Transfers in coarse homology

TL;DR

This paper extends the framework of coarse homology theories by incorporating transfer morphisms, enabling new algebraic and geometric insights, especially in the context of equivariant theories and Mackey functors.

Contribution

It introduces transfer morphisms into coarse homology theories and demonstrates their applications in extending algebraic and ordinary coarse homology, including the development of Mackey functors.

Findings

Equivariant coarse algebraic K-homology extended with transfers.

Equivariant coarse ordinary homology extended with transfers.

Transfers used to prove injectivity of assembly maps.

Abstract

We enlarge the category of bornological coarse spaces by adding transfer morphisms and introduce the notion of an equivariant coarse homology theory with transfers. We then show that equivariant coarse algebraic $K$-homology and equivariant coarse ordinary homology can be extended to equivariant coarse homology theories with transfers. In the case of a finite group we observe that equivariant coarse homology theories with transfers provide Mackey functors. We express standard constructions with Mackey functors in terms of coarse geometry, and we demonstrate the usage of transfers in order to prove injectivity results about assembly maps.