Adams spectral sequence
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.
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.
For everything below, once and for all, we fix a prime p. All spaces are assumed to be CW complexes.
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.
, these maps form the nth homotopy group of Y.
of maps (up to homotopy) that remain after we apply the suspension functor a large number of times.
(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.)
To figure out what this group is, we first isolate a prime p. In an attempt to compute the p-torsion of
This is a good idea because cohomology groups are usually tractable to compute.
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)).
The Ext groups are designed to measure the failure of Hom's preservation of algebraic structure, so this is a reasonable step.
The classical Adams spectral sequence can be stated for any connective spectrum
is a finitely generated Abelian group in each degree.
which has the convergence property of being isomorphic to the graded pieces of a filtration of the
, then the Adams spectral sequence has the convergence property
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 information[2]pp 23–25.
can be thought of how "deep" in the Adams resolution we go before we can find the generators.
The sequence itself is not an algorithmic device, but lends itself to problem solving in particular 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
Adams' original use for his spectral sequence was the first proof of the Hopf invariant 1 problem:
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.
we can compute several terms explicitly, giving some of the first stable homotopy groups of spheres.
Sometimes homotopy theorists like to rearrange these elements by having the horizontal index denote
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.
