An identification of the Baum-Connes and Davis-L\"uck assembly maps

TL;DR

This paper establishes an equivalence between two homotopy theoretical and category theoretical constructions of the assembly map in the Baum-Connes conjecture, extending previous results to arbitrary coefficients.

Contribution

It identifies the Davis-Lück homotopy construction with Meyer and Nest's category theoretical approach, generalizing prior work to include arbitrary coefficients.

Findings

Confirmed the equivalence of the two assembly map constructions

Extended the identification to arbitrary coefficients

Used abstract properties for the proof

Abstract

The Baum-Connes conjecture predicts that a certain assembly map is an isomorphism. We identify the homotopy theoretical construction of the assembly map by Davis and L\"uck with the category theoretical construction by Meyer and Nest. This extends the result of Hambleton and Pedersen to arbitrary coefficients. Our approach uses abstract properties rather than explicit constructions and is formally similar to Meyer's and Nest's identification of their assembly map with the original construction of the assembly map by Baum, Connes and Higson.