Assembly maps

Abstract

We introduce and analyze the concept of an assembly map from the original homotopy theoretic point of view. We give also interpretations in terms of surgery theory, controlled topology and index theory. The motivation is that prominent conjectures of Farrell-Jones and Baum-Connes about K-and L-theory of group rings and group C * -algebras predict that certain assembly maps are weak homotopy equivalences.

Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

“…is an equivalence, where Vcyc is the family of virtually cyclic subgroups and we calculated the colimit in the target of (1.1) using that * is the final object of G All Orb. The Farrell-Jones conjecture is known in many cases, see [Lüc,Sec. 7.7] for an overview.…”

On the Farrell-Jones conjecture for localising invariants

“…For example, it implies the Borel and the Novikov conjecture. For more background information on the linear Farrell-Jones conjecture we refer to the surveys [LR05,Lüc20,RV18] as well as Lück's ongoing book project [Lüc].…”

Coassembly is a homotopy limit map

“…It is formally dual to the assembly map of [WW95a,DL98], which by [HP04,DL98] coincides with the assembly map of the Farrell-Jones conjecture [FJ93]. A comprehensive recent survey on assembly maps is given in [Lüc19]. The coassembly map is also a close analog of the linear approximation map of embedding calculus [Wei99,GW99].…”