The reductive Borel--Serre compactification as a model for the K-theory space

Abstract: The reductive Borel--Serre compactification, introduced by Zucker in 1982, is a stratified space which is well suited for the study of $L^2$-cohomology of arithmetic groups and has come to play a central role in the theory of compactifications. We determine its stratified homotopy type (the exit path $\infty$-category) to be a $1$-category defined purely in terms of parabolic subgroups. This category makes sense in a much more general setting, in fact for any exact category, but in this talk we restrict ourselves to well-behaved rings. With direct sum, these naturally give rise to a monoidal category, and we show that (the loop space of the classifying space of) this monoidal category is a model for the K-theory space. For finite fields, we encounter much better homological stability properties than for the general linear groups.

commutative algebraalgebraic geometryalgebraic topologygeometric topologyK-theory and homology

Audience: researchers in the topic

electronic Algebraic K-theory Seminar

Series comments: Description: Research seminar on algebraic K-theory