Higson Compactification and Dimension Raising

Abstract

Let $X$ and $Y$ be proper metric spaces. We show that a coarsely $n$-to-$1$ map $f:X\to Y$ induces an $n$-to-$1$ map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if $f:X\to Y$ is a coarsely $n$-to-$1$ map between proper metric spaces $X$ and $Y$ then $asdim(Y)\le asdim(X)+n-1$. Furthermore we introduce coarsely open coarsely $n$-to-$1$ maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension.

An updated version (in 2021) containing a small correction, for details see Acknowledgments

References in corpus (1)

• The asymptotic dimension of quotients by finite groups

Cited by in corpus (2)

• Visual maps between coarsely convex spaces
• Linear algebra and unification of geometries in all scales