C algebraic Higher Signatures on Non-Witt 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. Non-Witt spaces need a refined intersection homology to hold Poincare duality. Followed by the combinatorial framework developed by Higson and Roe, this dissertation 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.