On the link of a stratum in a real algebraic set

Abstract

THE AIM of this paper is to exhibit some combinatorial topological properties of real algebraic sets. These properties are in the line of Sullivan’s condition on the local Euler characteristic of a real algebraic (or analytic) set. Let us recall this condition, in a way that will introduce the main result of this paper. Let X c R” be a real algebraic set, x a point of X. Let B(x, E) (resp. S(x, E)) denote the closed ball (resp. the sphere) with center x and radius E > 0. The local conic structure theorem says that, for E small enough, fi(x, c) n X is homeomorphic to the cone with vertex x and base S(x, E) n X, via a homeomorphism which preserves the distance to x. Hence the topological type of the space S(x, E) n X is independent of E, for E small enough; it is called rhe link of x in X, and dcnotcd by Ik(x, X). When X is a curve, then Ik(x, X) has two points for each real branch of X passing through x, and so Ik(x, X) consists of an even number of points. Sullivan’s condition says that this evenness is found in any dimension.