Algebraicity of Nash sets and of their asymmetric cobordism

TL;DR

This paper establishes a new algebraic-topological framework called asymmetric Nash cobordism to characterize when compact Nash sets are semialgebraically homeomorphic to real algebraic sets, leading to new classification results.

Contribution

It introduces asymmetric Nash cobordism as a criterion for algebraicity of Nash sets and develops approximation procedures, expanding the understanding of Nash set algebraicity.

Findings

A compact Nash set is algebraic iff it is asymmetric Nash cobordant to a point.

New classes of Nash sets are shown to be semialgebraically homeomorphic to algebraic sets.

Developed algebraic-topological approximation techniques for Nash sets.

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.