Thom Polynomials for Group Actions and Singularities

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

Associated with an invariant subvariety of a representation we consider its cohomology class in equivariant cohomology, and call it the Thom polynomial of the variety (``singularity''). Thus, the Thom polynomial is a G-equivariant characteristic class. Tracing back the definition of equivariant cohomology we obtain that the knowledge of the Thom polynomial of singularities reduces global singularity theoretic problems to homotopy theory (namely, to the computation of the characteristic classes of the underlying topological situation). Thom polynomials compute the cohomology class of the locus in a parameter space over which an object (map germ, differential form, arrangement, a bundle, a digaram of linear maps, etc) degenerates. A promising direction is the search for the corresponding good notion in extraordinary cohomology theories, the computation of these extraordinary Thom polynomials and finding topological applications. Another important direction is working out global singularity theory of geometrically or physically relevant representations, such as Dynkin quiver representations, surface bundles, multisingualrity loci over the complex and the real numbers, hyperplane arrangements, as well as their connections to degree, resultant and discriminant formulas. In many of these cases one expects different positivity properties of the Thom polynomials, which can contribute to the newly emerging bridge between geometry and combinatorics. The third goal is to explore the indirect usage of Thom polynomials in Geometric Invariant Theory. Here Thom polynomials can compute various non- stability loci of representations, hence they give natural, geometrically defined relations in the cohomology ring of the G.I.T. quotients. In various geometric and topological situations singularities occur for global reasons. That is, the global topology of a space, a manifold, a variety or a map forces some singularities to occur. (A trivial example is the global topology of the Klein bottle which forces double points when mapped into 3-space.) This global behaviour of singularities is governed by their Thom polynomials. If we substitute global topological invariants into the Thom polynomial we obtain the number of singularities forced by topology. This general point of view contains several mathematical areas as special cases, among others ``degeneracy loci formulas'' of algebraic geometry, immersion and general multiple point formulas of differentaial topology, and the theory of Schur, Schubert and quiver polynomials of algebraic combinatorics. Thom polynomials, however, are notoriously hard to compute. Although some powerful methods are known, some by the author, no universal method has beed found. We propose to study the basics of Thom polynomial theory, including possible generalizations to other cohomology theories. Another challenge is the understanding of the interior structure of natural infinite series of Thom polynomials. This direction promises to establish connections between the different areas that play roles in different computational approaches, e.g. algebraic topology, symmetric functions, interpolation theory, localizations, Groebner basis theory. Another main goal is to find applications of computed Thom polynomials in topology, geometry and invariant theory.