60 60 04 v 1 6 J un 1 99 6 ALGEBRAICALLY CONSTRUCTIBLE FUNCTIONS

An algebraic version of Kashiwara and Schapira’s calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and King’s numerical conditions for a stratified set of dimension three to be algebraic. These properties, which include generalizations of the invariants modulo 4, 8, and 16 of Coste and Kurdyka, are defined using the link operator on the ring of constructible functions. In 1970 Sullivan [Su] proved that if X is a real analytic set and x ∈ X, then the Euler characteristic of the link of x in X is even. Ten years later, Benedetti and Dedò [BD], and independently Akbulut and King [AK1], proved that Sullivan’s condition gives a topological characterization of real algebraic sets of dimension less than or equal to two. Using their theory of resolution towers, Akbulut and King introduced a finite set of local “characteristic numbers” of a stratified space X of dimension three, such that X is homeomorphic to a real algebraic set if and only if all of these numbers vanish [AK2]. In 1992 Coste and Kurdyka [CK] proved that if Y is an irreducible algebraic subset of the algebraic set X and x ∈ Y , then the Euler characteristic of the link of Y in X at x, which is even by Sullivan’s theorem, is generically constant mod 4. They also introduced invariants mod 2 for chains of k strata, and they showed how to recover the Akbulut-King numbers from their mod 4 and mod 8 invariants. The Coste-Kurdyka invariants were generalized and given a simpler description in [MP] using complexification and monodromy. We introduce a new approach to the Akbulut-King numbers and their generalizations which is motivated by the theory of Stiefel-Whitney homology classes, as was Sullivan’s original theorem. We use the ring of constructible functions on X, which has been systematically developed by Kashiwara and Schapira [KS] [Sch] in the subanalytic setting. Their calculus of constructible functions includes the fundamental operations of duality and pushforward, which correspond to standard operations in sheaf theory. Our primary object of study is the ring of algebraically constructible functions on the real algebraic set X. We say that the function φ : X → Z and the stratification S of X are compatible if φ is constant on each stratum of S. If X is a complex algebraic 1991 Mathematics Subject Classification. Primary: 14P25, 14B05. Secondary: 14P10, 14P20.