Large scale absolute extensors

TL;DR

This paper explores the dualization of dimension-theoretical results from small to large scale, focusing on the relationship between asymptotic dimension and the covering dimension of the Higson corona in proper metric spaces.

Contribution

It characterizes asymptotic dimension via extensions of slowly oscillating functions and clarifies its relation to the Higson corona's covering dimension.

Findings

Established the equivalence of asymptotic dimension and Higson corona dimension for finite asymptotic dimension spaces.

Reproduced key results of Dranishnikov-Keesling-Uspenskiy and Dranishnikov.

Provided a new characterization of asymptotic dimension in terms of function extensions.

Abstract

This paper is devoted to dualization of dimension-theoretical results from the small scale to the large scale. So far there are two approaches for such dualization: one consisting of creating analogs of small scale concepts and the other amounting to the covering dimension of the Higson corona $\nu(X)$ of $X$. The first approach was used by M.Gromov when defining the asymptotic dimension $\asdim(X)$ of metric spaces $X$. The second approach was implicitly contained in the paper \cite{Dran AsyTop} by Dranishnikov on asymptotic topology. It is not known if the two approaches yield the same concept. However, Dranishnikov-Keesling-Uspenskiy proved $\dim(\nu(X)\leq \asdim(X)$ and Dranishnikov established that $\dim(\nu(X)= \asdim(X)$ provided $\asdim(X) < \infty$. We characterize asymptotic dimension (for spaces of finite asymptotic dimension) in terms of extensions of slowly oscillating functions to spheres. Our approach is specifically designed to relate asymptotic dimension to the covering dimension of the Higson corona $\nu(X)$ in case of proper metric spaces $X$. As an application, we recover the results of Dranishnikov-Keesling-Uspenskiy and Dranishnikov.