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 stratiied 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 deened 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 2 X, then the Euler characteristic of the link of x in X is even. Ten years later, Benedetti and Dedd o 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 nite set of local \characteristic numbers" of a stratiied 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 2 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 k 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 complexiication 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 and by Viro V]. 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.