Equivariant Coarse (Co-)Homology Theories

TL;DR

This paper develops an axiomatic framework for equivariant coarse homology and cohomology theories, connecting them to topological theories, introducing a flexible coarse homotopy, and exploring group actions on coarse spaces.

Contribution

It introduces a new axiomatic setup for equivariant coarse (co-)homology, a general construction from topological theories, and a more flexible notion of coarse homotopy.

Findings

Well-established coarse (co-)homology theories fit into the new framework.

A new, more flexible coarse homotopy concept is proposed.

The paper explores topological actions of groups on coarse spaces.

Abstract

We present an Eilenberg-Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories, whose equivariant versions are either already known or will be introduced in this paper, fit into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology theories are even invariant under these new homotopies. They also led us to a meaningful concept of topological actions of locally compact groups on coarse spaces.