Asymptotic dimension of coarse spaces via maps to simplicial complexes

TL;DR

This paper characterizes the asymptotic dimension of coarse spaces using maps to simplicial complexes, extending classical dimension theory concepts to the coarse category with a new approach involving variation of maps.

Contribution

It introduces a novel characterization of asymptotic dimension for coarse spaces through maps to simplicial complexes, generalizing classical topological ideas.

Findings

Provides a new characterization of asymptotic dimension in coarse spaces.

Generalizes Property A to arbitrary coarse spaces.

Connects classical dimension theory with coarse geometry concepts.

Abstract

It is well-known that a paracompact space $X$ is of covering dimension at most $n$ if and only if any map $f\colon X\to K$ from $X$ to a simplicial complex $K$ can be pushed into its $n$-skeleton $K^{(n)}$. We use the same idea to characterize asymptotic dimension in the coarse category of arbitrary coarse spaces. Continuity of the map $f$ is replaced by variation of $f$ on elements of a uniformly bounded cover. The same way one can generalize Property A of G.Yu to arbitrary coarse spaces.