Injectivity results for coarse homology theories

Abstract

We show injectivity results for assembly maps using equivariant coarse homology theories with transfers. Our method is based on the descent principle and applies to a large class of linear groups or, more generally, groups with finite decomposition complexity. Contents

Split Injectivity ofA-Theoretic Assembly Maps

“…For more details on this and a slightly more general result see [BEKWb, Theorem 1.11]. We also obtain the following A-theoretic analog of [BEKWb,Theorem 1.15]. 1.4.…”

“…Recall from [BEKWb,Definition 1.5] that the family of subgroups FDC consists of those subgroups H of G such that the family {F \H | F ≤ H finite} has finite decomposition complexity as defined by Guentner, Tessera and Yu in [GTY12,GTY13].…”

On the Farrell-Jones conjecture for localising invariants

“…Throughout this article, we will make use of the language of bornological coarse spaces introduced in [BEKW20a]. See [BEKW20c,Sec. 3] for a concise summary of the key notions.…”

“…For the complete proof, we also require the concept of an equivariant coarse homology theory [BEKW20c,Def. 3.13], but this will only be used in Section 2.5.…”

“…In this remark we assume that E : GBC → M is an equivariant coarse homology theory. Then for Y in GBC the twist E Y : GBC → M by Y is again an equivariant coarse homology theory [BEKW20c,Lem. 3.16].…”