### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 057, 62 pages      arXiv:2006.02053      https://doi.org/10.3842/SIGMA.2022.057

### Equivariant Coarse (Co-)Homology Theories

Christopher Wulff
Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, D-37073 Göttingen, Germany

Received October 03, 2021, in final form July 15, 2022; Published online July 26, 2022

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.

Key words: equivariant coarse homology; equivariant coarse cohomology; equivariant coarse assembly; equivariant coarse coassembly; generalized coarse homotopies.

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