A twisted Version of controlled K-Theory

TL;DR

This paper develops a geometric approach to controlled K-theory on coarse spaces, demonstrating that Roe-algebras form a cosheaf and establishing a Mayer-Vietoris sequence for coarse covers, with computed examples.

Contribution

It introduces a geometric perspective on controlled K-theory, proving Roe-algebras form a cosheaf and deriving a Mayer-Vietoris sequence for coarse covers.

Findings

Roe-algebras restrict to subspaces under certain conditions

Roe-algebras form a cosheaf on the coarse topology

Mayer-Vietoris sequence established for coarse covers

Abstract

There are a number of (co-)homology theories on coarse spaces. Controlled operator K-theory is by far the most popular one of them. Our approach is geometric. We study when does the Roe-algebra of a space restrict to a subspace. Then we show the Roe-algebra is a cosheaf on the coarse topology. A result is a Mayer-Vietoris exact sequence in the presence of a coarse cover. We compute examples as an application.