Strong desingularization of semialgebraic sets and applications

Abstract: Hironaka's resolution of singularities of algebraic varieties (over a field of characteristic $0$) is a widespread celebrated discipline that has many applications in many areas of Mathematics. Bierstone and Parusi\'nski studied the desingularization of closed subanalytic sets and in particular of closed semialgebraic sets. Our starting point is Bierstone and Parusi\'nski's results, but in this work we study desingularization of semialgebraic sets with some purposes in mind: (1) to deal with arbitrary semialgebraic sets (and not only with the closed ones), (2) to use (as desingularizating models) Nash manifolds with corners (instead of closures of unions of connected components of the complements of normal crossing divisors of non-singular real algebraic sets), (3) to have (as desingularizating maps) surjective algebraic maps and (4) to preserve analytic path-connected components. We also show 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), which contains ${\mathcal Q}$ as a closed subset, by folding $M$ along the irreducible components of a normal-crossings divisor of $M$, which is the smallest Nash subset of $M$ that contains the boundary $\partial{\mathcal Q}$ of ${\mathcal Q}$. We propose several applications of the previous results: (1) Weak desingularization of closed semialgebraic sets using Nash manifolds with (smooth) boundary, (2) Representation of compact semialgebraic sets connected by analytic paths as images under Nash maps of closed unit balls, (3) Explicit construction of Nash models for compact orientable smooth surfaces of genus $g\geq0$, and (4) Nash approximation of continuous semialgebraic maps whose target spaces are Nash manifolds with corners.