# Piecewise-regular maps

## TL;DR

This paper introduces the concept of piecewise-regular maps between real algebraic varieties, demonstrating that continuous maps can be approximated or homotoped by such maps under certain conditions, with applications to vector bundle algebraization.

## Contribution

It establishes the approximation of continuous maps by piecewise-regular maps on algebraic varieties, extending the algebraization of topological vector bundles.

## Key findings

- Continuous maps into Grassmannians or spheres can be approximated by piecewise-regular maps.
- Every continuous map into a sphere on a compact nonsingular variety is homotopic to a C^k piecewise-regular map.
- Application to algebraization theorem for topological vector bundles.

## Abstract

Let V, W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and let X be a subset of V. A map f from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X. Furthermore, such a map is said to be piecewise-regular if there exists a stratification of V such that the restriction of f to the intersection of X with each stratum is a regular map. By a stratification of V we mean a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union is equal to V. Assuming that the subset X is compact, we prove that every continuous map from X into a Grassmann variety or a unit sphere can be approximated by piecewise-regular maps. As an application, we obtain a variant of the algebraization theorem for topological vector bundles. If the variety V is compact and nonsingular, we prove that each continuous map from V into a unit sphere is homotopic to a piecewise-regular map of class C^k, where k is an arbitrary nonnegative integer.