On cohomology of the Higson compactification of hyperbolic spaces

TL;DR

This paper proves that the cohomology groups of the Higson compactification of hyperbolic spaces are trivial in various dimensions depending on the coarse structure, advancing understanding in coarse geometry and topology.

Contribution

It establishes new results on the triviality of cohomology groups of Higson compactifications for hyperbolic spaces under different coarse structures.

Findings

Cohomology groups are trivial for dimensions >1 with the $C_0$ coarse structure.

Cohomology groups are trivial in all even dimensions with the bounded coarse structure.

Results apply specifically to hyperbolic spaces $ ext{H}^n$.

Abstract

We show that in dimensions $>1$ the cohomology groups of the Higson compactification of the hyperbolic space $\H^n$ with respect to the $C_0$ coarse structure are trivial. Also we prove that the cohomology groups of the Higson compactification of $\H^n$ for the bounded coarse structure are trivial in all even dimensions.