Singular Spaces, Characteristic Classes, and Intersection Homology

Abstract

global topological invariant to be studied will be the L-classes Li(X) E Hi(X; Q), X a stratified pseudomanifold with even-codimension strata. For manifolds, these characteristic classes are the Poincare duals of the Hirzebruch polynomials in the Pontrjagin classes; Goresky-MacPherson extended the definition to stratified spaces with even-codimension strata using intersection homology [GM1]. Cheeger [Ch] defined L-classes analytically using his development of L2cohomology theory of singular Riemannian spaces. See also [S], [CSW]. The importance of these classes is illustrated by the famous theorem of Browder and Novikov: The homeomorphism type of a compact simply connected manifold of dimension at least five is determined by its homotopy type and its L-classes, up to a finite number of possibilities. (For 4-manifolds, the homotopy type alone suffices [F].) By recent work of Cappell-Weinberger (see [We]), similar results hold for stratified spaces with simply connected even-codimension strata and simply connected links, with respect to isovariant homotopy type and L-classes of the space and its strata. The study of L-classes will take place in the context of a stratified pseudomanifold X7 of real dimension n, embedded, say piecewise linearly, in a manifold Mm' of dimension m = n + 2. For example X might be a hypersurface in a projective complex algebraic variety. For X a smoothly or PL locally flatly embedded submanifold, the L-classes are given by the classical formula L(X) = [X] n i*/(P(M) u (1 +X f the total Hirzebruch L-polynomial, P(M) E H*(M) the total Pontrjagin class of M, i the inclusion, and X the Poincare dual of i [X].