Higson Compactification and Dimension Raising

Let $X$ and $Y$ be proper metric spaces. We show that a coarsely $n$-to-$1$ map $f\colon X\to Y$ 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 $f \colon X\to Y$ is a coarsely $n$-to-$1$ map between proper metric spaces $X$ and $Y$ then $asdim(Y) \leq asdim(X) + n -1$. 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.