Equivariant coarse homotopy theory and coarse algebraic K-homology

Abstract

We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the category of equivariant coarse motivic spectra.As examples of equivariant coarse homology theories we discuss equivariant coarse ordinary homology and equivariant coarse algebraic K-homology.Moreover, we discuss the cone functor, its relation with equivariant homology theories in equivariant topology, and assembly and forget-control maps. This is a preparation for applications in subsequent papers aiming at split-injectivity results for the Farrell-Jones assembly map.

Coarse homology theories and finite decomposition complexity

“…Many examples of equivariant coarse homology theories are strongly additive. The following has been shown in [BEKW17]. The arguments of the present paper use a weaker form of additivity we will now introduce.…”

“…We set I j := I \ {j}. Then (X j , free i∈I j X i ) is a coarsely excisive decomposition (see [BEKW17,Def. 4.12]) of free i∈I X i .…”

“…2]. In [BEKW17,Sec. 3 & 4] we introduced the notion of an equivariant coarse homology theory and constructed the universal equivariant coarse homology theory Yo s : GBornCoarse → GSpX whose target is the presentable stable ∞-category of equivariant coarse motivic spectra.…”

“…To every G-bornological coarse space X one can functorially associate the motivic forgetcontrol map β X : F ∞ (X) → ΣF 0 (X) (1.1) which is a morphism in GSpX (see [BEKW17,Def. 11.10] and (1.3) below).…”

“…In the case that X is the group G with the canonical bornological coarse structure and the action by left multiplication (we often denote this object by G can,min ), the forget-control map is closely related to the assembly map which appears in Farrell-Jones type conjectures, see [BEKW17,Sec. 11.2] for details.…”