Algebra/Topology seminar

Speaker: Akhil Matthew (University of California, Berkeley) 

Title: Derived induction and restriction theory

Abstract:  Let G be a finite group. Then it is a classical result of Quillen that the group cohomology of G can be recovered, up to nilpotence (more precisely, a relation called F-isomorphism), from the cohomology of its elementary abelian subgroups. Using ideas from descent theory, we introduce a class of genuinely equivariant spectra that can be recovered in a strong homotopical sense from their restrictions to a given family of subgroups. We obtain large classes of examples of such equivariant spectra and show in particular show that complex-oriented theories have this property with respect to the family of abelian subgroups. As a result, we obtain (by decategorifying) analogs of the F-isomorphism theorem as well as analogs of Artin and Brauer induction for complex-oriented theories. As one of our applications, we obtain Galois-type descent statements in "multiplicative" theories such as algebraic K-theory and THH.