The Adams spectral sequence was invented as a tool for computing stable homo-

topy groups of spheres, and more generally the stable homotopy groups of any space.
Let us begin by explaining the underlying idea of this spectral sequence.
As a rst step toward computing the set [X, Y] of homotopy classes of maps
XY one could consider induced homomorphisms on homology. This produces a
map [X, Y]Hom(H

(X), H

(Y)). The rst interesting instance of this is the notion

of degree for maps S
n
S
n
, where it happens that the degree computes [S
n
, S
n
]
completely. For maps between spheres of different dimension we get no information
this way, however, so it is natural to look for more sophisticated structure. For a
start we can replace homology by cohomology since this has cup products and their
stable outgrowths, Steenrod squares and powers. Changing notation by switching the
roles of X and Y for convenience, we then have a map [Y, X]Hom
A
(H

(X), H

(Y))
where A is the mod p Steenrod algebra and cohomology is taken with Z
p
coefcients.
Since cohomology and Steenrod operations are stable under suspension, it makes
sense to change our viewpoint and let [Y, X] now denote the stable homotopy classes
of maps, the direct limit under suspension of the sets of maps
k
Y
k
X. This has
the advantage that the map [Y, X]Hom
A
(H

(X), H

(Y)) is a homomorphism of
abelian groups, where cohomology is now to be interpreted as reduced cohomology
since we want it to be stable under suspension.
Since Hom
A
(H

(X), H

(Y)) is just a subgroup of Hom(H

(X), H

(Y)), we are
not yet using the real strength of the A module structure. To do this, recall that
Hom
A
is the n = 0 case of a whole sequence of functors Ext
n
A
. Since A has such a
complicated multiplicative structure, these higher Ext
n
A
groups could be nontrivial and
might carry quite a bit more information than Hom
A
by itself. As evidence that there
may be something to this idea, consider the functor Ext
1
A
. This measures whether
short exact sequences of A modules split. For a map f : S
k
S

with k > one can

form the mapping cone C
f
, and then associated to the pair (C
f
, S

) there is a short
exact sequence of A modules
0H

(S
k+1
)H

(C
f
)H

(S

)0
Additively this splits, but whether it splits over A is equivalent to whether A acts
trivially in H

(C
f
) since it automatically acts trivially on the two adjacent terms in
2 Chapter 2 The Adams Spectral Sequence
the short exact sequence. Since A is generated by the squares or powers, we are
therefore asking whether some Sq
i
or P
i
is nontrivial in H

(C
f
). For p = 2 this is
the mod 2 Hopf invariant question, and for p > 2 it is the mod p analog. The answer
for p = 2 is the theorem of Adams that Sq
i
can be nontrivial only for i = 1, 2, 4, 8.
For odd p the corresponding statement is that only P
1
can be nontrivial.
Thus Ext
1
A
does indeed detect some small but nontrivial part of the stable ho-
motopy groups of spheres. One could hardly expect the higher Ext
n
A
functors to give
a full description of stable homotopy groups, but the Adams spectral sequence says
that, rather miraculously, they give a reasonable rst approximation. In the case that
Y is a sphere, the Adams spectral sequence will have the form
E
s,t
2
= Ext
s,t
A
(H

(X), Z
p
) converging to
s

(X)/non p torsion
Here the second index t in Ext
s,t
A
denotes merely a grading of Ext
s
A
arising from
the usual grading of H

(X). The fact that torsion of order prime to p is factored

out should be no surprise since one would not expect Z
p
cohomology to give any
information about non- p torsion.
More generally if Y is a nite CW complex and we dene
Y
k
(X) = [
k
Y, X], the
stable homotopy classes of maps, then the Adams spectral sequence is
E
s,t
2
= Ext
s,t
A
(H

(X), H

(Y)) converging to
Y

(X)/non p torsion
Taking Y = S
0
gives the earlier case, which sufces for the more common applica-
tions, but the general case illuminates the formal machinery, and is really no more
difcult to set up than the special case. For the space X a modest hypothesis is needed
for convergence, that it is a CW complex with nitely many cells in each dimension.
The Adams spectral sequence breaks the problem of computing stable homotopy
groups of spheres up into three steps. First there is the purely algebraic problem of
computing Ext
s,t
A
(Z
p
, Z
p
). Since A is a complicated ring, this is not easy, but at least
it is pure algebra. After this has been done through some range of values for s and t
there remain the two problems one usually has with a spectral sequence, computing
differentials and resolving ambiguous extensions. In practice it is computing differ-
entials that is the most difcult. As with the Serre spectral sequence for cohomology,
there will be a product structure that helps considerably.
The fact that the Steenrod algebra tells a great deal about stable homotopy groups
of spheres should not be quite so surprising if one recalls the calculations done in
1.3. Here the Serre spectral sequence was used repeatedly to gure out successive
stages in a Postnikov tower for a sphere. The main step was computing differen-
tials by means of computations with Steenrod squares. One can think of the Adams
spectral sequence as streamlining this process. There is one spectral sequence for all
the p torsion rather than one spectral sequence for the p torsion in each individual
homotopy group, and the algebraic calculation of the E
2
page replaces much of the
A Sketch of the Construction 3
calculation of differentials in the Serre spectral sequences. As we will see, the rst
several stable homotopy groups of spheres can be computed completely without hav-
ing to do any nontrivial calculations of differentials in the Adams spectral sequence.
Eventually, however, hard work is involved in computing differentials, but we will stop
well short of that point in the exposition here.
A Sketch of the Construction
Our approach to constructing the Adams spectral sequence will be to try to realize
the algebraic denition of the Ext functors topologically. Let us recall how Ext
n
R
(M, N)
is dened, for modules M and N over a ring R. The rst step is to choose a free
resolution of M, an exact sequence
F
2
F
1
F
1
F
0
M 0
with each F
i
a free R module. Then one applies the functor Hom
R
(, N) to the free
resolution, dropping the term Hom
R
(M, N), to obtain a chain complex
Hom
R
(F
2
, N)Hom
R
(F
1
, N)Hom
R
(F
0
, N)0
Finally, the homology groups of this chain complex are by denition the groups
Ext
n
R
(M, N). It is a basic lemma that these do not depend on the choice of the free
resolution of M.
Now we take R to be the Steenrod algebra A for some prime p and M to be
H

(X), the reduced cohomology of a space X with Z

p
coefcients, and we ask
whether it is possible to construct a sequence of maps
X K
0
K
1
K
2

that induces a free resolution of H

(X) as an A module:
H

(K
2
) H

(K
1
) H

(K
0
) H

(X) 0
Stated in this way, this is impossible because no space can have its cohomology a free
A module. For if H

(K) were free as an A module then for each basis element we

would have Sq
i
nonzero for all i in the case p = 2, or P
i
nonzero for all i when
p is odd, but this contradicts the basic property of squares and powers that Sq
i
= 0
for i > and P
i
= 0 for i > /2.
The spaces whose cohomology is closest to being free over A are Eilenberg-
MacLane spaces. The cohomology H

(K(Z
p
, n)) is free over A in dimensions less
than 2n, with one basis element, the fundamental class in H
n
. This follows from
the calculations in 1.3 since below dimension 2n there are only linear combinations
of admissible monomials, and the condition that the monomials have excess less than
n is automatically satised in this range. Alternatively, if one denes A as the limit
of H

(K(Z
p
, n)) as n goes to innity, the freeness below dimension 2n is automatic
from the Freudenthal suspension theorem. More generally, by taking a wedge sum of
4 Chapter 2 The Adams Spectral Sequence
K(Z
p
, n
i
)s with n
i
n and only nitely many n
i
s belowany given N we would have
a space with cohomology free over A below dimension 2n. Instead of the wedge sum
we could just as well take the product since this would have the same cohomology as
the wedge sum below dimension 2n.
Free modules have the good property that every module is the homomorphic
image of a free module, and products of Eilenberg-MacLane spaces have an analogous
property: For every space X there is a product K of Eilenberg-MacLane spaces and
a map XK inducing a surjection on cohomology. Namely, choose some set of
generators
i
for H

(X), either as a group or more efciently as an A module, and

then there are maps f
i
: XK(Z
p
,
i
) sending fundamental classes to the
i
s, and
the product of these maps induces a surjection on H

.
Using this fact, we construct a diagram

X
X X
K
0
K
1
K
2
1
K
0
= /

X X
2
K
1 1
= /

X X
3
K
2 2
= /
. . .
by the following inductive procedure. Start with a map XK
0
to a product of
Eilenberg-MacLane spaces inducing a surjection on H

. Then after replacing this

map by an inclusion via a mapping cylinder, let X
1
= K
0
/X and repeat the process
with X
1
in place of X = X
0
, choosing a map X
1
K
1
to another product of Eilenberg-
MacLane spaces inducing a surjection on H

, and so on. Thus we have a diagram of

short exact sequences
0
0 0 0

0












. . .
X H 0 ( )

K H ( )

1
K H ( )

2
K H ( )

1
X H ( )

2
X H ( )

3
X H ( )

The sequence across the top is exact, so we have a resolution of H

(X) which would

be a free resolution if the modules H

(K
i
) were free over A.
Since stable homotopy groups are a homology theory, when we apply them to the
cobrations X
i
K
i
K
i
/X
i
= X
i+1
we obtain a staircase diagram


t 1 + t 1 +
X
X X

t
s s 1 +
s
X
t
X X

t 1 t 1 s 2
s 1

s
s 2 s 1
s 1 +
s
s 1
s 1
X

t 1 +
K
t
K
t 1
K
X
t 1 +
K
s 2 +
s
s 1 + t
K
t 1
K
t 1 +

t 1
X
and hence a spectral sequence. Since it is stable homotopy groups we are interested
in, we may assume X has been suspended often enough to be highly connected, say
n connected, and then all the spaces K
i
and X
i
can be taken to be n connected as
well. Then below dimension 2n the cohomology H

(K
i
) is a free A module and the
A Sketch of the Construction 5
stable homotopy groups of K
i
coincide with its ordinary homotopy groups, hence
are very simple. As we will see, these two facts allow the E
1
terms of the spectral
sequence to be identied with Hom
A
groups and the E
2
terms with Ext
A
groups,
at least in the range of dimensions below 2n. The Adams spectral sequence can be
obtained from the exact couple above by repeated suspension and passing to a limit
as n goes to innity. In practice this is a little awkward, and a much cleaner and more
elegant way to proceed is to do the whole construction with spectra instead of spaces,
so this is what we will do instead.
6 Chapter 2 The Adams Spectral Sequence
2.1 Spectra
The derivation of the Adams spectral sequence will be fairly easy once we have
available some basic facts about spectra, so our rst task will be to develop these
facts. The theme here will be that spectra are much like spaces, but are better in a
few key ways, behaving more like abelian groups than spaces.
A spectrum consists of a sequence of basepointed spaces X
n
, n 0, together
with basepoint-preserving maps X
n
X
n+1
. In the realm of spaces with basepoints
the suspension X
n
should be taken to be the reduced suspension, with the basepoint
cross I collapsed to a point. The two examples of spectra we will have most to do
with are:
The suspension spectrum of a space X. This has X
n
=
n
X with X
n
X
n+1
the identity map.
An Eilenberg-MacLane spectrumfor an abelian group G. Here X
n
is a CWcomplex
K(G, n) and K(G, n)K(G, n+1) is the adjoint of a map giving a CW approx-
imation K(G, n)K(G, n +1). More generally we could shift dimensions and
take X
n
= K(G, m+n) for some xed m, with maps K(G, m+n)K(G, m+
n+1) as before.
The idea of spectra is that they should be the objects of a category that is the natu-
ral domain for stable phenomena in homotopy theory. In particular, the homotopy
groups of the suspension spectrum of a space X should be the stable homotopy
groups of X. With this aim in mind, one denes
i
(X) for an arbitrary spectrum
X = X
n
to be the direct limit of the sequence

i+n
(X
n
)


i+n+1
(X
n
)
i+n+1
(X
n+1
)


i+n+2
(X
n+1
)
Here the unlabeled map is induced by the map X
n
X
n+1
that is part of the struc-
ture of the spectrum X. For a suspension spectrumthese are the identity maps, so the
homotopy groups of the suspension spectrum of a space X are the stable homotopy
groups of X. For the Eilenberg-MacLane spectrumwith X
n
= K(G, m+n) the Freuden-
thal suspension theorem implies that the map K(G, m+n)K(G, m+n+1) in-
duces an isomorphism on ordinary homotopy groups up to dimension approximately
2(m+n), so the spectrum has
i
equal to G for i = m and zero otherwise, just as
for an Eilenberg-MacLane space.
The homology groups of a spectrum can be dened in the same way, and in this
case the suspension maps are isomorphisms on homology. For cohomology, how-
ever, this denition in terms of limits would involve inverse limits rather than direct
limits, and inverse limits are not as nice as direct limits since they do not generally
preserve exactness, so we will give a different denition of cohomology for spectra.
For suspension spectra and Eilenberg-MacLane spectra the denition in terms of in-
verse limits turns out to give the right thing since the limits are achieved at a nite
stage. But for the construction of the Adams spectra sequence we have to deal with
Spectra Section 2.1 7
more general spectra than these, so we need a general denition of the cohomology a
spectrum. The denition should be such that the fundamental property of CW com-
plexes that H
n
(X; G) is homotopy classes of maps XK(G, n) remains valid for
spectra. Our task then is to give good denitions of CW spectra, their cohomology,
and maps between them, so that this result is true.
CW Spectra
For a spectrum X whose spaces X
n
are CWcomplexes it is always possible to nd
an equivalent spectrum of CW complexes for which the structure maps X
n
X
n+1
are inclusions of subcomplexes, since one can rst deform the structure maps to be
cellular and then replace each X
n
by the union of the reduced mapping cylinders of
the maps

n
X
0

n1
X
1
X
n1
X
n
This leads us to dene a CWspectrumto be a spectrum X consisting of CWcomplexes
X
n
with the maps X
n
X
n+1
inclusions of subcomplexes. The basepoints are
assumed to be 0 cells. For example, the suspension spectrum associated to a CW
complex is certainly a CW spectrum. An Eilenberg-MacLane CW spectrum with X
n
a
K(G, m+n) can be constructed inductively by letting X
n+1
be obtained from X
n
by
attaching cells to kill
i
for i > m+n+1. By the Freudenthal theorem the attached
cells can be taken to have dimension greater than 2m+2n, approximately.
In a CW spectrum X each nonbasepoint cell e
i

of X
n
becomes a cell e
i+1

of
X
n+1
. Regarding these two cells as being equivalent, one can dene the cells of X
to be the equivalence classes of nonbasepoint cells of all the X
n
s. Thus a cell of X
consists of cells e
k+n

of X
n
for all n n

for some n

. The dimension of this cell

is said to be k. The terminology is chosen so that for the suspension spectrum of a
CW complex the denitions agree with the usual ones for CW complexes.
The cells of a spectrum can have negative dimension. A somewhat articial ex-
ample is the CW spectrum X with X
n
the innite wedge sum S
1
S
2
for each
n and with X
n
X
n+1
the evident inclusion. In this case there is one cell in every
dimension, both positive and negative. There are other less articial examples that
arise in some contexts, but for the Adams spectral sequence we will only be concerned
with CW spectra whose cells have dimensions that are bounded below. Such spectra
are called connective. For a connective spectrum the connectivity of the spaces X
n
goes to innity as n goes to innity.
The homology and cohomology groups of a CW spectrum X can be dened
in terms of cellular chains and cochains. If one considers cellular chains relative
to the basepoint, then the inclusions X
n
X
n+1
induce inclusions C

(X
n
; G)

(X
n+1
; G) with a dimension shift to account for the suspension. The union C

(X; G)
of this increasing sequence of chain complexes is then a chain complex having one G
summand for each cell of X, just as for CW complexes. We dene H
i
(X; G) to be the
8 Chapter 2 The Adams Spectral Sequence
i
th
homology group of this chain complex C

(X; G). Since homology commutes with

direct limits, this is the same as the direct limit of the homology groups H
i+n
(X
n
; G).
Note that this can be nonzero for negative values of i, as in the earlier example having
X
n
=

k
S
k
for each n, which has H
i
(X; Z) = Z for all i Z.
For cohomology we dene C

(X; G) to be simply the dual cochain complex, so

C
i
(X; G) is Hom(C
i
(X; Z), G), the functions assigning an element of G to each cell of
X, and H

(X; G) is dened to be the homology of this cochain complex. This assures

that the universal coefcient theorem remains valid, for example.
A CW spectrum is said to be nite if it has just nitely many cells, and it is of
nite type if it has only nitely many cells in each dimension. If X is of nite type
then for each i there is an n such that X
n
contains all the i cells of X. It follows that
H
i
(X; G) = H
i
(X
n
; G) for all sufciently large n, and the same is true for cohomology.
The corresponding statement for homotopy groups is not always true, as the example
with X
n
=

k
S
k
for each n shows. In this case the groups
i+n
(X
n
) never stabilize
since, for example, there are elements of
2p
(S
3
) of order p that are stably nontrivial,
for all primes p. But for a connective CWspectrumof nite type the groups
i+n
(X
n
)
do eventually stabilize by the Freudenthal theorem.
Maps between CW Spectra
Now we come to the slightly delicate question of how to dene a map between CW
spectra. Areasonable goal would be that a map f : XY of CWspectra should induce
maps f

:
i
(X)
i
(Y), and likewise for homology and cohomology. Certainly a
sequence of basepoint-preserving maps f
n
: X
n
Y
n
forming commutative diagrams
as at the right would induce maps on homotopy groups, and also

X
n
n
X

n 1 +
n 1 +

Y
n
Y
n 1 +
f f
on homology and cohomology groups if the individual f
n
s were
cellular. Let us call such a map f a strict map, since it is not
the most general sort of map that works. For example, it would
sufce to have the maps f
n
dened only for all sufciently large n. This would be
enough to yield an induced map on
i
, thinking of
i
(X) as lim

i+n
(X
n
) and
i
(Y)
as lim

i+n
(Y
n
). If the maps f
n
were cellular there would also be an induced chain
map C

(X)C

(Y) and hence induced maps on H

and H

.
It turns out that a weaker condition will sufce: For each cell e
i

of an X
n
, the map
f
n+k
is dened on
k
e
i

for all sufciently large k. Here each f

n
should be dened
on a subcomplex X

n
X
n
such that X

n
X

n+1
. Such a sequence of subcomplexes
is called a subspectrum of X. The condition that for each n and each cell e
i

of
X
n
the cell
k
e
i

belongs to X

n+k
for all sufciently large k is what is meant by
saying that X

is a conal subspectrum of X. Thus we dene a map of CW spectra

f : XY to be a strict map X

Y for some conal subspectrum X

of X. If the
maps f
n
: X

n
Y
n
dening f are cellular it is clear that there is an induced chain map
f

: C

(X)C

(Y) and hence induced maps on homology and cohomology. A map of

CWspectra f : XY also induces maps f

:
i
(X)
i
(Y) since each map S
i+n
X
n
Spectra Section 2.1 9
has compact image contained in a nite union of cells, whose k fold suspensions lie
in X

n+k
for sufciently large k, and similarly for homotopies S
i+n
IX
n
.
Two maps of CW spectra XY are regarded as the same if they take the same
values on a common conal subspectrum. Since the intersection of two conal sub-
spectra is a conal subspectrum, this amounts to saying that replacing the conal
subspectrum on which a spectrum map is dened by a smaller conal subspectrum
is regarded as giving the same map.
It needs to be checked that the composition of two spectrum maps X
f
Y
g
Z
is dened. If f and g are given by strict maps on subspectra X

and Y

, let X

be
the subspectrum of X

consisting of the cells of the complexes X

n
mapped by f to
Y

n
. Then X

is also conal in X

and hence in X since f takes each cell e

i

of X

n
to a union of nitely many cells of Y
n
, suspending to cells of Y

n+k
for some k since
Y

is conal in Y , and then f

n+k
takes
k
e
i

to Y

n+k
so
k
e
i

is in X

n+k
. Thus X

is
conal in X and the composition gf is a strict map X

Z.
The inclusion of a subspectrum X

into a spectrum X is of course a map of

spectra, in fact a strict map. If X

is conal in X then the identity maps X

n
X

n
dene a map XX

which is an inverse to the inclusion X

X, in the sense that

the compositions of these two maps, in either order, are the identity. This means that
a spectrum is always equivalent to any conal subspectrum.
For example, for any spectrum X the subspectrum X

with X

n
dened to be
X
n1
X
n
is conal and hence equivalent to X. This means that every spectrum
X is equivalent to the suspension of another spectrum. Namely, if we dene the
suspension Y of a spectrum Y by setting (Y)
n
= Y
n
, then a given spectrum X
is equivalent to Y for Y the spectrum with Y
n
= X
n1
. It is reasonable to denote
this spectrum Y by
1
X, so that X = (
1
X). More generally we could dene
k
X
for any k Z by setting (
k
X)
n
= X
n+k
, where X
n+k
is taken to be the basepoint if
n+k < 0. (Alternatively, we could dene spectra in terms of sequences X
n
for n Z,
and then use the fact that such a spectrum is equivalent to the conal subspectrum
obtained by replacing X
n
for n < 0 with the basepoint.)
A homotopy of maps between spectra is dened as one would expect, as a map
XIY , where XI is the spectrum with (XI)
n
= X
n
I , this being the reduced
product, with basepoint cross I collapsed to a point, so that (X
n
I) = X
n
I .
The set of homotopy classes of maps XY is denoted [X, Y]. When X is S
i
, by
which we mean the suspension spectrum of the sphere S
i
, we have [S
i
, Y] =
i
(Y)
since spectrum maps S
i
Y are space maps S
i+n
Y
n
for some n, and spectrum
homotopies S
i
IY are space homotopies S
i+n
IY
n
for some n.
One way in which spectra are better than spaces is that [X, Y] is always a group,
in fact an abelian group, since as noted above, every CW spectrum X is equivalent
to a suspension spectrum, hence also to a double suspension spectrum, allowing an
abelian sum operation to be dened just as in ordinary homotopy theory. The sus-
10 Chapter 2 The Adams Spectral Sequence
pension map [X, Y][X, Y] is a homomorphism, and in fact an isomorphism, as
one can see in the following way. To show surjectivity, start with a map f : XY ,
which we may assume is a strict map. For clarity write this as f : X S
1
Y S
1
,
consisting of map f
n
: X
n
S
1
Y
n
S
1
. Passing to conal subspectra, we can replace
this by its restriction f
n1
: (X
n1
S
1
)(Y
n1
S
1
). The parentheses here are
redundant and can be omitted. This map is independent of the suspension coordinate
, and we want it to be independent of the last coordinate S
1
. This can be achieved
by a homotopy rotating the sphere S
1
by 90 degrees. So f
n1
is homotopic to a
map h
n
11, as desired, proving surjectivity. Injectivity is similar using XI in place
of X.
The homotopy extension property is valid for CW spectra as well as for CW com-
plexes. Given a map f : XY and a homotopy F : AIY of f
A for a subspec-
trum A of X, we may assume these are given by strict maps, after passing to conal
subspectra. Assuming inductively that F has already been extended over X
n
I , we
suspend to get a map X
n
IY
n
Y
n+1
, then extend the union of this map with
the given A
n+1
IY
n+1
over X
n+1
I .
The cellular approximation theorem for CW spectra can be proved in the same
way. To deform a map f : XY to be cellular, staying xed on a subcomplex A
where it is already cellular, we may assume we are dealing with strict maps, and that
f is already cellular on X
n
, hence also its suspension X
n
Y
n+1
. Then we deform
f to be cellular on X
n+1
, staying xed where it is already cellular, and extend this
deformation to all of X to nish the induction step.
Whiteheads theorem also translates to spectra:
Proposition 2.1. A map between CW spectra that induces isomorphisms on all ho-
motopy groups is a homotopy equivalence.
Proof: Without loss we may assume the map is cellular. We will use the same scheme
as in the standard proof for CW complexes, showing that if f : XY induces isomor-
phisms on homotopy groups, then the mapping cylinder M
f
deformation retracts
onto X as well as Y . First we need to dene the mapping cylinder of a cellular map
f : XY of CW spectra. This is the CW spectrum M
f
obtained by rst passing to
a strict map f : X

Y for a conal subspectrum X

of X, then taking the usual re-

duced mapping cylinders of the maps f
n
: X

n
Y
n
. These form a CW spectrum since
the mapping cylinder of f
n
is the suspension of the mapping cylinder of f
n
. Replac-
ing X

by a conal subspectrum replaces the spectrum M

f
by a conal subspectrum,
so M
f
is independent of the choice of X

, up to equivalence. The usual deformation

retractions of M
f
n
onto Y
n
give a deformation retraction of the spectrum M
f
onto
the subspectrum Y .
If f induces isomorphisms on homotopy groups, the relative groups

(M
f
, X)
are zero, so the proof of the proposition will be completed by applying the following
Spectra Section 2.1 11
result to the identity map of (M
f
, X): L
Lemma 2.2. If (Y, B) is a pair of CW spectra with

(Y, B) = 0 and (X, A) is an

arbitrary pair of CW spectra, then every map (X, A)(Y, B) is homotopic, staying
xed on A, to a map with image in B.
Proof: The corresponding result for CW complexes is proved by the usual method
of induction over skeleta, but if we lter a CW spectrum by its skeleta there may
be no place to start the induction unless the spectrum is connective. To deal with
nonconnective spectra we will instead use a different ltration. In a CW complex the
closure of each cell is compact, hence is contained in a nite subcomplex. There is in
fact a unique smallest such subcomplex, the intersection of all the nite subcomplexes
containing the given cell. Dene the width of the cell to be the number of cells in this
minimal subcomplex. In the basepointed situation we do not count the basepoint
0 cell, so cells that attach only to the basepoint have width 1. Reduced suspension
preserves width, so we have a notion of width for cells of a CW spectrum. The key
fact is that cells of width k attach only to cells of width strictly less than k, if k > 1.
Thus a CW spectrum X is ltered by its subspectra X(k) consisting of cells of width
at most k.
Using this ltration by width we can now prove the lemma. Suppose inductively
that for a given map f : (X, A)(Y, B), which we may assume is a strict map, we have
a conal subspectrum X

(k) of X(k) for which we have constructed a homotopy of

f
X

(k) to a map to B, staying xed on A X

(k). Choose a conal subspectrum

X

(k + 1) of X(k + 1) with X

(k + 1) X(k) = X

(k). This is possible since each

cell of width k + 1 will have some sufciently high suspension that attaches only to
cells in X

(k). Extend the homotopy of f

X

(k) to a homotopy of f
X

(k +1). The
restriction of the homotoped f to each cell of width k + 1 then denes an element
of

(Y, B). Since

(Y, B) = 0, this restriction will be nullhomotopic after some

number of suspensions. Thus after replacing X

(k + 1) by a conal subspectrum
that still contains X

(k), there will be a homotopy of the restriction of f to the new

X

(k+1) to a map to B. We may assume this homotopy is xed on cells of A, so this

nishes the induction step. In the end we have a conal subspectrum X

of X, the
union of the X

(k)s, with a homotopy of f on X

to a map to B, xing A. L
Proposition 2.3. If a CW spectrum X is n connected in the sense that
i
(X) = 0 for
i n, then X is homotopy equivalent to a CW spectrum with no cells of dimension
n.
In particular this says that a CW spectrum that is n connected for some n is ho-
motopy equivalent to a connective CW spectrum, so one could broaden the denition
of a connective spectrum to mean one whose homotopy groups vanish below some
dimension.
12 Chapter 2 The Adams Spectral Sequence
Another consequence of this proposition is the Hurewicz theoremfor CWspectra:
If a CW spectrum X is n connected, then the Hurewicz map
n+1
(X)H
n+1
(X) is
an isomorphism. This follows since if X has no cells of dimension n then the
Hurewicz map
n+1
(X)H
n+1
(X) is the direct limit of the Hurewicz isomorphisms

n+1+k
(X
k
)H
n+1+k
(X
k
), hence is also an isomorphism.
Proof the Proposition: We can follow the same procedure as for CW complexes, con-
structing the desired CW spectrum Y and a map YX inducing isomorphisms on
all homotopy groups by an inductive process. To start, choose maps S
n+1+k

X
k

representing generators of
n+1
(X). These give a map of spectra

S
n+1

X induc-
ing a surjection on
n+1
. Next choose generators for the kernel of this surjection and
represent these generators by maps from suitable suspensions of S
n+1
to the cor-
responding suspensions of

S
n+1

. Use these maps to attach cells to the wedge of

spheres, producing a spectrum Y
1
with a map Y
1
X that induces an isomorphism
on
n+1
. Now repeat the process for
n+2
and each successive
n+i
. L
Notice that if X has nitely generated homotopy groups, then we can choose
the CW spectrum Y to be of nite type. Thus a connective CW spectrum with nitely
generated homotopy groups is homotopy equivalent to a connective spectrumof nite
type.
Cobration Sequences
We have dened the mapping cylinder M
f
for a map of CWspectra f : XY , and
the mapping cone C
f
can be constructed in a similar way, by rst passing to a strict
map on a conal subspectrum X

and then taking the mapping cones of the maps

f
n
: X

n
Y
n
. For an inclusion A

X the mapping cone can be written as XCA. We

would like to say that the quotient map X CAX/A collapsing CA is a homotopy
equivalence, but rst we need to specify what X/A means for a spectrum X and
subspectrum A. In order for the quotients X
n
/A
n
to form a CW spectrum we need
to assume that A is a closed subspectrum of X, meaning that if a cell of an X
n
has
an iterated suspension lying in A
n+k
for some k, then the cell is itself in A
n
. Any
subspectrum is conal in its closure, the subspectrum consisting of cells of X having
some suspension in A, so in case A is not closed we can rst pass to its closure before
taking the quotient X/A.
When A is closed in X the quotient map X CAX/A is a strict map con-
sisting of the quotient maps X
n
CA
n
X
n
/A
n
, which are homotopy equivalences
of CW complexes. Whiteheads theorem for CW spectra then implies that the map
X CAX/A is a homotopy equivalence of spectra. (This could also be proved
directly.)
Thus for a pair (X, A) of CW spectra we have a cobration sequence just like the
one for CW complexes:
A

X X CA A

X
Spectra Section 2.1 13
This implies that, just as for CWcomplexes, there is an associated long exact sequence
[A, Y] [X, Y] [X/A, Y] [A, Y] [X, Y]
But unlike for CW complexes, there is also an exact sequence
[Y, A] [Y, X] [Y, X/A] [Y, A] [Y, X]
To derive this it sufces to show that [Y, A][Y, X][Y, X CA] is exact. The
composition of these two maps is certainly zero, so to prove exactness consider a
map f : YX which becomes nullhomotopic after we include X in X CA. A null-
homotopy gives a map CYX CA making a commutative square with f in the
following diagram:

Y
A CA X X A X
Y

CY

Y Y

1 1 1 1
f
i
f
i
We can then automatically ll in the next two vertical maps to make homotopy-
commutative squares. We observed earlier that the suspension map [Y, A][Y, A]
is an isomorphism, so we can take the map YA in the diagram to be a suspen-
sion g for some g : YA. Commutativity of the right-hand square gives f =
(i)(g) = (ig), and this implies that f = ig since suspension is an isomorphism.
This gives the desired exactness.
If we were dealing with spaces instead of spectra, the analog of the exactness of
[Y, A][Y, X][Y, X/A] would be the exactness of [Y, F][Y, E][Y, B] for a
bration FEB. This exactness follows immediately from the homotopy lifting
property. Thus when one is interested in homotopy properties of spectra, cobrations
can also be regarded as brations. For a cellular map f : AX of CW spectra with
mapping cone C
f
, the sequence [Y,
1
(C
f
)][Y, A][Y, X] is exact, so
1
C
f
can be thought of as the ber of f .
The second long exact sequence associated to a cobration, in the case of a pair
(AB, A), has the form
[Y, A] [Y, AB] [Y, B]
and this sequence splits, so we deduce that the natural map [Y, AB][Y, A][Y, B]
is an isomorphism. By induction this holds more generally for wedge sums of nitely
many factors.
Cohomology and Eilenberg-MacLane Spectra
The long exact sequences we have constructed can be extended indenitely in
both directions since spectra can always be desuspended. In the case of the rst
long exact sequence this means that for a xed spectrum Y the functors h
i
(X) =
[
i
(X), Y] dene a reduced cohomology theory on the category of CW spectra. The
wedge axiom h
i
(

) =

h
i
(X

) is obvious.
14 Chapter 2 The Adams Spectral Sequence
In particular, we have a cohomology theory associated to the Eilenberg-MacLane
spectrum K = K(G, m) with K
n
= K(G, m+ n), and this coincides with ordinary
cohomology:
Proposition 2.4. There are natural isomorphisms H
m
(X; G) [X, K(G, m)] for all
CW spectra X.
The proof of the analogous result for CW complexes given in 4.3 of [AT] works
equally well for CW spectra, and is in fact a little simpler since there is no need to talk
about loopspaces since spectra can always be desuspended. It is also possible to give
a direct proof that makes no use of generalities about cohomology theories, analogous
to the direct proof for CW complexes. One takes the spaces K
n
= K(G, m+n) to have
trivial (m+ n 1) skeleton, and then each cellular map f : XK gives a cellular
cochain c
f
in X with coefcients in
m
(K) = G sending an m cell of X to the
element of
m
(K) determined by the restriction of f to this cell. One checks that
this association f

c
f
satises several key properties: The cochain c
f
is always a
cocycle since f extends over (m+ 1) cells; every cellular cocycle occurs as c
f
for
some f ; and c
f
c
g
is a coboundary iff f is homotopic to g.
The identication H
m
(X; G) = [X, K(G, m)] allows cohomology operations to
be dened for cohomology groups of spectra by taking compositions of the form
XK(G, m)K(H, k). Taking coefcients in Z
p
, this gives an action of the Steenrod
algebra A on H

(X), making H

(X) a module over A. This uses the fact that com-

position of maps of spectra satises the distributivity properties f(g +h) = fg +fh
and (f +g)h = fh +gh, the latter being valid when h is a suspension, which is no
loss of generality if we are only interested in homotopy classes of maps. For spectra
X of nite type this denition of an A module structure on H

(X) agrees with the

denition using the usual A module structure on the cohomology of spaces and the
identication of H

(X) with the inverse limit lim

H
+n
(X
n
) since Steenrod opera-
tions are stable under suspension.
For use in the Adams spectral sequence we need a version of the splitting [Y, A
B] = [Y, A][Y, B] for certain innite wedge sums. Here the distinction between
innite direct sums and innite direct products becomes important. For an innite
wedge sum

the group [Y,

] can sometimes be the direct sum

[Y, X

],
for example if Y is a nite CW spectrum. This follows from the case of nite wedge
sums by a direct limit argument since the image of any map Y

lies in the
wedge sumof only nitely many factors by compactness. However, we will need cases
when Y is not nite and [Y,

] is instead the direct product

[Y, X

]. There
is always a natural map [Y,

[Y, X

] whose coordinates are obtained by

composing with the projections of

onto its factors.

Proposition 2.5. The natural map [X,

i
K(G, n
i
)]

i
[X, K(G, n
i
)] is an isomor-
phism if X is a connective CW spectrum of nite type and n
i
as i.
Spectra Section 2.1 15
Proof: When X is nite the result is obviously true since we can omit the factors
K(G, n
i
) with n
i
greater than the maximum dimension of cells of X without af-
fecting either [X,

i
K(G, n
i
)] or

i
[X, K(G, n
i
)]. For the general case we use a
limiting argument, expressing X as the union of its skeleta X
k
, which are nite.
Let h

(X) be the cohomology theory associated to the spectrum

i
K(G, n
i
), so
h
n
(X) = [
n
X,

i
K(G, n
i
)]. There is a short exact sequence
0
lim

1
h
n1
(X
k
) h
n
(X)


lim

h
n
(X
k
) 0
whose derivation for CW complexes in Theorem 3F.8 of [AT] applies equally well to
CW spectra. The term lim

h
n
(X
k
) is just the product

i
[X, K(G, n
i
)] from the nite
case, since the inverse limit of the nite products is the innite product. So it remains
to show that the lim

1
term vanishes.
We will use the Mittag-Lefer criterion, which says that lim

1
G
k
vanishes for a
sequence of homomorphisms of abelian groups G
2

2
G
1

1
G
0
if for each k
the decreasing chain of subgroups of G
k
formed by the images of the compositions
G
k+n
G
k
is eventually constant once n is sufciently large. This holds in the present
situation since the images of the maps H
i
(X
k+n
; G)H
i
(X
k
; G) are independent of
n when k +n > i. (When G = Z
p
these cohomology groups are nite so the groups
G
k
are all nite and the Mittag-Lefer condition holds automatically.)
The proof of the Mittag-Lefer criterion was relegated to the exercises in [AT], so
here is a proof. Recall that lim

G
k
and lim

1
G
k
are dened as the kernel and cokernel
of the map :

k
G
k

k
G
k
given by (g
k
) =

g
k

k+1
(g
k+1
)

, or in other words
as the homology groups of the two-term chain complex
0

k
G
k

k
G
k
0
Let H
k
G
k
be the image of the maps G
k+n
G
k
for large n. Then
k
takes
H
k
to H
k1
, so the short exact sequences 0H
k
G
k
G
k
/H
k
0 give rise to a
short exact sequence of two-term chain complexes and hence a six-term associated
long exact sequence of homology groups. The part of this we need is the sequence
lim

1
H
k

lim

1
G
k

lim

1
(G
k
/H
k
). The rst of these three terms vanishes since the
maps
k
: H
k
H
k1
are surjections, so it sufces to show that the third term van-
ishes. For the sequence of quotients G
k
/H
k
the associated groups H
k
are zero, so
it is enough to check that lim

1
G
k
= 0 when the groups H
k
are zero. In this case is
surjective since a given sequence (g
k
) is the image under of the sequence obtained
by adding to each g
k
the sum of the images in G
k
of g
k+1
, g
k+2
, , a nite sum if
H
k
= 0. L
16 Chapter 2 The Adams Spectral Sequence
2.2 The Spectral Sequence
Having established the basic properties of CW spectra that we will need, we begin
this section by lling in details of the sketch of the construction of the Adams spectral
sequence given in the introduction to this chapter. Then we examine the spectral
sequence as a tool for computing stable homotopy groups of spheres.
Constructing the Spectral Sequence
We will be dealing throughout with CW spectra that are connective and of nite
type. This assures that all homotopy and cohomology groups are nitely generated.
The coefcient group for cohomology will be Z
p
throughout, with p a xed prime. A
comment on notation: We will no longer have to consider the spaces X
n
that make up
a spectrum X, so we will be free to use subscripts to denote different spectra, rather
than the spaces in a single spectrum.
Let X be a connective CW spectrum of nite type. We construct a diagram

X
X X
K
0
K
1
K
2
1
K
0
= /

X X
2
K
1 1
= /

X X
3
K
2 2
= /
. . .
in the following way. Choose generators
i
for H

(X) as an A module, with at most

nitely many
i
s in each group H
k
(X). These determine a map XK
0
where K
0
is a wedge of Eilenberg-MacLane spectra, and K
0
has nite type. Replacing the map
XK
0
by an inclusion, we form the quotient X
1
= K
0
/X. This is again a connective
spectrum of nite type, so we can repeat the construction with X
1
in place of X. In
this way the diagram is constructed inductively. Note that even if X is the suspension
spectrum of a nite complex, as in the application to stable homotopy groups of
spheres, the subsequent spectra X
s
will no longer be of this special form.
The associated diagram of cohomology
0
0 0 0

0












. . .
X H 0 ( )

K H ( )

1
K H ( )

2
K H ( )

1
X H ( )

2
X H ( )

3
X H ( )

then gives a resolution of H

(X) by free A modules, by Proposition 2.5.

Now we x a nite spectrum Y and consider the functors
Y
t
(Z) = [
t
Y, Z].
Applied to the cobrations X
s
K
s
X
s+1
these give long exact sequences forming
a staircase diagram
The Spectral Sequence Section 2.2 17


t 1 + t 1 +
X
X X

t
s s 1 +
s
X
t
X X

t 1 t 1 s 2
s 1

s
s 2 s 1
s 1 +
s
s 1
s 1
X

t 1 +
K
t
K
Y Y Y Y Y
Y Y Y Y Y
Y Y Y Y Y

t 1
K
X
t 1 +
K
s 2 +
s
s 1 + t
K
t 1
K
t 1 +

t 1
X
so we have a spectral sequence, the Adams spectral sequence. The spectrum Y plays
a relatively minor role in what follows, and the reader is free to take it to be the
spectrum S
0
so that
Y
t
(Z) =
t
(Z). The groups
Y
t
(Z) are nitely generated when
Z is a connective spectrum of nite type, as one can see by induction on the number
of cells of Y .
There is another way of describing the construction of the spectral sequence
which provides some additional insight, although it involves nothing
more than a change in notation really. Let X
n
=
n
X
n
and K
n
=

n
K
n
. Then the earlier horizontal diagram starting with X can be
rewritten as a vertical tower as at the right. The spectra K
n
are again

X K

X K

X K
0
1
2
1
2
.
.
.
wedges of Eilenberg-MacLane spectra, so this tower is reminiscent of a
Postnikov tower. Let us call it an Adams tower for X. The staircase
diagram can now be rewritten in the following form:

t s
X
X X

s s 1 +
s
X
t s
X X

t s
s 2
X

t s 1 +
t s 1 +
+
K
s
K
Y Y Y Y Y
Y Y Y Y Y
Y Y Y Y Y

t s 1
s 2 s 1
s 1 s 1
s 1
K
X
K
s 1 + s 2 +
s
s
s 1 +
K
K
t s 1

t s + t s 1
t s t s 1 +
t s 1 t s t s 1 +
t s 1
X
This has the small advantage that the groups
Y
i
in each column all have the same
index i.
The E
1
and E
2
terms of the spectral sequence are easy to identify. Since K
s
is a
wedge of Eilenberg-MacLane spectra K
s,i
, elements of [Y, K
s
] are tuples of elements of
H

(Y), one for each summand K

s,i
, in the appropriate group H
n
i
(Y). Since H

(K
s
)
is free over A this means that the natural map [Y, K
s
]Hom
0
A
(H

(K
s
), H

(Y)) is
an isomorphism. Here Hom
0
denotes homomorphisms that preserve degree, i.e.,
dimension. Replacing Y by
t
Y , we obtain a natural identication
[
t
Y, K
s
] = Hom
0
A
(H

(K
s
), H

(
t
Y)) = Hom
t
A
(H

(K
s
), H

(Y))
where the superscript t denotes homomorphisms that lower degree by t . Thus if we
set E
s,t
1
=
Y
t
(K
s
), we have E
s,t
1
= Hom
t
A
(H

(K
s
), H

(Y)).
The differential d
1
:
Y
t
(K
s
)
Y
t
(K
s+1
) is induced by the map K
s
K
s+1
in the
resolution of X constructed earlier. This implies that the E
1
page of the spectral
18 Chapter 2 The Adams Spectral Sequence
sequence consists of the complexes
0 Hom
t
A
(H

(K
0
), H

(Y)) Hom
t
A
(H

(K
1
), H

(Y))
The homology groups of this complex are by denition Ext
s,t
A
(H

(X), H

(Y)), so we
have E
s,t
2
= Ext
s,t
A
(H

(X), H

(Y)).
Theorem 2.6. For X a connective CW spectrum of nite type, this spectral sequence
converges to
Y

(X) modulo torsion of order prime to p. In other words,

(a) For xed s and t the groups E
s,t
r
are independent of r once r is sufciently
large, and the stable groups E
s,t

are isomorphic to the quotients F

s,t
/F
s+1,t+1
for the ltration of
Y
ts
(X) by the images F
s,t
of the maps
Y
t
(X
s
)
Y
ts
(X),
or equivalently the maps
Y
ts
(X
s
)
Y
ts
(X).
(b)

n
F
s+n,t+n
is the subgroup of
Y
ts
(X) consisting of torsion elements of order
prime to p.
Thus we are ltering
Y
ts
(X) by how far its elements pull back in the Adams
tower. Unlike in the Serre spectral sequence this ltration is potentially innite, and
in fact will be innite if
Y
ts
(X) contains elements of innite order since all the
terms in the spectral sequence are nite-dimensional Z
p
vector spaces. Namely E
s,t
1
=
Hom
t
A
(H

(K
s
), H

(Y)) is certainly a nite-dimensional Z

p
vector space, so E
s,t
r
is as
well.
Throughout the proof we will be dealing only with connective CW spectra of nite
type, so we make this a standing hypothesis that will not be mentioned again.
A key ingredient in the proof will be an analog for spectra of the algebraic lemma
(Lemma 3.1 in [AT]) used to show that Ext is independent of the choice of free res-
olution. In order to state this we introduce some terminology. A sequence of maps
of spectra ZL
0
L
1
will be called a complex on Z if each composition of
two successive maps is nullhomotopic. If the L
i
s are wedges of Eilenberg-MacLane
spectra K(Z
p
, m
ij
) we call it an Eilenberg-MacLane complex. A complex for which
the induced sequence 0H

(Z)H

(L
0
) is exact is a resolution of Z.
Lemma 2.7. Suppose we are given the solid arrows in a diagram

Z L
0
0

L
1
L
2
. . .

X K
0
K
1
K
2
. . .

f f
1

f
2

f
where the rst row is a resolution and the second row is an Eilenberg-MacLane com-
plex. Then the dashed arrows can be lled in by maps f
i
: L
i
K
i
forming homotopy-
commutative squares.
Proof: Since the compositions in a complex are nullhomotopic we may start with an
enlarged diagram
The Spectral Sequence Section 2.2 19

Z
Z
L
0
L
1
L
2
L
0
/

Z L
1 1
Z
1
/

. . .

X
X
K
0
K
1
K
2
K
0
/ X K
1 1
X
1
/
. . .
. . .
. . .

Z
2
=
=

X
2
=
where the triangles are homotopy-commutative. The map XK
0
is equivalent to a
collection of classes
j
H

(X). Since H

(L
0
)H

(Z) is surjective by assumption,

there are classes
j
H

(L
0
) mapping to the classes f

(
j
) H

(Z). These
j
s
give a map f
0
: L
0
K
0
making a homotopy-commutative square with f . This square
induces a map L
0
/ZK
0
/X making another homotopy commutative square. The
exactness property of the upper row implies that the map H

(L
1
)H

(L
0
/Z) is
surjective, so we can repeat the argument with Z and X replaced by Z
1
= L
0
/Z and
X
1
= K
0
/X to construct the map f
1
, and so on inductively for all the f
i
s. L
Proof of Theorem 2.6: First we show statement (b). As noted earlier, all the terms
E
s,t
1
= Hom
t
A
(H

(K
s
), H

(Y)) in the staircase diagram are Z

p
vector spaces, so by
exactness all the vertical maps in the diagram are isomorphisms on non- p torsion.
This implies that the non- p torsion in
Y
ts
(X) is contained in

n
F
s+n,t+n
.
To prove the opposite inclusion we rst do the special case that

(X) is entirely
p torsion. These homotopy groups are then nite since we are dealing only with
connective spectra of nite type. We construct a special Eilenberg-MacLane complex
(not a resolution) of the form XL
0
L
1
in the following way. Let
n
(X) be
the rst nonvanishing homotopy group of X. Then let L
0
be a wedge of K(Z
p
, n)s
with one factor for each element of a basis for H
n
(X), so there is a map XL
0
inducing an isomorphismon H
n
. This map is also an isomorphismon H
n
, so on
n
it
is the map
n
(X)
n
(X)Z
p
by the Hurewicz theorem, which holds for connective
spectra. After converting the map XL
0
into an inclusion, the cober Z
1
= L
0
/X
then has
i
(Z
1
) = 0 for i n and
n+1
(Z
1
) is the kernel of the map
n
(X)
n
(L
0
),
which has smaller order than
n
(X). Now we repeat the process with Z
1
in place of
X to construct a map Z
1
L
1
inducing the map
n+1
(Z
1
)
n+1
(Z
1
)Z
p
on
n+1
,
so the cober Z
2
= L
1
/Z
1
has its rst nontrivial homotopy group
n+2
(Z
2
) of smaller
order than
n+1
(Z
1
). After nitely many steps we obtain Z
n+k
with
n+k
(Z
n+k
) = 0
as well as all the lower homotopy groups. At this point we switch our attention to

n+k+1
(Z
n+k
) and repeat the steps again. This innite process yields the complex
XL
0
L
1
.
It is easier to describe what is happening in this complex if we look at the associ-
ated tower Z
2
Z
1
X where Z
k
=
k
Z
k
. Here the rst map Z
1
X induces
an isomorphism on all homotopy groups except
n
, where it induces an inclusion of
a proper subgroup. The same is true for the next map Z
2
Z
1
, and after nitely
20 Chapter 2 The Adams Spectral Sequence
many steps this descending chain of subgroups
n
(Z
k
) becomes zero and we move
on to
n+1
(X), eventually reducing this to zero, and so on up the tower, killing each

i
(X) in turn. Thus for each i the groups
i
(Z
k
) are zero for all sufciently large
k. The same is true for the groups
Y
i
(Z
k
) when Y is a nite spectrum, since a map

i
YZ
k
can be homotoped to a constant map one cell at a time if all the groups

j
(Z
k
) vanish for j less than or equal to the largest dimension of the cells of
i
Y .
By the lemma the complex used to dene the spectral sequence maps to the com-
plex we have just constructed. This is equivalent to a map of towers, inducing a
commutative diagram
X X
1 2
X

. . .
X Z
1 2
Z

. . .

=
=
Y
i

Y
i

( )
( )
Y
i

Y
i

( )
( )
Y
i
( )
Y
i
( )
If an element of
Y
i
(X) pulled back arbitrarily far in the rst row, it would also pull
back arbitrarily far in the second row, but we have just seen this is impossible. Hence

n
F
s+n,t+n
is empty, which proves (b) in the special case that

(X) is all p torsion.

In the general case let be an element of
Y
n
(X) whose order is either innite
or a power of p. Then there is a positive integer k such that is not divisible by p
k
,
meaning that is not p
k
times any element of
Y
n
(X). Consider the map X
p
k
X
obtained by adding the identity map of X to itself p
k
times using the abelian group
structure in [X, X]. This map ts into a cobration X
p
k
X Z inducing a long
exact sequence
i
(X)
p
k

i
(X)
i
(Z) where the map p
k
is multipli-
cation by p
k
. From exactness it follows that

(Z) consists entirely of p torsion.

By the lemma the map XZ induces a map from the given Adams tower on X to a
chosen Adams tower on Z. The map
Y
n
(X)
Y
n
(Z) sends to a nontrivial element

Y
n
(Z) by our choice of and k, using exactness of
Y
n
(X)
p
k

Y
n
(X)
Y
n
(Z).
If pulled back arbitrarily far in the tower on X then would pull back arbitrarily
far in the tower on Z. This is impossible by the special case already proved. Hence
(b) holds in general.
To prove (a) consider the portion of the r
th
derived couple shown
in the diagram at the right. We claim rst that if r is sufciently

r
E
r
r
i
r
k
t s,
A
r

A
r

A
r

A
r
large then the vertical map i
r
is injective. For nontorsion
and non p torsion this follows from exactness since the E
columns are Z
p
vector spaces. For p torsion it follows from
part (b) that a term A
s,t
r
contains no p torsion if r is
sufciently large since A
s,t
r
consists of the elements
of A
s,t
1
that pull back r 1 units vertically.
Since i
r
is injective for large r , the preceding map
k
r
is zero, so the differential d
r
starting at E
s,t
r
is
zero for large r . The differential d
r
mapping to E
s,t
r
is also zero for large r since it
The Spectral Sequence Section 2.2 21
originates at a zero group, as all the terms in each E column of the initial staircase
diagram are zero below some point. Thus E
s,t
r
= E
s,t
r+1
for r sufciently large.
Since the map k
r
starting at E
s,t
r
is zero for large r , exactness implies that for
large r the group E
s,t
r
is the cokernel of the vertical map in the lower left corner of
the diagram. This vertical map is just the inclusion F
s+1,t+1

F
s,t
when r is large,
so the proof of (a) is nished. L
Stable Homotopy Groups of Spheres
For a rst application of the Adams spectral sequence let us consider the special
case that was one of the primary motivations for its construction, the problem of
computing stable homotopy groups of spheres. Thus we take X and Y both to be S
0
,
in the notation of the preceding section. We will focus on the prime p = 2, but we
will also take a look at the p = 3 case as a sample of what happens for odd primes.
Fixing p to be 2, here is a picture of an initial portion of the E
2
page of the
spectral sequence (the musical score to the harmony of the spheres?):
0
0
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
t s
s

h
0
h
2
h
2
2
h
3
2
h
2
3
h
3
h
4
h
2
h
4
h
The horizontal coordinate is t s so the i
th
column is giving information about
s
i
.
Each dot represents a Z
2
summand in the E
2
page, so in this portion of the page
there are only two positions with more than one summand, the (15, 5) and (18, 4)
positions. Referring back to the staircase diagram, we see that the differential d
r
goes one unit to the left and r units upward. The nonzero differentials are drawn
as lines sloping upward to the left. For t s 20 there are thus only six nonzero
differentials, but if the diagramwere extended farther to the right one would see many
more nonzero differentials, quite a jungle of them in fact.
For example, in the t s = 15 column we see six dots that survive to E

, which
says that the 2 torsion in
s
15
has order 2
6
. In fact it is Z
32
Z
2
, and this information
about extensions can be read off from the vertical line segments which indicate multi-
22 Chapter 2 The Adams Spectral Sequence
plication by 2 in
s

. So the fact that this column has a string of ve dots that survive
to E

and are connected by vertical segments means that there is a Z

32
summand of

s
15
, and the other Z
2
summand comes from the remaining dot in this column. In the
t s = 0 column there is an innite string of connected dots, corresponding to the
fact that
s
0
= Z, so iterated multiplication of a generator by 2 never gives zero. The
individual dots in this column are the successive quotients 2
n
Z/2
n+1
Z in the ltration
of Z by the subgroups 2
n
Z.
The line segments sloping upward to the right indicate multiplication by the el-
ement h
1
in the (1, 1) position of the diagram. We have drawn them mainly as a
visual aid to help tie together some of the dots into recognizable patterns. There is in
fact a graded multiplication in each page of the spectral sequence that corresponds to
the composition product in
s

. (This is formally like the multiplication in the Serre

spectral sequence for cohomology.) For example in the t s = 3 column we can read
off the relation h
3
1
= 4h
2
. To keep the diagram uncluttered we have not used line
segments to denote any other nonzero products, such as multiplication by h
2
, which
is nonzero in a number of cases.
The s = 1 row of the E
2
page consists of just the elements h
i
in the position
(2
i
1, 1). These are related to the Hopf invariant, and in particular h
1
, h
2
, and h
3
correspond to the classical Hopf maps. The next one, h
4
does not survive to E

, and
in fact the differential d
2
h
4
= h
0
h
2
3
is the rst nonzero differential in the spectral
sequence. It is easy to see why this differential must be nonzero: The element of

s
14
corresponding to h
2
3
must have order 2 by the commutativity property of the
composition product, since h
3
has odd degree, and there is no other term in the E
2
page except h
4
that could kill h
0
h
2
3
. No h
i
for i > 4 survives to E

either, but this

is a harder theorem, equivalent to Adams theorem on the nonexistence of elements
of Hopf invariant one.
There are only a few differentials to the left of the t s = 14 column that could
be nonzero since d
r
goes r units upward and r 2. It is easy to use the derivation
property d(xy) = x(dy) + (dx)y to see that these differentials must vanish. For
the element h
1
, if we had d
r
h
1
= h
r+1
0
then we would have d(h
0
h
1
) = h
r+2
0
nonzero
as well, but h
0
h
1
= 0. The only other differential which could be nonzero is d
2
on the element h
1
h
3
in the t s = 8 column, but d
2
h
1
and d
2
h
3
both vanish so
d
2
(h
1
h
3
) = 0.
Computing the E
2
page of the spectral sequence is a mechanical process, as we
will see, although its complexity increases rapidly as t s increases, so that even with
computer assistance the calculations that have been made only extend to values of
t s on the order of 100. Computing differentials is much harder, and not a purely
mechanical process, and the known calculations only go up to t s around 60.
Let us rst showthat for computing Ext
s,t
A
(H

(X), Z
p
) it sufces just to construct
The Spectral Sequence Section 2.2 23
a minimal free resolution of H

(X), that is, a free resolution

F
2

2
F
1

1
F
0

0
H

(X) 0
where at each step of the inductive construction of the resolution we choose the min-
imum number of free generators for F
i
in each degree.
Lemma 2.8. For a minimal free resolution, all the boundary maps in the dual complex
Hom
A
(F
2
, Z
p
)Hom
A
(F
1
, Z
p
)Hom
A
(F
0
, Z
p
)0
are zero, hence Ext
s,t
A
(H

(X), Z
p
) = Hom
t
A
(F
s
, Z
p
).
Proof: Let A
+
be the ideal in A consisting of all elements of strictly positive degree,
or in other words the kernel of the augmentation map AZ
p
given by projection onto
the degree zero part A
0
of A. Observe that Ker
i
A
+
F
i
since if we express an
element x Ker
i
of some degree in terms of a chosen basis for F
i
as x =

j
a
j
x
ij
with a
j
A, then if x is not in A
+
F
i
, some a
j
is a nonzero element of A
0
= Z
p
and
we can solve the equation 0 =
i
(x) =

j
a
j

i
(x
ij
) for
i
(x
ij
), which says that the
generator x
ij
was superuous.
Since
i1

i
= 0, we have
i
(x) Ker
i1
for each x F
i
, so from the
preceding paragraph we obtain a formula
i
(x) =

j
a
j
x
i1,j
with a
j
A
+
. Hence
for each f Hom
A
(F
i1
, Z
p
) we have

i
(f(x)) = f
i
(x) =

j
a
j
f(x
i1,j
) = 0
since a
j
A
+
and f(x
i1,j
) lies in Z
p
which has a trivial A module structure. L
Let us describe how to compute Ext
s,t
A
(Z
2
, Z
2
) by constructing a minimal resolu-
tion of Z
2
as an A module. An initial portion of the resolution is shown in the chart
on the next page. For the rst stage of the resolution F
0
Z
2
we must take F
0
to be a
copy of A with a generator in degree 0 mapping to the generator of Z
2
. This copy
of A forms the rst column of the table, which consists of the elements Sq
I
as Sq
I
ranges over the admissible monomials in A. The kernel of the map F
0
Z
2
consists
of everything in the rst column except , so we want the second column, which rep-
resents F
1
, to map onto everything in the rst column except . To start, we need an
element
1
at the top of the second column mapping to Sq
1
. (We will use subscripts
to denote the degree t , so
i
will have degree t = i, and similarly for the later gener-
ators
i
,
i
, .) Once we have
1
in the second column, we also have all the terms
Sq
I

1
for admissible I lower down in this column. To see what else we need in the
second column we need to compute how the terms in the second column map to the
rst column. Since
1
is sent to Sq
1
, we know that Sq
I

1
is sent to Sq
I
Sq
1
. The
product Sq
I
Sq
1
will be admissible unless I ends in 1, in which case Sq
I
Sq
1
will be
0 because of the Adem relation Sq
1
Sq
1
= 0. In particular, Sq
1

1
maps to 0. This
means we have to introduce a new generator
2
to map to Sq
2
. Then Sq
I

2
maps to
Sq
I
Sq
2
and we can use Adem relations to express this in terms of admissibles. For
24 Chapter 2 The Adams Spectral Sequence
example Sq
1

2
maps to Sq
1
Sq
2
= Sq
3
and Sq
2

2
maps to Sq
2
Sq
2
= Sq
3
Sq
1
.
Sq
t s
s
1
1
1
1 2 3 4 5 0
0
1
2
3
4
5
6
7

2 3

Sq
2

Sq
1

1
1
Sq
3

2
Sq
3

2
4

Sq
2

2
Sq
1

2
Sq
2

3
Sq
2

4
Sq
2

5
Sq
2

2
Sq
1

1
Sq
2

1
Sq
4

Sq
3

Sq
3 1

Sq
4

Sq
5

Sq
6

4
Sq
1

4
3
Sq
1

5
4
Sq
1

5
Sq
1

,
Sq
2 1

,
Sq
2 1 ,
2
2
Sq
3

4
Sq
1

5
Sq
1

2
Sq
4

4
Sq
2

Sq
2 1 ,
3
3
Sq
3

6
Sq
1

Sq
2 1 ,
3
3
Sq
4

Sq
3 1 ,
4
4
Sq
3

Sq
2 1 ,
5
5
Sq
3

Sq
2 1 ,
4
4
Sq
4

Sq
3 1 ,
5
5
Sq
4

Sq
3 1 ,
6
Sq
2

6
Sq
3

3
3
Sq
5

Sq
4 1 ,
3
3
Sq
6

Sq
4 2 ,
3

10

Sq
6 1 ,
3
Sq
5 1 ,
6
Sq
2 1 ,
4
4
Sq
5

Sq
4 1 ,
5
5
Sq
5

Sq
4 1 ,
4
4
Sq
6

Sq
4 2 ,
5
5
Sq
6

Sq
4 2 ,
4

Sq
5 1 ,
4

11

Sq
6 1 ,
5

Sq
5 1 ,
1
Sq
3 1 ,
2
Sq
3 1 ,
5
Sq
2

2
Sq
5

2
Sq
6

2
Sq
7

4
Sq
3

4
Sq
4

2
Sq
4 1 ,
2
Sq
5 1 ,
2
Sq
6 1 ,
2
Sq
4 2 ,
4
Sq
2 1 ,
5
Sq
3

4
Sq
3 1 ,
5
Sq
3 1 ,
5
Sq
2 1 ,
4
Sq
5

4
Sq
4 1 ,
2
Sq
2 1 ,
1
Sq
5

2
Sq
4

4
Sq
2

1
Sq
4 1 ,
2
Sq
3 1 ,
1
Sq
6

2
Sq
5

4
Sq
3

1
Sq
4 2 ,
1
Sq
4 2 , 1 ,
2
Sq
4 2 , 1 ,
3
Sq
4 2 , 1 ,
4
Sq
4 2 , 1 ,
1
Sq
5 1 ,
1
Sq
7

2
Sq
6

4
Sq
4

1
Sq
5 2 ,
1
Sq
6 1 ,
2
Sq
4 1 ,
2
Sq
4 2 ,
2
Sq
5 1 ,
4
Sq
2 1 ,
4
Sq
3 1 ,
Sq
4 1

,
Sq
4 2

,
Sq
5 1

,
Sq
7

Sq
6 1

,
Sq
4 2

, 1 ,
Sq
5 2

,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Spectral Sequence Section 2.2 25
Some of the simpler Ademrelations, enough to do the calculations shown in the chart,
are listed in the following chart.
Sq
1
Sq
2n
= Sq
2n+1
Sq
3
Sq
4n
= Sq
4n+3
Sq
1
Sq
2n+1
= 0 Sq
3
Sq
4n+1
= Sq
4n+2
Sq
1
Sq
2
Sq
4n
= Sq
4n+2
+Sq
4n+1
Sq
1
Sq
3
Sq
4n+2
= 0
Sq
2
Sq
4n+1
= Sq
4n+2
Sq
1
Sq
3
Sq
4n+3
= Sq
4n+5
Sq
1
Sq
2
Sq
4n+2
= Sq
4n+3
Sq
1
Sq
4
Sq
3
= Sq
5
Sq
2
Sq
2
Sq
4n+3
= Sq
4n+5
+Sq
4n+4
Sq
1
Sq
4
Sq
4
= Sq
7
Sq
1
+Sq
6
Sq
2
Note that the relations for Sq
3
Sq
i
follow from the relations for Sq
2
Sq
i
and Sq
1
Sq
i
since Sq
3
= Sq
1
Sq
2
.
Moving down the s = 1 column we see that we need a new generator
4
to map
to Sq
4
. In fact it is easy to see that the only generators we need in the second column
are
2
n s mapping to Sq
2
n
. This is because Sq
i
is indecomposable iff i = 2
n
, which
implies inductively that every Sq
I
except Sq
2
n
will be hit by previously introduced
terms, while Sq
2
n
will not be hit.
Now we start to work our way down the third column, introducing the minimum
number of generators necessary to map onto the kernel of the map from the second
column to the rst column. Thus, near the top of this column we need
2
mapping to
Sq
1

1
,
4
mapping to Sq
3

1
+Sq
2

2
, and
5
mapping to Sq
4

1
+Sq
2
Sq
1

2
+Sq
1

4
.
One can see that things are starting to get more complicated here, and it is not easy
to predict where new generators will be needed.
Subsequent columns are computed in the same way. Near the top, the structure
of the columns soon stabilizes, each column looking just the same as the one before.
This is fortunate since it is the rows, with t s constant, that we are interested in for
computing
s
ts
. The most obvious way to proceed inductively would be to compute
each diagonal with t constant by induction on t , moving up the diagonal from left
to right. However, this would require innitely many computations to determine a
whole row. To avoid this problem we can instead proceed row by row, moving across
each row from left to right assuming that higher rows have already been computed.
To determine whether a new generator is needed in the (s, t s) position we need to
see whether the map from the (s 1, t s +1) position to the (s 2, t s +2) position
is injective. These two positions are below the row we are working on, so we do not
yet know whether any new generators are required in these positions, but if they are,
they will have no effect on the kernel we are interested in since minimality implies
that new generators always generate a subgroup that maps injectively. Thus we have
enough information to decide whether new generators are needed in the (s, t s)
position, and so the induction can continue.
The chart shows the result of carrying out the row-by-row calculation through
the row t s = 5. As it happens, no new generators are needed in this row or the
26 Chapter 2 The Adams Spectral Sequence
preceding one. In the next row t s = 6 one new generator
8
will be needed, but the
chart does not show the computations needed to justify this. And in the t s = 7 row
four new generators
8
,
9
,
10
, and
11
will be needed. The reader is encouraged
to do some of these calculations to get a real feeling for what is involved. Most of the
work involves applying Ademrelations, and then when the maps have been computed,
their kernels need to be determined.