Dimension-Raising Maps in a Large Scale

TL;DR

This paper extends classical dimension-raising theorems to large scale geometry by introducing a new finite-to-one like map concept, establishing dimension-raising results for asymptotic and Assouad-Nagata dimensions.

Contribution

It introduces a novel notion of finite-to-one like maps in large scale geometry and formulates corresponding dimension-raising theorems for asymptotic and Assouad-Nagata dimensions.

Findings

Established a dimension-raising theorem for asymptotic dimension.

Formulated a similar theorem for asymptotic Assouad-Nagata dimension.

Extended classical finite-to-one mapping results to large scale setting.

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.