Adams–Novikov spectral sequence
In mathematics, the Adams spectral sequence is a spectral sequence introduced by Adams (1958). 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 H*(X) are understood to mean H*(X; Z/pZ).
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 Sn, these maps form the nth homotopy group of Y. A more reasonable (but still very difficult!) goal is to understand [X, Y], the 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.)
[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], we look at cohomology: send [X, Y] to Hom(H*(Y), H*(X)). This is a good idea because cohomology groups are usually tractable to compute.
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