Dimension-raising maps in a large scale

Takahisa Miyata¹ ; Žiga Virk²

¹ Department of Mathematics and Informatics Graduate School of Human Development and Environment Kobe University Kobe, 657-8501 Japan
² Faculty of Mathematics and Physics University of Ljubljana Jadranska 21 Ljubljana, 1000 Slovenia

Fundamenta Mathematicae, Tome 223 (2013) no. 1, pp. 83-97

Résumé

Hurewicz's dimension-raising theorem states that $\dim Y \leq \dim X + n$ for every $n$-to-$1$ map $f: X\rightarrow Y$. 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 asymptotic dimension and asymptotic Assouad–Nagata dimension. It is also well-known (Hurewicz's finite-to-one mapping theorem) that $\dim X \leq n$ 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 asymptotic dimension and asymptotic Assouad–Nagata dimension.

Détail

DOI : 10.4064/fm223-1-6

Keywords: hurewiczs dimension raising theorem states dim leq dim every n to map rightarrow paper introduce notion finite to one map large scale setting using notion formulate dimension raising type theorem asymptotic dimension asymptotic assouad nagata dimension well known hurewiczs finite to one mapping theorem dim leq only there exists to map dimensional space formulate finite to one mapping type theorem asymptotic dimension asymptotic assouad nagata dimension