On spaces with connected Higson coronas

TL;DR

This paper characterizes metric spaces with connected Higson coronas using categorical and coarse cohomology methods, and explores implications for finitely generated groups and coarse space decompositions.

Contribution

It provides a new categorical characterization of spaces with connected Higson coronas and links this to coarse cohomology and $ ext{ω}$-excisive decompositions.

Findings

Characterization via categorical conditions in the coarse category

Connection between Higson coronas and coarse cohomology

Application to finitely generated groups and coarse decompositions

Abstract

In this paper, we characterise metric spaces which have topologically connected Higson coronas. The characterisation is given by a natural categorical condition applied in the coarse category. We also give a characterisation in terms of coarse cohomology, and consider the special case of finitely generated groups and the more general case of abstract coarse spaces. Along the way, we exhibit some connections to the notion of $\omega$-excisive decomposition introduced by Higson, Roe and Yu, and give a categorical characterisation of such decompositions.