Adams spectral sequence

In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank 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. In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank 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 ) {displaystyle H^{*}(X)} are understood to mean H ∗ ( X ; Z / p Z ) {displaystyle H^{*}(X;mathbb {Z} /pmathbb {Z} )} . 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 , Y ] {displaystyle } 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 } 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 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 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 , 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. For X and Y spaces of finite type, with X a finite dimensional CW-complex, there is a spectral sequence, called the classical Adams spectral sequence, converging to the p-torsion in , with E2-term given by and differentials of bidegree (r, r − 1).