, Y ] [X, Z]is a bijection if dim(X) < n and

a surjection if dim(X) = n, where [X, Y ]stands for the homotopy classes of
unbased maps. In particular if e is weak equivalence then the result holds for
any CW-complex X.
Proof. Given a map f : X Z, an application of last theorem to the
pair (X, ), gives at the same time a map g : Y Z and a homotopy
h : X Z satisfying h
0
= f and h
1
= e g, hence [f] = [e g] in
[X, Z], and therefore e

is surjective. Given two maps

0
,
1
[X, Y ] with
[e
0
] = [e
1
] in [X, Z], so there is a homotopy f : X I Z with
f
i
=
i
, i = 0, 1, now apply last theorem to the pair (X I, X I), with
h : (X I) I Z the constant homotopy
h(x, t
i
, s) = e
i
(x), i = 0, 1, and s I.
This gives the required homotopy between
0
and
1
, thus e

is injective.
Theorem 5.21 (Whitehead). An n-equivalence between CW-complexes
of dimension less than n is a homotopy equivalence. A weak equivalence
between CW-complexes is a homotopy equivalence.
Proof. Let e : Y Z satisfy either hypothesis, then from last theorem
we have that e

: [Z, Y ] [Z, Z] is a bijection and so there is map f :

Z Y such that e f id
Z
, thus e f e e and since the map e

:
[Y, Y ] [Y, Z] is also a bijection from last theorem, and the elements [id
Y
]
and [f e] are mapped to the same element [e] we deduce that f e idY.
Remark 5.22. This theorem says that an n-equivalence between CW-
complexes of dimension less than n is a weak equivalence since any homotopy
equivalence is a weak equivalence, so we have only to check bijectivity of the
rst n1 homotopy groups to have the isomorphism on all homotopy groups
40
Whitehead theorem states that the homotopy groups are, in a sense, a
complete homotopy invariants for a CW-complex. So it would be of great
interest to be able to approximate a topological space with a CW-complex
and to work in the homotopy category of CW-complexes in which the weak
equivalences are inversible, rather than the larger category of topological
spaces. Surprisingly, this approximation exists and is functorial as we will
see, but before we prove a result, that says that any map between CW-
complexes is homotopic to a cellular map. This result is very helpful, since
the cellular maps behave much better that general maps.
Denition 5.23. A space X is said to be n-connected if
q
(X, x) = 0 for
0 q n and all x X. A pair (X, A) is said to be n-connected if

0
(A)
0
(X) is surjective and
q
(X, A, a) = 0 for 1 q n and all
a A. Equivalently the pair (X, A) is n-connected if the inclusion A X
is an n-equivalence.
Proposition 5.24. A relative CW-complex (X, A) with no m-cells for m
n is n-connected. In particular, (X, X
n
) is n-connected for any CW-complex
X.
Theorem 5.25 (Cellular Approximation). Any map f : (X, A)
(Y, B) between relative CW-complexes is homotopic relative to A to a cellular
map.
Proof. We construct this map inductively, that is we construct succes-
sively for all n a map g
n
: X
n
Y
n
that is homotopic to the restriction
of f to X
n
. Since the CW-complex Y is constructed from Y
0
by attaching
cells, any point of Y can be connected to a point in Y
0
by a path, now for
each point of X
0
A pick a point in Y
0
that is connected to it via a path,
this gives a map from g
0
: X
0
Y
0
, if we dene the images of points in
A as their images under f. This map is homotopic relative to A to f [ X
0
,
this was the rst step of the induction. Now suppose that we have con-
structed a map g
n
: X
n
Y
n
and a homotopy h
n
: X
n
I Y such
that h
n
: f [ X
n

n
g
n
, where
n
: Y
n
Y is the inclusion. For an
attaching map j : S
n
X
n
of a cell

j : D
n+1
X, we apply HELP to
the following diagram, since
n+1
: Y
n
Y
n+1
is an (n + 1)-equivalence
according to last proposition
41
S
n
i
0

S
n
I

h
n
(jid)
.u
u
u
u
u
u
u
u
u
u
S
n
i
1

g
n
j
.u
u
u
u
u
u
u
u
u
Y Y
n+1

n+1

D
n+1
i
0

z
z
z
z
z
z
z
z
D
n+1
I
h
n+1
H
H
H
H
H
D
n+1
.
i
1

g
n+1
I
I
I
I
I
where : Y
n
Y
n+1
is the inclusion. We could apply the same argu-
ment for an arbitrary family of attaching map. Since the (n+1)-skeleton is
the pushout of attaching maps and inclusions S
n
D
n+1
, the maps and
homotopies so far constructed induce the required map and homotopy on
(n + 1)-skeleton X
n+1
. Note that, since the map g
n
and the homotopy h
n
are relative to A the map and homotopy g
n+1
, h
n+1
are also relative to A.
The maps g
n
induce the required cellular map on X.
As consequence we have the following proposition.
Proposition 5.26. Any map f : X Y between CW-complexes is homo-
topic to a cellular map, and any two homotopic cellular maps are cellularly
homotopic.
Proof. The rst statement is the direct consequence of last proposition.
To see the second statement, observe that a homotopy h : f g is a map
h : (XI, X I) (Y, Y ), and vice versa. Applying last proposition to
this map, gives the desired cellular homotopy.
We have a sequence of approximation theorems for spaces, pair of spaces
and triads.
Theorem 5.27 (Approximation by CW-complexes). For any space
X, there is CW-complex X and a weak equivalence
X
: X X. For
a map f : X Y and another such CW approximation
Y
: Y Y ,
there is a map f : X Y , unique up to homotopy, such that the
following diagram is homotopy commutative
42
X
f

X
f

Y.
If X is n-connected, with n 1, then X can be chosen to have a unique
vertex and no q-cells for 1 q n.
Proof. First we show the existence of such approximation.It is sucient
to consider the case X path-connected, since the procedure can be ap-
plied to each path component separately.Again the construction is recur-
sively.Fix a basepoint y
0
Y and let X
1
be the wedge

(q,j)
S
q
, q 1, where
j : S
q
X represents a generator of the group
q
(X).On the (q, j)th wedge
summand, the map
1
is the given map j.X
1
has the obvious CW struc-
ture.The way we constructed X
1
shows that the induced homomorphism

q
(
1
) is surjective for all q.Assume that we have constructed CW-complexes
X
m1
, cellular inclusions i
m
: X
m1
X
m
and maps
m
: X
m
X for
m n such that
m
i
m1
=
m1
and (
m
)

:
q
(X
m
)
q
(X) is a
surjection for all q and a bijection for q < m.We may attache (n+1)-cells to
X
n
to vanish the kernel of the map (
n
)

, and then extend over these cells

to dene a map

n+1
: X
n+1
= X
n
(

D
n+1

) X.
The cellular approximation theorem then implies that
q
(
n+1
) is injective
for q n, and it is obviously surjective for all q.Now the CW approxima-
tion X is the colimit of the inclusions i
n
: X
n
X
n+1
and the weak
equivalence
X
: X X is the induced map on X from the maps

n
: X
n
X.If X is n-connected, then we have used no q-cells for q n
in the construction.The uniqueness and existence of f is immediate since
the Whitehead theorem gives a bijection (
Y
)

: [X, Y ] [X, Y ].
Remark 5.28. This theorem say that the CW approximation is a functor
from the homotopy category of topological spaces to itself, and that there is
natural transformation : Id such that the morphism
X
: X X
is a weak equivalence.
There is a relative generalization of last the theorem, as follows.
43
Theorem 5.29. For any pair of spaces (X, A) and any CW approximation

A
: A A, there is a CW approximation
X
: X such that A is
a subcomplex of X and
X
restricts to
A
.If f : (X, A) (Y, B) is a map
of pairs and : (Y, B) (Y, B) is another such CW approximation of
pairs, there is a map f : (X, A) (Y, B), unique up to homotopy,
such that the following diagram of pairs is homotopy commutative
(X, A)
f

(Y, B)

(X, A)
f

(Y, B).
If (X, A) is n-connected, then (X, A) can be chosen to have no q-cells for
q n.
Proof. The idea of the construction of the pair (X, A) is the same as
that of last theorem, we have only to construct in the same fashion the CW
approximation X over the given approximation A in order to guarantee
that A is a subcomplex of X. To construct f, we rst construct it on A
by mean of the last theorem, that is we have a map (f [ A) : A B
by the last theorem, and then extend this map to all X by use of HELP
A
i
0

C
C
C
C
C
C
C
C
AI
h
.v
v
v
v
v
v
v
v
v

A
(f|A)
.y
y
y
y
y
y
y
y

i
1

A
f|A

X
f

Y Y

{
{
{
{
{
{
{
{
i
0

X I

h
H
H
H
H
H
X.
i
1

f
E
E
E
E
The uniqueness up to homotopy of f is proved similarly.
Denition 5.30. A triad(X; A, B) is a space X together with subspaces A
and B. A triad is said to be excisive if X is the union of the interiors of
A and B. A CW-triad (X; A, B) is a CW-complex X with subcomplexes A
and B such that X = A B.
44
Remark 5.31. A traid (X; A, B) must not be confused with a triple (X, A, B),
which would require B A X.
Theorem 5.32. If e : (X; A, B) (X

; A

, B

) is a map of excisive triads,

such that the maps
e : C C

, e : A A

and e : B B

are weak equivalences, where C = AB and C

= A

, then e : X X

is a weak equivalence.
Remark 5.33. A CW-triad (X; A, B) is not excisive, since A and B are
closed in X, but a simple argument shows that it is equivalent to an excisive
triad. More generally, suppose that maps i : C A and j : C B are
given. Dene the double mapping cylinder
M(i, j) = A
i
(C I)
j
B
to be the space obtained from C I by attaching A to C 0 along i and
attaching B to C 1 along j. Let A
C
B denote the pushout of i and j
and observe that we have a natural quotient map q : M(i, j) A
C
B by
collapsing the cylinder, sending (c, t) to the image of c in the pushout.
The following proposition says more about the quotient map q : M(i, j)
A
C
B.
Proposition 5.34. For a cobration i : C A and any map j : C B,
the quotient map q : M(i, j) A
C
B is a homotopy equivalence.
When i is a cobration and j is an inclusion, with X = A B and
C = A B, we can think of q as giving a map of triads
q : (M(i, j); A
i
(C [0, 2/3)), (C (1/3, 1])
j
B) (A
C
B; A, B).
The domain is excisive, and q restricts to homotopy equivalences from the do-
main spaces and their intersection to the target subspaces A, B, and C.This
applies when (X; A, B) is CW-triad with C = AB. Now we can state and
prove the theorem of approximation of excisive triads by CW-triads
Theorem 5.35. Let (X; A, B) be an excisive triad and let C = AB. Then
there is a CW-triad (X; A, B) and a map of triads
: (X; A, B) (X; A, B)
45
such that, with C = A B, the maps
: C C, : A A, : B B, and : X X
are all weak equivalences. If (A, C) is n-connected, then (A, C) can be
chosen to have no q-cells for q n, and similarly for (B, C). Up to homo-
topy, CW approximation of excisive triads is functorial in such a way that
is natural.
Proof. Chose a CW approximation : C C and use the CW approx-
imation of patrs to extend it to a CW approximation
: (A, C) (A, C) and : (B, C) (B, C).
We then dene X to be the pushout A
C
B and let : X X be
given by the universal property of pushouts. The way we have constructed
A and B shows at once that C = AB. To see that : X X is a
weak equivalence, dene X

= M(i, j), where i : C A and j : C

B are the inclusions of subcomplexes, and let A

= A
i
(C [0, 2/3))
and B

= (C (1/3, 1])
j
B and C

= A

, then we know that

the map q : (X

; A

, B

) (X; A, B) described above is a homotopy

equivalence. Now dene e : (X

; A

, B

) (X; A, B) to be the composite

e = q, this satises the conditions of the Theorem 5.32, thus e : X

X
is a weak equivalence which implies that : X X is too. Other
armations are direct consequences of the theorem for pairs. This ends the
proof.
46
6 The Homotopy Excision And Suspension
Theorems
Denition 6.1. A map f : (A, C) (X, B) of pairs is an n-equivalence,
n 1, if
(f
1

)(im(
o
(B)
0
(X))) = im(
0
(C)
0
(A))
and for all choices of basepoints in C,
f

:
q
(A, C)
q
(X, B)
is a bijection for q < n and a surjection for q = n.
Remark 6.2. The rst condition in the denition holds whenever A and X
are path connected.
Proposition 6.3. For a triple (X, A, B) and any basepoint in B, the fol-
lowing sequence is exact

q
(A, B)
i

q
(X, B)
j

q
(X, A)
k

q1
(A, B)
where
i : (A, B) (X, B), j : (X, B) (X, A) and k : (A, ) (A, B)
are inclusions.
Proof. The proof is a purely algebraic argument on the exact sequences of
pairs, for instance to show the exactness of

q
(A, B)
i

q
(X, B)
j

q
(X, A)
one has only to chase the following commutative diagram



q
(B)
j

id

q
(A)

i

q
(A, B)

i

q1
(B)
id

q
(B)
j

q
(X)
id

q
(X, B)
j

q1
(B)

j

q
(A)
i

q
(X)


q
(X, A)


q1
(A)

.
Note that the sequence in question is represented vertically in this diagram.
47
Denition 6.4. For a triad (X; A, B) with basepoint C = AB, dene

q
(X; A, B) =
q1
(P(X; , B), P(A; , C))
where q 2.
Remark 6.5. More explicitly,
q
(X; A, B) is the set of homotopy classes of
maps of tetrads
(I
q
; I
q2
1 I, I
q1
1, J
q1
I
q1
0) (X; A, B, )
where again J
q1
= I
q2
I I
q2
0.
Remark 6.6. The long exact sequence of the pair (P(X; , B), P(A; , C))
is the following sequence, which is clear from the denitions of homotopy
groups of pairs and triads

q
(A, C)
q
(X, B)
q
(X; A, B) .
The following theorem has many consequences and applications especially
in homology theory. One of its consequences is the Freudenthal suspension
theorem, which is the starting point of the stable homotopy theory.
Theorem 6.7 (Homotopy Excision). Let (X; A, B) be an excisive triad
such that C = AB is not empty. Assume that (A, C) is (m1)-connected
and (B, C) is (n1)-connected, where m 2 and n 1. Then the inclusion
(A, C) (X, B) is an (m + n 2)-equivalence.
Sketch of Proof. We admit the following technical result and show that
it is equivalent to the homotopy excision theorem. Under the hypothesis of
the excision theorem we have

q
(X; A, B) = 0 for 2 q m + n 2
and all choices of basepoints in C. Now we return to the homotopy excision
theorem. The conditions that m 1 and n 1 give the sujectivity of the
maps
0
(C)
0
(A) and
0
(C)
0
(B), so it remains to show that
the induced map f

:
q
(A, C)
q
(X, B) is a bijection for q < n and a
48
surjection for q = n. The fact that m 2 implies that (A, C) is 1-connected,
that is

1
(A, C) =
0
(P(A; , C)) = 0.
It is now easy to see that this implies also that (X, B) is 1-connected. The
idea is to replace any fragment of paths that lie in A by a path that is
entirely in C and this is possible since (A, C) is 1-connected. This shows
that (X, B) is 1-connected, since (X; A, B) is excisive. The Remark 6.6 says
exactly that
q
(A, C)

=
q
(X, B) if and only if
q+1
(X; A, B) = 0 and

n
(f) :
n
(A, C)
n
(X, B)
is surjective if and only if
n
(X; A, B) = 0, hence the last result is equivalent
to the homotopy excision theorem.
Now we see some applications of this theorem.
Theorem 6.8. Let f : X Y be an (n 1)-equivalence between (n 2)-
connected spaces, where n 2. Then the quotient map : (M
f
, X)
(C
f
, ) is a (2n 2)-equivalence, where C
f
is the unreduced cober M
f
/X.
In particular, C
f
is (n 1)-connected. If X and Y are (n 1)-connected,
then : (M
f
, X) (C
f
, ) is (2n 1)-equivalence.
Proof. We have the excisive triad(C
F
; A, B), where
A = Y (X [0, 2/3]) and B = (X [1/3, 1])/(X 1).
Thus C := A B = X [1/3, 2/3]. It is easy to check that is homotopic
to the composite
(M
f
, X) (A, C)

(C
f
, B) (C
f
, ),
where the rst and last maps are homotopy equivalences of pairs. From
Proposition 1.10 f can be written as the composite X
j
M
f
r
, where
r is a homotopy equivalence, thus
q
(r) is an isomorphism and
q
(r j) =

q
(r)
q
(j) =
q
(f) for all q. So
q
(f) is an epimorphism(or isomorphism)
if and only if
q
(j) is an epimorphism(or isomorphism). Now consider the
exact sequence of the pair (M
f
, X)

q
(X)

q
(j)

q
(M
f
)
q
(M
f
, X)
q1
(X)
49
if q n 1 then
q
(f) and therefore
q
(j) are epimorphisms, and since X
is (n2)-connected this exact sequence shows that
q
(Mf, X) is the trivial
group. Thus (M
f
, X) is (n1)-connected, this together with the homotopy
equivalence (M
f
, X)

= (A, C) show at once that (A, C) is (n1)-connected
as well. The exact sequence of the pair (CX, X) and the fact that the cone
CX is contractible(and therefore

(CX) = 0), show that the connecting

homomorphism :
q+1
(CX, X)
q
(X) is in fact an isomorphism, thus
(CX, X) is (n1)-connected, and it is n-connected if X is (n1)-connected.
There is an evident homotopy equivalence (CX, X)

= (B, C), hence (B, C)
is (n 1)-connected, and it is n-connected if X is (n 1)-connected. Now
the result follows from the homotopy excision theorem.
Theorem 6.9. Let i : A X be a cobration and an (n 1)-equivalence
between (n 2)-connected spaces, where n 2. Then the quotient map
(X, A) (X/A, ) is a (2n2)-equivalence, and it is a (2n1)-equivalence
if A and X are (n 1)-connected.
Proof. The verticals arrows are homotopy equivalences of pairs in the fol-
lowing commutative diagram
(M
i
, A)
r

(C
i
, )

(X, A)

(X/A, )
where r : (M
i
, A) (X, A) is the retraction dened in section 1, and
: (C
i
, ) (X/A, ) is the homotopy equivalence given in Proposition
3.22 .The result follows from the previous theorem.
Denition 6.10. For a based space X, dene the suspension homomor-
phism
:
q
(X)
q+1
(X)
by letting
f = f id : S
q+1

= S
q
S
1
X S
1
= X.
Remark 6.11. We can dene more explicitly as follows. Let C

X be the
reversed cone on X, that is the space
C

X = X I/(X 0 I).
50
A map f
q
(X) can be regarded as a map of pairs f : (I
q
, I
q
) (X, ),
then the product f id : I
q+1
= I
q
I X I passes to quotients to
give a map of triples
(I
q+1
, I
q+1
, J
q+1
) (C

X, X, )
whose restriction to I
q
1 is f and which induces f when we quotient
out X 1. Note that we see X as the subspace X 1 of C

X.
Theorem 6.12 (Freudenthal Suspension). Assume that X is nonde-
generately base and(n 1)-connected, where n 1. Then the suspension
homomorphism is a bijection if q < 2n1 and a surjection if q = 2n1.
Proof. According to the last remark the following diagram is commutative,
where : C

X X is the quotient map

q+1
(C

X, X, )

o
o
o
o
o
o
o
o
o
o
o

Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q

q
(X)

q+1
(X).
As we have seen in the proof of Theorem 6.8, since C

X is contractible, is
an isomorphism. Up to degree n1, both of
q
(X) and
q
(C

X) are trivial
and
n
(C

X) = 0, so the inclusion X C

X is an n-equivalence between
(n1)-connected spaces. From Proposition 3.17 we know that this inclusion
is a cobration, therefore the quotient map
: (C

X, X) (C

X/X, )

= (X, )
is a 2n-equivalence by the last theorem. The conclusion now follows from
the bijectivity of and the last diagram.
Proposition 6.13.
1
(S
1
) = Z and :
1
(S
1
)
2
(S
2
) is an isomor-
phism.
Theorem 6.14. For all n 2,
n
(S
n
) = Z and :
n
(S
n
)
n+1
(S
n+1
)
is an isomorphism.
Proof. S
n
is an (n 1)-connected space by Proposition 4.20 and since
n > 1 the Freudenthal theorem applies and implies the second statement.
For the rst statement, we use the last proposition.
51
Denition 6.15. The q-th stable homotopy group
s
q
(X) of a space X, is
the colimit of the following diagram

q
(X)
q+1
(X)
q+2
(
2
X) .
Remark 6.16. We can show by induction on n 1 and using the argument
and diagram in the proof of Freudenthal theorem, that the space
n
(X) is
(n 1)-connected,thus for q < n 1 there is an isomorphism

q+n
(
n
X)

=
q+n+1
(
n+1
X)
by Freudenthal theorem, and therefore for any xed q, we have

q+n
(
n
X)

=
q+n+1
(
n+1
X)

=
q+n+2
(
n+2
X)

=
for all n > q + 1. This implies that the q-th stable homotopy group is
isomorphic to the group
q+n
(
n
X) for n > q + 1.
52
References
[1] Y. Felix, S. Halperin, J. Thomas, Rational Homotopy Theory, Springer-
Verlag (2000)
[2] A. Hatcher, Algebraic Topology, Cambridge University Press (2001)
[3] A.T. Lundell, S. Weingram, The Topology of CW Complexes, Van Nos-
trand Reinhold (1969)
[4] W.S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag
(1991)
[5] C.R.F. Maunder, Algebraic Topology, Dover (1996)
[6] J.P. May, A Concise Course in Algebraic Topology, Chicago lectures in
mathematics series (1999)
[7] R.A. Piccinini, Lectures on Homotopy Theory, North-Holland mathe-
matics studies (1992)
[8] P. Selick, Introduction to Homotopy Theory, Fields Institute Mono-
graphs No.9 AMS (1997)
53