# Algebraicity of Nash sets and of their asymmetric cobordism

### Riccardo Ghiloni

Università di Trento, Povo (Trento), Italy### Alessandro Tancredi

Università degli Studi di Perugia, Italy

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## Abstract

This paper deals with the existence of algebraic structures on compact Nash sets. We introduce the algebraic-topological notion of asymmetric Nash cobordism between compact Nash sets, and we prove that a compact Nash set is semialgebraically homeomorphic to a real algebraic set if and only if it is asymmetric Nash cobordant to a point or, equivalently, if it is strongly asymmetric Nash cobordant to a real algebraic set. As a consequence, we obtain new large classes of compact Nash sets semialgebraically homeomorphic to real algebraic sets. To prove our results, we need to develop new algebraic-topological approximation procedures. We conjecture that every compact Nash set is asymmetric Nash cobordant to a point, and hence semialgebraically homeomorphic to a real algebraic set.

## Cite this article

Riccardo Ghiloni, Alessandro Tancredi, Algebraicity of Nash sets and of their asymmetric cobordism. J. Eur. Math. Soc. 19 (2017), no. 2, pp. 507–529

DOI 10.4171/JEMS/672