Equivariant coarse homotopy theory and coarse algebraic $\boldsymbol{K}$-homology

TL;DR

This paper develops an axiomatic framework for equivariant coarse homology theories, introduces a universal theory, and explores examples like equivariant coarse algebraic K-homology, setting the stage for applications in algebraic topology.

Contribution

It introduces the category of equivariant bornological coarse spaces and constructs the universal equivariant coarse homology theory valued in equivariant coarse motivic spectra.

Findings

Defined the category of equivariant bornological coarse spaces.

Constructed the universal equivariant coarse homology theory.

Discussed applications to Farrell-Jones assembly map.

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.