Approximation of maps from algebraic polyhedra to real algebraic varieties
Marcin Bilski, Wojciech Kucharz
TL;DR
The paper addresses the problem of approximating continuous maps from algebraic polyhedra to real algebraic varieties by $\mathcal{K}$-regular maps. It introduces uniformly retract rational varieties and a Whitney-type interpolation framework to enable local regular approximations that glue across algebraic complexes. The main result shows that, for $0\le l\le k$, any $\mathcal{C}^l$ map into a uniformly retract rational variety $Y$ can be approximated in the $\mathcal{C}^l$ topology by $\mathcal{K}$-regular maps of class $\mathcal{C}^k$, generalizing prior work on piecewise-regular maps. The approach leverages strong dominating sprays and a detailed cell-wise extension/approximation scheme, yielding applicability to classical varieties like spheres, Grassmannians, and orthogonal groups, and enabling corollaries for broader source sets.
Abstract
Given a finite simplicial complex $\mathcal{K}$ in $\mathbb{R}^n$ and a real algebraic variety $Y,$ by a $\mathcal{K}$-regular map $|\mathcal{K}|\rightarrow Y$ we mean a continuous map whose restriction to every simplex in $\mathcal{K}$ is a regular map. A simplified version of our main result says that if $Y$ is a uniformly retract rational variety and if $k, l$ are integers satisfying $0\leq l\leq k,$ then every $\mathcal{C}^l$ map $|\mathcal{K}|\rightarrow Y$ can be approximated in the $\mathcal{C}^l$ topology by $\mathcal{K}$-regular maps of class $\mathcal{C}^k.$ By definition, $Y$ is uniformly retract rational if for every point $y\in Y$ there is a Zariski open neighborhood $V\subset Y$ of $y$ such that the identity map of $V$ is the composite of regular maps $V\rightarrow W\rightarrow V,$ where $W\subset\mathbb{R}^p$ is a Zariski open set for some $p$ depending on $y.$