Motivic Euler characteristics and the Motivic Segal-Becker theorem (Remotely)

Roy Joshua (The Ohio State University)

A well-known and very useful result in algebraic topology is the statement that the Euler characteristic of G/N(T) in singular cohomology is 1, where G is a compact Lie group and N(T) is the normalizer of a maximal torus. In the presence of a transfer map as constructed by Becker and Gottlieb the above result shows that in any generalized cohomology theory the classifying space of G is a split summand of the classifying space of N(T).

Based on this, Fabien Morel made a conjecture that an analogous motivic Euler characteristic for a split reductive group G over a field k and N(T) the normalizer of a split maximal torus is 1. We will sketch a proof of this conjecture in the first part of the talk under the assumption the base field has a square root of -1. In the second part of the talk we will apply this result to prove what we call a motivic Segal-Becker theorem for Algebraic K-Theory.

All of this is based on joint work with Gunnar Carlsson and Pablo Pelaez.