Cohomological approach to asymptotic dimension

TL;DR

This paper introduces a cohomological framework for understanding asymptotic dimension, demonstrating its equivalence to classical asymptotic dimension for finite cases and exploring new properties through examples.

Contribution

It defines asymptotic cohomology based on bounded cohomology and shows its equivalence to asymptotic dimension when finite, also constructing a space with unique dimension properties.

Findings

Asymptotic cohomology agrees with asymptotic dimension for finite cases

Constructed a space with finite asymptotic dimension where the product with R does not increase dimension

Coarse asymptotic dimension can be strictly less than asymptotic dimension in certain spaces

Abstract

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension $\as_{\Z} X$ of metric spaces. We show that it agrees with the asymptotic dimension $\as X$ when the later is finite. Then we use this fact to construct an example of a metric space $X$ of bounded geometry with finite asymptotic dimension for which $\as(X\times\R)=\as X$. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe's coarse cohomology is strictly less than its asymptotic dimension.