Dimension-Raising Maps in a Large Scale
Abstract
Hurewicz's dimension-raising theorem states that for every n-to-1 map f : X \to Y, dim Y =< dim X + n holds. In this paper we introduce a new notion of finite-to-one like map in a large scale setting. Using this notion we formulate a dimension-raising type theorem for the asymptotic dimension and the asymptotic Assouad-Nagata dimension. It is also well-known as Hurewicz's finite-to-one mapping theorem that dim X =< n if and only if there exists an (n + 1)-to-1 map from a 0-dimensional space onto X. We formulate a finite-to-one mapping type theorem for the asymptotic dimension and the asymptotic Assouad-Nagata dimension.
Cite
@article{arxiv.1307.6625,
title = {Dimension-Raising Maps in a Large Scale},
author = {Takahisa Miyata and Ziga Virk},
journal= {arXiv preprint arXiv:1307.6625},
year = {2021}
}
Comments
17 pages. A minor update in 2021: the assumption of coarse surjectivity is now explicitly stated in the main results. The obviously needed assumption has been used in the proofs but not stated explicitly in the previously published version