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. For everything below, once and for all, we fix a prime p. All spaces are assumed to be CW complexes. The ordinary cohomology groups are understood to mean . 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 , these maps form the nth homotopy group of Y. A more reasonable (but still very difficult!) goal is to understand the set 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 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 , we look at cohomology: send to Hom(H*(Y), H*(X)). This is a good idea because cohomology groups are usually tractable to compute. The key idea is that 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.