Equivariant stable homotopy methods in the algebraic K-theory of infinite groups

TL;DR

This paper reviews the development of equivariant homotopy methods in algebraic K-theory, highlighting their role in proving conjectures like the Borel and Farrell-Jones conjectures for infinite groups.

Contribution

It provides a comprehensive overview of the evolution and advantages of equivariant homotopy techniques in algebraic K-theory and their application to major conjectures.

Findings

Outline of the proof of the Borel conjecture for groups of finite asymptotic dimension

Discussion of the relation to the Farrell-Jones conjecture

Explanation of the evolution of equivariant homotopy methods

Abstract

Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the topological rigidity of compact aspherical manifolds. Our goal is to strip the basic idea to the core and follow the evolution over time in order to explain the advantages of the flexible state that exists today. We end with an outline of the proof of the Borel conjecture in algebraic K-theory for groups of finite asymptotic dimension. We also discuss the relation of these methods to the recent work on the Farrell-Jones conjecture.