Coarse cohomology theories

TL;DR

This paper introduces the concept of coarse cohomology theories, explores examples like coarse ordinary cohomology and stable cohomotopy, and proves that the dualizing spectrum of a torsion-free group is a coarse invariant.

Contribution

It defines coarse cohomology theories, studies their examples, and proves the dualizing spectrum of a torsion-free group depends only on its coarse motivic spectrum.

Findings

Dualizing spectrum of a torsion-free group is a coarse invariant

Coarse cohomology theories include coarse ordinary cohomology and stable cohomotopy

Dualizing spectrum depends only on the coarse motivic spectrum

Abstract

We propose the notion of a coarse cohomology theory and study the examples of coarse ordinary cohomology, coarse stable cohomotopy and of coarse cohomology theories obtained by dualizing coarse homology theories. We show that the dualizing spectrum of a finitely generated torsion-free group only depends on the coarse motivic spectrum represented by the underlying bornological coarse space of the group. This in particular implies a conjecture of J. R. Klein that the dualizing spectrum of a group is a coarse invariant.