# On Nash images of Euclidean spaces

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Josè Fernando Galvan

created by collari on 05 May 2015

modified on 14 May 2015

27 may 2015
-- 18:00

Sala Seminari, Dipartimento di Matematica, Pisa

Seminari dei Baby-Geometri

**Abstract.**

We have recently characterized the subsets of $\mathbb{R}^n$ that are images of Nash maps $f:\mathbb{R}^m \to \mathbb{R}^{n}$. This implies to prove Shiota's conjecture and show that a subset S of $\mathbb{R}^n$ is the image of a Nash map $f:\mathbb{R}^m\to \mathbb{R}^n$ if and only if S is semialgebraic, pure dimensional of dimension dâ‰¤m and there exists an analytic path $\alpha:[0,1]\to S$ whose image meets all the connected components of the set of regular points of S. Some remarkable consequences of the previous cahracterization are the following: (1) pure dimensional irreducible semialgebraic sets of dimension d with arc-symmetric closure are Nash images of $\mathbb{R}^d$; (2) semialgebraic sets are projections of irreducible algebraic sets whose connected components are Nash diffeomorphic to Euclidean spaces; and (3) compact d-dimensional smooth manifolds with boundary are smooth images of $\mathbb{R}^d$. In the seminar we will provide a sketch of the previous characterization in the two dimensional case.