On asymptotic extension dimension

The aim of this paper is to introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes a relation between the asymptotic extensional dimension of a proper metric space and extension dimension of its Higson corona.

💡 Research Summary

The paper introduces a “asymptotic extension dimension” (as‑ext‑dim) for proper metric spaces, which is a large‑scale analogue of Dranishnikov’s classical extension dimension. The authors work within the asymptotic category A, whose objects are proper metric spaces and whose morphisms are asymptotically Lipschitz maps (maps satisfying a (λ, ε)‑Lipschitz condition for some λ > 0, ε ≥ 0). They also define coarse uniform maps and metric proper maps, and recall that the Higson compactification (\bar X) and its corona νX are functorial with respect to coarse maps.

A central construction is the open cone O L over a compact absolute Lipschitz neighborhood Euclidean extensor (ALNER) L. The cone carries the metric
(d(