All compact manifolds are homeomorphic to totally algebraic real algebraic sets
A real algebraic set V is said to be totally algebraic if every \({\mathbb{Z}}/2{\mathbb{Z}}\) homology class of V is represented by a compact algebraic subset. In this paper, the authors prove that any closed smooth manifold is homeomorphic to a totally algebraic real algebraic set. It should be noted that this algebraic set is in general singular; this might appear to diminish the usefulness of the result, but in practice one can often resolve the singularities and apply existing approximation theorems to the resolved manifold to obtain the results one wants. The proofs use blowing-up techniques developed in previous papers of the authors. [For some other recent results of these authors, see Bull. Am. Math. Soc., New Ser. 23, No.2, 441-446 (1990; Zbl 0723.14039)].
scientific article; zbMATH DE number 4208275