A cohomological characterisation of Yu's Property A for metric spaces

TL;DR

This paper provides a homological characterization of Yu's Property A for metric spaces by constructing analogues of cohomology theories, showing that Property A is equivalent to the vanishing of these cohomologies.

Contribution

It introduces new cohomology theories for metric spaces and proves that Property A corresponds to their vanishing, answering Higson's question.

Findings

Property A is equivalent to vanishing cohomology.

Constructed analogues of group cohomology for metric spaces.

Characterized Property A via asymptotically invariant means.

Abstract

Property A was introduced by Yu as a non-equivariant analogue of amenability. Nigel Higson posed the question of whether there is a homological characterisation of property A. In this paper we answer Higson's question affirmatively by constructing analogues of group cohomology and bounded cohomology for a metric space X, and show that property A is equivalent to vanishing cohomology. Using these cohomology theories we also give a characterisation of property A in terms of the existence of an asymptotically invariant mean on the space.