Algebraic equations for a class of P. L. spaces

Abstract

Our main purpose is to discuss the question: "Can closed P. L. manifolds be represented as real algebraic varieties ?" It is known by Nash that all closed smooth manifolds are diffeomorphic to components o f real algebraic variables [7]. He also conjectured that they are diffeomorphic to real algebraic varieties. Since all closed P. L. manifolds of dimension less than 8 have smooth structures, they are P. L. homeomorphic to components of real algebraic varieties. For closed P. L. manifolds of dimension 8, Kuiper proves that they are represented by components of real algebraic varieties [3]. Then recently Tognoli proved the Nash conjecture [10]. Here we define a class of polyhedra and prove that the elements of this class are represented as real algebraic varieties (i.e., we prove a P. L. version of the Nash conjecture for these polyhedra). Elements of this class roughly are: Singular spaces which are smooth in the complements of disjoint union of smooth manifolds with certain nice properties. In particular, it contains all P. L. 8-manifolds. All our methods produce algebraic varieties, which are locally complete intersections. We produce a nonsmoothable P. L. 9-mainfold, which is not such a variety.