Smooth approximations in PL geometry

TL;DR

This paper establishes new approximation techniques showing that certain triangulable sets in Euclidean space can be approximated by smooth maps, extending to various classes including polyhedra, definable sets, and analytic sets.

Contribution

It proves that weakly $ ext{C}^r$ triangulable sets are $ ext{C}^r$-approximation target spaces, broadening the scope of smooth approximation in PL geometry.

Findings

Locally compact polyhedra are $ ext{C}^ abla$-ats.

Sets locally $ ext{C}^r$ equivalent to polyhedra are $ ext{C}^r$-ats.

Locally compact definable sets in o-minimal structures are $ ext{C}^1$-ats.

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 $\mathscr{C}^r$-approximation target space, or a $\mathscr{C}^r\text{-}\mathtt{ats}$ 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 $\mathscr{C}^r$ maps $g:X\to Y$ with respect to the strong $\mathscr{C}^0$ Whitney topology. Taking advantage of new approximation techniques we prove: if $Y$ is weakly $\mathscr{C}^r$ triangulable, then $Y$ is a $\mathscr{C}^r\text{-}\mathtt{ats}$. This result applies to relevant classes of triangulable sets, namely: (1) every locally compact polyhedron is a $\mathscr{C}^\infty\text{-}\mathtt{ats}$, (2) every set that is locally $\mathscr{C}^r$ equivalent to a polyhedron is a $\mathscr{C}^r\text{-}\mathtt{ats}$, and (3) every locally compact locally definable set of an arbitrary o-minimal structure is a $\mathscr{C}^1\text{-}\mathtt{ats}$ (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 $\mathscr{C}^\infty$ maps $g:X\to Y$. Explicit examples show the sharpness of our results.