
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 of . The first approach was used by M.Gromov when defining the asymptotic dimension of metric spaces . 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 and Dranishnikov established that provided . We characterize asymptotic dimension (for spaces of finite asymptotic dimension) in terms of extensions of slowly oscillating…
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