Smooth approximations in PL geometry

Abstract

Let $Y\subset {\mathbb{R}}^n$ be a triangulable set and let $r$ be either a positive integer or $r=\infty$. We say that $Y$ is a ${\mathcal{C}}^r$-approximation target space, or a ${\mathcal{C}}^r\text{-}\mathtt{ats}$ for short, if it has the following universal approximation property: For each $m\in \mathbb{N}$ and each locally compact subset $X$ of~${\mathbb{R}}^m$, any continuous map $f:X\to Y$ can be approximated by ${\mathcal{C}}^r$ maps $g:X\to Y$ with respect to the strong ${\mathcal{C}}^0$ Whitney topology. Taking advantage of new approximation techniques we prove: if $Y$ is weakly ${\mathcal{C}}^r$ triangulable, then $Y$ is a ${\mathcal{C}}^r\text{-}\mathtt{ats}$. This result applies to relevant classes of triangulable sets, namely: (1) every locally compact polyhedron is a ${\mathcal{C}}^{\infty }\text{-}\mathtt{ats}$, (2) every set that is locally ${\mathcal{C}}^r$ equivalent to a polyhedron is a ${\mathcal{C}}^r\text{-}\mathtt{ats}$, and (3) every locally compact locally definable set of an arbitrary o-minimal structure is a ${\mathcal{C}}^1\text{-}\mathtt{ats}$ (this includes locally compact locally semialgebraic sets and locally compact subanalytic sets). In addition, we prove: if $Y$ is a global analytic set, then each proper continuous map $f:X\to Y$ can be approximated by proper ${\mathcal{C}}^{\infty }$ maps $g:X\to Y$. Explicit examples show the sharpness of our results.

29 pages