Date:
Mon, 15/12/2014 - 16:30 - 18:15
Let G be a finite group. Artin's theorem says that we can recover the complex representation ring of G from the representations of the cyclic subgroups of G up to torsion, or additive nilpotence. Quillen's F-isomorphism theorem says that we can recover the mod-p cohomology of G from the mod-p cohomology of the elementary abelian p-subgroups of G up to multiplicative nilpotence.
This analysis fits into Dress's framework for induction/restriction theory for Mackey functors. These particular Mackey functors appear as the homotopy groups of G-spectra and it is relatively straightforward to lift Dress's framework to G-spectra. Here the central question is: can we recover a G-spectrum from it's underlying H-spectra as H varies over some fixed (small) family of subgroups of G.
There are well-known characterizations of when such a result holds, but by asking for a stronger condition we can deduce induction and restriction results for the homotopy groups as well. In particular we can prove generalizations of the theorems of Artin and Quillen. Our theory applies to genuine equivariant complex and real K-theory (extending Artin's theorem and a result of Fausk), and the Borel equivariant cohomology theories associated to mod-p cohomology (extending the result of Quillen), integral cohomology (extending a result of Carlson), complex oriented theories (extending a result of Hopkins-Kuhn-Ravenel), ko, the many variants of topological modular forms, L_n-local spectra, and classical cobordism theories.
Along the way, we show that the class of G-spectra such that our strong induction/restriction theorems hold is a triangulated subcategory closed under retracts and (co)tensoring with an arbitrary G-spectrum.
This is joint work with Akhil Mathew and Niko Naumann.
