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

2014/12/10 by Gunnar Carlsson, Carlsson, Gunnar, Boris Goldfarb +1
Mathematics · #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.1412.3483

openalex publication_date 2014/12/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

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