Midwest Topology Seminar

Northwestern University

April 24, 2010

Abstract: I will survey several different approaches to explicit computations of motivic stable homotopy groups: geometric constructions, Toda brackets, and Adams spectral sequences. The motivic Adams spectral sequence over C presents some curious exotic phenomena starting in the 25-stem. Z/2-equivariant stable homotopy groups will play a role in the discussion, since they are related to motivic stable homotopy groups over R.

Abstract: This a report on work in progress with K. Hess. Blumberg, Gepner, and Tabuada, in have constructed a category which seems to accomodate both the motives of algebraic geometry (defined via derived categories of qcoh sheaves) and the objects of differential topology (defined in terms of categories of modules over the functional duals of finite complexes). Their category is naturally enriched over the K-theory of the sphere spectrum S and it is plausible that some version of it is homotopically Tannakian over the category of S-modules, with an appropriate Hopf ring-spectrum as `motivic group'. Over the rationals, this can be identified with (the dual to) the universal enveloping algebra of a free Lie algebra on odd-degree generators, called the `cosmic Galois group' by arithmetic geometers