Normally nonsingular maps between singular spaces possess transfer homomorphisms on general multiplicative homology theories such as bordism, L-homology, KO-homology. We will first discuss the behavior of orientation and characteristic classes of singular spaces under such transfers. In the context of an abstract framework of Gysin coherence, we obtain a general ambient uniqueness theorem for characteristic classes associated to complex algebraic varieties embedded in Grassmannians. The theorem implies for instance that for Cohen-Macaulay or Gorenstein varieties with algebraic homology basis, the Goresky-MacPherson L-class equals the intersection generalized Todd class at parameter 1, defined using motivic and Hodge-theoretic methods by Brasselet, Schürmann and Yokura. For Schubert varieties, these classes are therefore equal.

We consider in various settings the problem of the title. In some settings (e.g., with many strata allowed) the answers depend only upon such elementary invariants as Euler characteristics, even for non-simply connected spaces. This is part of a series of projects on transformation groups with Shmuel Weinberger of the U. of Chicago and Min Yan of Hong Kong U. of Science & Tech.

The Milnor fibration gives a well-defined notion of the smooth local fiber of a holomorphic function at a critical point. Milnor's work in the isolated case suggests that this fiber's topology should be controlled by the scheme-theoretic invariants of the critical locus; we give results which demonstrate that this is true in a relative sense. Specifically, we show that the local smooth fiber varies nicely in families where the embedded critical locus satisfies certain algebraic consistency requirements and discuss implications for homogeneous polynomials and other special cases.

Using equivariant Todd (resp. Hirzebruch) classes of toric varieties, we translate the equivariant Hirzebruch-Riemann-Roch theorem for simplicial projective toric varieties into various (weighted) Euler-Maclaurin type formulae for simple lattice polytopes. This is joint work with S. Cappell, L. Maxim and J. Shaneson.

Kodaira's classification for elliptic surface singularities in terms of ADE algebras uses a resolution process. A similar, very slightly more refined invariant for this classification can be obtained via deformations. There are various approches to extending the Kodaira approach to three folds, always with some hypothesis, e.g. Grassi-Morrison (smooth case) and then Grassi-Wiegand, via a complete or partial resolution that is still Calabi Yau and with some control over the type of singularity to obtain the sufficient Poncare duality needed. This talk on joint efforts with Grassi, Halverson (and some other physicists), will try to outline something similar, possibly more general and with a richer invariant, using deformations and monodromy.