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.

It is proved that the assembly map in algebraic K‐ and L‐theory with respect to the family of finite subgroups is injective for groups normalΓ with finite quotient finite decomposition complexity (a strengthening of finite decomposition complexity introduced by Guentner, Tessera and Yu) that admit a finite‐dimensional model for normalE̲Γ and have an upper bound on the order of their finite subgroups. In particular, this applies to finitely generated linear groups over fields with characteristic zero with a finite‐dimensional model for normalE̲Γ.

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

We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w_2-type and their equivariant intersection forms are stably isometric. We also find explicit algebraic invariants that determine the stable classification for spin manifolds in this class.Comment: 55 pages, 2 figure

Abstract. Let X be a proper metric space and let F be a finite group acting on X by isometries. We show that the asymptotic dimension of F \X is the same as the asymptotic dimension of X.