Applications of the Nash double of a Nash manifold with corners

TL;DR

This paper explores properties and applications of Nash manifolds with corners, introducing a method to construct such manifolds from existing Nash manifolds and utilizing the Nash double concept for various geometric and topological applications.

Contribution

It introduces a novel construction of Nash manifolds with corners using folding along divisors and leverages the Nash double to facilitate multiple applications in semialgebraic geometry.

Findings

Constructed Nash manifolds with corners from Nash manifolds via folding.

Demonstrated the use of Nash double in modeling and approximation.

Applied results to Nash ramified coverings, uniformization, and surface modeling.

Abstract

In this work we study some properties and applications of Nash manifolds with corners. Our first main result shows how to `build' a Nash manifold with corners ${\mathcal Q}\subset{\mathbb R}^n$ from a suitable Nash manifold $M\subset{\mathbb R}^n$ (of its same dimension), that contains ${\mathcal Q}$ as a closed subset, by folding $M$ along the irreducible components of a normal-crossings divisor of $M$ (the smallest Nash subset of $M$ that contains the boundary $\partial{\mathcal Q}$ of ${\mathcal Q}$). Our second main results shows that we can choose as the Nash manifold $M$ the Nash `double' $D({\mathcal Q})$ of ${\mathcal Q}$, which is the analogous to the Nash double of a Nash manifold with (smooth) boundary, but $D({\mathcal Q})$ takes into account the peculiarities of the boundary of a Nash manifold with corners. We propose several applications of the previous results: (1) Nash ramified coverings of closed semialgebraic sets, (2) Weak Nash uniformization of closed semialgebraic sets using Nash manifolds with (smooth) boundary, (3) Representation of compact semialgebraic sets connected by analytic paths as images under Nash maps of closed unit balls, (4) Explicit construction of Nash models for compact orientable smooth surfaces of genus $g\geq0$, and (5) Nash approximation of continuous semialgebraic maps whose target spaces are Nash manifolds with corners.