Adams 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.

Motivation

For everything below, we need to once and for all 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.

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 [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

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 [X, Y], with E₂-term given by

E₂^t,s = Ext_A^t,s(H^*(Y), H^*(X)),

and differentials of bidegree (r, r − 1).

Calculations

The sequence itself is not an algorithmic device, but lends itself to problem solving in particular cases.

Adams' original use for his spectral sequence was the first proof of the Hopf invariant 1 problem: $\mathbb{R}^n$ 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, Milnor and Novikov used the Adams spectral sequence to compute the coefficient ring of complex cobordism. Further, Milnor and 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.

Generalizations

The Adams–Novikov spectral sequence is a generalization of the Adams spectral sequence introduced by Novikov (1967) 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.

References

Adams, J. Frank (1958), "On the structure and applications of the Steenrod algebra", Commentarii Mathematici Helvetici 32 (1): 180–214, doi:10.1007/BF02564578, ISSN 0010-2571, MR 0096219
Adams, J. Frank (1964), Stable homotopy theory, Lecture notes in mathematics 3, Berlin–Göttingen–Heidelberg–New York: Springer-Verlag, MR 0185597
Botvinnik, Boris (1992), Manifolds with Singularities and the Adams–Novikov Spectral Sequence, London Mathematical Society Lecture Note Series, Cambridge: Cambridge Univ. Press, ISBN 0-521-42608-1
McCleary, John (February 2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics 58 (2nd ed.), Cambridge University Press, doi:10.2277/0521567599, ISBN 978-0-521-56759-6, MR 1793722
Novikov, S. (1967), "Methods of algebraic topology from the point of view of cobordism theory", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya (in Russian) 31: 855–951
Ravenel, Douglas C. (1978), "A novice's guide to the Adams–Novikov spectral sequence", in Barratt, M. G.; Mahowald, Mark E., Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math. 658, Berlin, New York: Springer-Verlag, pp. 404–475, doi:10.1007/BFb0068728, ISBN 978-3-540-08859-2, MR 513586
Ravenel, Douglas C. (2003), Complex cobordism and stable homotopy groups of spheres (2nd ed.), AMS Chelsea, ISBN 978-0-8218-2967-7, MR 0860042 .

External links

Hatcher, Book chapter (PDF)