Smooth atlas stratified spaces, K-Homology Orientations, and Gysin maps
Pierre Albin, Markus Banagl, Paolo Piazza
TL;DR
This work develops a unified analytic framework for singular spaces by introducing smooth atlas stratified spaces and proving their equivalence with Thom-Mather and Whitney stratifications, enabling a robust theory of Witt spaces in analytic K-homology. It constructs analytic orientation classes via the signature operator on Witt spaces, defines analytic Gysin transfers for Witt-fiber bundles and normally non-singular inclusions, and proves functoriality and base-change properties within Kasparov KK-theory. A key achievement is showing that the analytic signature orientation is preserved by analytic transfer and inclusion maps, and that these analytic constructions agree with topological Siegel-Sullivan orientations under precise Adams-operation and complexification identifications. The results bridge topological and analytic approaches to singular spaces, unify several existing orientation theories, and provide tools for calculating L-classes and signatures on complex algebraic varieties and related Witt spaces, with potential applications in index theory and noncommutative geometry.
Abstract
We introduce smooth atlas stratified spaces. We show that this class is closed under cartesian products; consequently, it is possible to define fiber bundles of smooth atlas stratified spaces. We describe the resolution of such a space to a manifold with fibered corners and use this result in order to prove that the class of smooth atlas stratified spaces coincides with that of Thom-Mather stratified spaces. We then consider Witt pseudomanifolds (such as singular complex algebraic varieties) where it is well-known that a bordism invariant signature is available and equal to the Fredholm index of a realization of the signature operator. To each oriented fiber bundle of stratified spaces, with Witt fibers, we assign a class in bivariant KK-theory (with 2 inverted). Kasparov multiplication by this element defines a Gysin map in analytic K-homology and one of our main results is that this Gysin map preserves the analytic signature class of Witt spaces. We prove in fact a more general result: functoriality for fiber bundles in the sense that if one fiber bundle is the composition of two others then the KK-class of the former is the Kasparov product of the classes of the latter. We also discuss this result for other Dirac-type operators satisfying an analytic Witt condition, for example the spin-Dirac operator on a fibration of psc-Witt spin pseudomanifolds. We next define the analytic Gysin map associated to an oriented normally non-singular inclusion of Witt spaces and prove that it also preserves the signature class. Finally, we relate the analytic signature class of a Witt space with the topological Siegel-Sullivan orientation. Specifically we show that if one applies the inverse of the second Adams operation to the Sullivan orientation and complexifies then one obtains our analytic signature class under the natural identification between analytic and topological K-homology.