Selfdual Sheaves on Singular Spaces

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

DMS-0072550 Marcus Banagl Complexes of sheaves which are self-dual in the sense of Verdier have become indispensable tools in studying the topology of singular spaces, a prime example being the Goresky-MacPherson-Deligne intersection chain sheaf on a stratified space with only even-codimensional strata. On spaces that include strata of odd codimension as well, self-duality of the latter sheaf fails to hold in general. In recent research, we have shown that "Lagrangian structures" can be employed to construct self-dual intersection homology on such spaces. We propose to study the variance of the associated characteristic classes as the Lagrangian structure changes. In many situations, the effective manipulation of self-dual sheaves places heavy demands on the stratification of the underlying space. We propose to investigate the potential existence of a good "sheaf package" on weakly stratified spaces (the homotopically stratified sets of Quinn), using the technology of teardrop neighborhoods and manifold stratified approximate fibrations. Further, in collaboration with Sylvain Cappell, we are interested in developing characteristic class formulae for the signature of singular spaces in the presence of non-trivial monodromy, via a synthesis of Atiyah's formula for manifolds and the Cappell-Shaneson signature formula. Topology is the study of geometrical objects focusing on the neighborhood relations between points rather than on measurement of distances. In the last century, considerable effort has been directed towards studying "manifolds" -- spaces that locally look uniform, at each point and in each direction. This effort has been immensely successful; a central insight was that crucial information about a manifold is carried by one number: its "signature," measuring intersections of geometric sub-objects within the manifold. In the last two decades, topologists have studied "singular" spaces with increasing interest, due to their numerous occurrences and applications within pure mathematics (algebraic geometry, number theory) and outside pure mathematics (mathematical physics). In contrast to a manifold, a singular space may locally look different from point to point. The proposed research tries to define, understand, and compute topological invariants for singular spaces.