In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.
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In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.
Contents
Motivation
For everything below, once and for all, we fix a prime p. All spaces are assumed to be CW complexes. The ordinary cohomology groups
The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces X and Y. This is extraordinarily ambitious: in particular, when X is
S
n
{\displaystyle S^{n}}
, these maps form the nth homotopy group of Y. A more reasonable (but still very difficult!) goal is to understand the set
[
X
Classical formulation
Formulation for computing homotopy groups of spectra
The classical Adams spectral sequence can be stated for any connective spectrum
X
{\displaystyle X}
of finite type, meaning
π
i
(
X
)
=
0
{\displaystyle \pi _{i}(X)=0}
for
i
<
0
{\displaystyle i<0}
and
π
i
(
X
)
{\displaystyle \pi _{i}(X)}
is a finitely generated Abelian group in each degree. Then, there is a spectral sequence
E
∗
∗
,
∗
(
Spectral sequence for the stable homotopy groups of spheres
For example, if we let both spectra be the sphere spectrum, so
X
=
Y
=
S
{\displaystyle X=Y=\mathbb {S} }
, then the Adams spectral sequence has the convergence property
Some of the simplest calculations are with Eilenberg–Maclane spectra such as
X
=
H
Z
{\displaystyle X=H\mathbb {Z} }
and
X
=
H
Z
/
(
p
k
)
{\displaystyle X=H\mathbb {Z} /(p^{k})}
. For the first case, we have the
E
1
{\displaystyle E_{1}}
page
E
1
s
,
t
=
{
Z
/
p
if
Other applications
Adams' original use for his spectral sequence was the first proof of the Hopf invariant 1 problem:
R
n
{\displaystyle \mathbb {R} ^{n}}
admits a division algebra structure only for n = 1, 2, 4, or 8. He subsequently found a much shorter proof using cohomology operations in K-theory.
The Thom isomorphism theorem relates differential topology to stable homotopy theory, and this is where the Adams spectral sequence found its first major use: in 1960, John Milnor and Sergei Novikov used the Adams spectral sequence to compute the coefficient ring of complex cobordism. Further, Milnor and C. T. C. Wall used the spectral sequence to prove Thom's conjecture on the structure of the oriented cobordism ring: two oriented manifolds are cobordant if and only if their Pontryagin and Stiefel–Whitney numbers agree.
Stable homotopy groups of spheres
Using the spectral sequence above for
X
=
Y
=
S
{\displaystyle X=Y=\mathbb {S} }
we can compute several terms explicitly, giving some of the first stable homotopy groups of spheres. For
p
=
2
{\displaystyle p=2}
this amounts to looking at the
E
2
{\displaystyle E_{2}}
-page with
E
2
s
,
t
=
Ext
A
2
s
,
t
(
Z
/
Generalizations
The Adams–Novikov spectral sequence is a generalization of the Adams spectral sequence introduced by Novikov (1967) where ordinary cohomology is replaced by a generalized cohomology theory, often complex bordism or Brown–Peterson cohomology. This requires knowledge of the algebra of stable cohomology operations for the cohomology theory in question, but enables calculations which are completely intractable with the classical Adams spectral sequence.
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,
Y
]
{\displaystyle [X,Y]}
of maps (up to homotopy) that remain after we apply the suspension functor a large number of times. We call this the collection of stable maps from X to Y. (This is the starting point of stable homotopy theory; more modern treatments of this topic begin with the concept of a spectrum. Adams' original work did not use spectra, and we avoid further mention of them in this section to keep the content here as elementary as possible.)
The set
[
X
,
Y
]
{\displaystyle [X,Y]}
turns out to be an abelian group, and if X and Y are reasonable spaces this group is finitely generated. To figure out what this group is, we first isolate a prime p. In an attempt to compute the p-torsion of
[
X
,
Y
]
{\displaystyle [X,Y]}
, we look at cohomology: send
[
X
,
Y
]
{\displaystyle [X,Y]}
to Hom(H*(Y), H*(X)). This is a good idea because cohomology groups are usually tractable to compute.
The key idea is that
H
∗
(
X
)
{\displaystyle H^{*}(X)}
is more than just a graded abelian group, and more still than a graded ring (via the cup product). The representability of the cohomology functor makes H*(X) a module over the algebra of its stable cohomology operations, the Steenrod algebra A. Thinking about H*(X) as an A-module forgets some cup product structure, but the gain is enormous: Hom(H*(Y), H*(X)) can now be taken to be A-linear! A priori, the A-module sees no more of [X, Y] than it did when we considered it to be a map of vector spaces over Fp. But we can now consider the derived functors of Hom in the category of A-modules, ExtAr(H*(Y), H*(X)). These acquire a second grading from the grading on H*(Y), and so we obtain a two-dimensional "page" of algebraic data. The Ext groups are designed to measure the failure of Hom's preservation of algebraic structure, so this is a reasonable step.
The point of all this is that A is so large that the above sheet of cohomological data contains all the information we need to recover the p-primary part of [X, Y], which is homotopy data. This is a major accomplishment because cohomology was designed to be computable, while homotopy was designed to be powerful. This is the content of the Adams spectral sequence.
giving a technical tool for approaching a computation of the stable homotopy groups of spheres. It turns out that many of the first terms can be computed explicitly from purely algebraic informationpp 23–25. Also note that we can rewrite
{\displaystyle E_{\infty }^{s,t}={\begin{cases}\mathbb {Z} /p^{k}&{\text{ if }}t=s\\0&{\text{ otherwise }}\end{cases}}}
.The only way for this spectral sequence to converge to this page is if is there are non-trivial differentials supported on every element with Adams grading
