C*-Algebraic higher Signature on Non-Witt space

TL;DR

This paper develops a $C^*$-algebraic higher signature theory for non-Witt spaces using noncommutative geometry, extending previous work on Witt spaces and analyzing signatures in singular stratified spaces.

Contribution

It introduces a new $C^*$-algebraic higher signature framework for non-Witt spaces, expanding the scope of geometric invariants in singular spaces.

Findings

Constructed $C^*$-signature on non-Witt spaces using noncommutative methods.

Compared analytical signatures of stratified non-Witt spaces with existing models.

Extended the signature theory beyond Witt spaces to more general singular spaces.

Abstract

Signature plays an important role in geometry and topology. In the space with singularity, Goresky and MacPherson extend the signatures to oriented pseudomanifolds with only even codimensional stratums by using generalized Poincare duality of intersection homology. After that Siegel extended the signature on Witt spaces. Higson and Xie study the $C^*$- higher signature on Witt space. Followed by the combinatorial framework developed by Higson and Roe, this paper construct the $C^*$-signature on non Witt space with noncommutative geometric methods. In conical singular case, we compare analytical signature of smooth stratified non Witt space by Albin, Leichtnam, Mazzeo and Piazza.