Adams spectral sequence

In mathematics, the Adams spectral sequence is a spectral sequence introduced by 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.
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 formula_1 are understood to mean formula_2.
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 formula_3, these maps form the "n"th homotopy group of "Y". A more reasonable (but still very difficult!) goal is to understand the set formula_4 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 formula_4 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 formula_4, we look at cohomology: send formula_4 to Hom("H"*("Y"), "H"*("X")). This is a good idea because cohomology groups are usually tractable to compute.
The key idea is that formula_1 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 F"p". But we can now consider the derived functors of Hom in the category of "A"-modules, Ext"A""r"("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.
Classical formulation.
Formulation for computing homotopy groups of spectra.
The classical Adams spectral sequence can be stated for any connective spectrum formula_9 of finite type, meaning formula_10 for formula_11 and formula_12 is a finitely generated Abelian group in each degree. Then, there is a spectral sequence formula_13 such that
Note this implies for formula_20, this computes the p-torsion of the homotopy groups of the sphere spectrum, i.e. the stable homotopy groups of the spheres. Also, because for any CW-complex formula_21 we can consider the suspension spectrum formula_22, this gives the statement of the previous formulation as well.
This statement generalizes a little bit further by replacing the formula_23-module formula_24 with the cohomology groups formula_25 for some connective spectrum formula_21 (or topological space formula_21). This is because the construction of the spectral sequence uses a "free" resolution of formula_1 as an formula_23-module, hence we can compute the Ext groups with formula_25 as the second entry. We therefore get a spectral sequence with formula_31-page given byformula_32which has the convergence property of being isomorphic to the graded pieces of a filtration of the formula_16-torsion of the stable homotopy group of homotopy classes of maps between formula_9 and formula_21, that isformula_36
Spectral sequence for the stable homotopy groups of spheres.
For example, if we let both spectra be the sphere spectrum, so formula_37, then the Adams spectral sequence has the convergence propertyformula_38giving a technical tool for approaching a computation of the stable homotopy groups of spheres. It turns out many of the first terms can be computed explicitly from purely algebraic informationpp 23–25. Also note we can rewrite formula_39, so the formula_31-page isformula_41We include this calculation information below for formula_42.
Ext terms from the resolution.
Given the adams resolutionformula_43we have the formula_44 terms asformula_45for the graded Hom groups. Then, the formula_44-page can be written asformula_47so the degree of formula_48 can be thought of how "deep" in the Adams resolution we go before we can find the generators.
Calculations.
The sequence itself is not an algorithmic device, but lends itself to problem solving in particular cases.
Examples with Eilenberg–Maclane spectra.
Some of the simplest calculations are with Eilenberg–Maclane spectra such as formula_49 and formula_50. For the first case, we have the formula_44 pageformula_52giving a collapsed spectral sequence, hence formula_53. This can be rewritten asformula_54giving the formula_31-page. For the other case, note there is a cofiber sequenceformula_56which ends up giving a splitting in cohomology, so formula_57 as formula_23-modules. Then, the formula_31-page of formula_60 can be read asformula_61Interestingly, with this computation the only way for the spectral sequence to converge to the expected formula_62-page havingformula_63is if there are non-trivialformula_64for every formula_65.
Other applications.
Adams' original use for his spectral sequence was the first proof of the Hopf invariant 1 problem: formula_66 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 formula_37 we can compute several terms explicitly, giving some of the first stable homotopy groups of spheres. For formula_42 this amounts to looking at the formula_31-page withformula_70This can be done by first looking at the Adams resolution of formula_71. Since formula_71 is in degree formula_73, we have a surjectionformula_74where formula_75 has a generator in degree formula_73 denoted formula_77. The kernel formula_78 consists of all elements formula_79 for admissible monomials formula_80 generating formula_75, hence we have a mapformula_82and we denote each of the generators mapping to formula_83 in the direct sum as formula_84, and the rest of the generators as formula_85 for some formula_86. For example,formula_87Notice that the last two elements of formula_84 map to the same element, which follows from the Adem relations. Also, there are elements in the kernel, such as formula_89 sinceformula_90because of the Adem relation. We call the generator of this element in formula_91, formula_92. We can apply the same process and get a kernel formula_93, resolve it, and so on. When we do, we get an formula_44-page which looks likeformula_95
which can be expanded by computer up to degree formula_96 with relative ease. Using the found generators and relations, we can calculate the formula_31-page with relative easy. Sometimes homotopy theorists like to rearrange these elements by having the horizontal index denote formula_48 and the vertical index denote formula_99 giving a different type of diagram for the formula_31-pagepg 21. See the diagram above for more information.
Generalizations.
The Adams–Novikov spectral sequence is a generalization of the Adams spectral sequence introduced by 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.