Equivariant Coarse (Co-)Homology Theories

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.

arXiv commentv2: added Lemma 6.10 about a nice metrization of the Rips complex and further minor changes. 55 + $\epsilon$ pages v3: Numerous corrections and improvements. Also, the introduction has been rewritten to clarify the purpose of this paper. v4: published version

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Cited by in corpus (4)

• Topological equivariant coarse K-homology
• Secondary cup and cap products in coarse geometry
• Paschke duality and assembly maps
• Coarse cohomology of configuration space and coarse embedding