Stratifiable maps and topological invariants

Let X and Y be spaces and f: X -Y be a map between them. Then a large family of problems in topology (and geometry) involve the attempt to relate the invariants of X and Y via f. This attempt has been successful often for the case X and Y manifolds and f the projection of a fibre bundle, for many types of important invariants. For example, the result of ChernHirzebruch-Serre [CHS] asserts that for Y simply connected, the signature of X is the product of the signatures of Y and of the fibre f 1 (y) . In this paper we undertake the study of maps in which the fiber may vary from point to point. The spaces X and Y will be any Whitney stratified spaces with even codimension strata (e.g., algebraic varieties) and f will be a stratified map. This means essentially that for V a component of a stratum of Y (thus V is an open manifold), f 1V is a union of strata of X and the restriction of f to f71V is a locally trivial fibre bundle. For example, subanalytic maps of subanalytic sets can be stratified, with respect to Whitney