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Applications of the Nash double of a Nash manifold with corners

Antonio Carbone, José F. Fernando

TL;DR

The paper introduces folding techniques to convert Nash manifolds into Nash manifolds with corners and defines the Nash double to handle corner structures, establishing a versatile framework for embedding and approximating semialgebraic sets. The main technical results—folding along Nash normal-crossings divisors and constructing the Nash double—enable a suite of applications: Nash ramified coverings, weak Nash uniformization by manifolds with boundary, and expressing compact semialgebraic sets as Nash images of closed balls, among others. The approach yields explicit, constructive procedures for modeling orientable compact surfaces and for approximating semialgebraic maps into Nash manifolds with corners, deepening the toolbox for semialgebraic geometry. Overall, the work provides new canonical tools to translate between corners, boundaries, and doubles in the Nash category, with broad implications for uniformization, model construction, and approximation in real algebraic geometry.

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.

Applications of the Nash double of a Nash manifold with corners

TL;DR

The paper introduces folding techniques to convert Nash manifolds into Nash manifolds with corners and defines the Nash double to handle corner structures, establishing a versatile framework for embedding and approximating semialgebraic sets. The main technical results—folding along Nash normal-crossings divisors and constructing the Nash double—enable a suite of applications: Nash ramified coverings, weak Nash uniformization by manifolds with boundary, and expressing compact semialgebraic sets as Nash images of closed balls, among others. The approach yields explicit, constructive procedures for modeling orientable compact surfaces and for approximating semialgebraic maps into Nash manifolds with corners, deepening the toolbox for semialgebraic geometry. Overall, the work provides new canonical tools to translate between corners, boundaries, and doubles in the Nash category, with broad implications for uniformization, model construction, and approximation in real algebraic geometry.

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 from a suitable Nash manifold (of its same dimension), that contains as a closed subset, by folding along the irreducible components of a normal-crossings divisor of (the smallest Nash subset of that contains the boundary of ). Our second main results shows that we can choose as the Nash manifold the Nash `double' of , which is the analogous to the Nash double of a Nash manifold with (smooth) boundary, but 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 , and (5) Nash approximation of continuous semialgebraic maps whose target spaces are Nash manifolds with corners.
Paper Structure (33 sections, 27 theorems, 100 equations, 3 figures, 1 table)

This paper contains 33 sections, 27 theorems, 100 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Semialgebraic setting
  3. Whitney's semialgebraic topology
  4. Folding Nash manifolds to build Nash manifolds with corners
  5. Applications
  6. Nash ramified covering of closed semialgebraic sets
  7. Weak Nash uniformization of closed semialgebraic sets by Nash manifolds with boundary.
  8. Nash images of the closed unit ball
  9. Explicit Nash models for orientable compact smooth surfaces
  10. Approximation of continuous semialgebraic maps by Nash maps
  11. Structure of the article
  12. Preliminaries
  13. Mostowski's trick
  14. Nash manifolds with boundary
  15. Nash doubles of Nash manifolds with boundary.
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let ${\EuScript Q}\subset{\mathbb R}^n$ be a $d$-dimensional Nash manifold with corners that is a closed (semialgebraic) subset of ${\mathbb R}^n$. Then there exist: In addition, for each $x\in\partial{\EuScript Q}$ there exist open semialgebraic neighborhoods $U,V\subset M$ of $x$ equipped with Nash diffeomorphisms $\varphi:U\to{\mathbb R}^d$ and $\psi:V\to{\mathbb R}^d$ and $1\leq s\leq d$ such

Figures (3)

  • Figure 2.1: Nash double of ${\EuScript H}$ and the projection $\pi$ (figure borrowed from fe3).
  • Figure 2.2: The teardrop.
  • Figure 3.1: Graphs of $f_a$ for $a=\frac{1}{2}$ and $k=2$ (left) and $k=6$ (right).

Theorems & Definitions (53)

  • Theorem 1.1: Folding Nash manifolds
  • Remark 1.2
  • Corollary 1.3: Nash ramified covering map
  • Theorem 1.4: Weak Nash uniformization of closed semialgebraic sets
  • Theorem 1.5: Compact Nash images
  • Theorem 1.6: Nash approximation I
  • Theorem 1.7: Nash approximation II
  • Proposition 2.1: Mostowski's trick
  • proof
  • Lemma 2.2
  • ...and 43 more