  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.
