On the Hodge theory of stratified spaces

Abstract

This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of a perversity in the sense of Goresky-MacPherson to a mezzoperversity. We define Cheeger spaces and their signatures and describe how to carry out the analytic proof of the Novikov conjecture on these spaces. Finally we review the reductive Borel-Serre compactification of a locally symmetric space to a stratified space and describe its resolution to a manifold with corners.

C*-Algebraic higher Signature on Non-Witt space

“…for the chain in the stratum of x ∈ X n−2s 1 −1 . We know W m * (X) [1] and W n * (X) [1] are two equivalent chain and it does not affect the Witt condition of higher codimensional stratum. Hence we need to consider the next odd codimensional stratum s 2 where the non Witt conditions fails.…”

“…In [11], Cheeger consider the Cheeger boundary condition carries the Cauchy information of the link, he prove a condition of self dual L 2 cohomology on pseudomanifold with conical singularity. In this section, I will follow Albin, Leichtnam, Mazzeo, and Piazza's framework to explain analytical signature on smooth stratified pseudomanifold [1]. Let us consider the stratified space X with only one singular stratum Y .…”

“…The L 2 differential form with D max or D min is a Hilbert complex [1]. We know the de Rham cohomology H i DR is isomorphic to simlicial homology H i for smooth manifold.…”

“…Let ( X, g, B) be a Cheeger space with a self-dual mezzoperversity W, and B are the associated Cheeger ideal boundary conditions with W. The signature operator ð + sign is closed and Fredholm in Theorem 6.6 [1], then the Fredholm index is the signature of the quadratic form with generalized Poincare duality…”

“…Remark 6.19. In [1], Albin, Leichtnam, Mazzeo and Piazza define the smoothly stratified space with depth k in mathematical induction. Let the Smooth manifold be depth 0 and assume space with depth K has been defined.…”