A Cheeger space is a smoothly stratified pseudomanifold which is in general non-Witt but that admits an additional structure along the strata that allows for the definition of ideal boundary conditions. An interesting example is given by the reductive Borel-Serre compactification of a Hilbert modular surface. In this talk I will explain recent results, in collaboration with Albin, Leichtnam and Mazzeo and, in part, with Banagl, concerning the topology and the analysis of Cheeger spaces. I will concentrate on the geometric consequences of our analysis; in particular I will explain how it is possible to define on a Cheeger space a homology L-class and thus higher signatures à la Novikov. One of our main result is the stratified homotopy invariance of the higher signatures for Cheeger spaces with fundamental group satisfying the rational injectivity of the assembly map in K-theory; put it differently, I will show how the usual Strong Novikov Conjecture implies the stratified homotopy invariance of these higher signatures.
The lecture was held within the framework of the Hausdorff Trimester Program Non-commutative Geometry and its Applications. (18.09.2014)
This video was created and edited with kind support from eCampus Bonn and is also available at [ Ссылка ].
