# Equivariant Coarse (Co-)Homology Theories

@article{Wulff2022EquivariantC, title={Equivariant Coarse (Co-)Homology Theories}, author={Christopher Wulff}, journal={Symmetry, Integrability and Geometry: Methods and Applications}, year={2022} }

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…

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## References

SHOWING 1-10 OF 73 REFERENCES

### DUALIZING THE COARSE ASSEMBLY MAP

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2005

We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly…

### Princeton

- New Jersey,
- 1952

### Simultaneous metrizability of coarse spaces

- Mathematics
- 2011

A metric space can be naturally endowed with both a topology and a coarse structure. We examine the converse to this. Given a topology and a coarse structure we give necessary and sufficient…

### Homotopy Theory with Bornological Coarse Spaces

- MathematicsLecture Notes in Mathematics
- 2020

We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the…

### Coarse and equivariant co-assembly maps

- Mathematics
- 2008

We study an equivariant co-assembly map that is dual to the usual Baum-Connes assembly map and closely related to coarse geometry, equivariant Kasparov theory, and the existence of dual Dirac…

### Jussieu

- 5(2):161–186,
- 2006

### C*-Algebras and Controlled Topology

- Mathematics
- 1997

This paper is an attempt to explain some aspects of the relationship between the K-theory of C-algebras, on the one hand, and the categories of modules that have been developed to systematize the…

### Cambridge Studies in Advanced Mathematics

- Cambridge University Press,
- 2020