Signature operators on Lipschitz manifolds and Novikov's theorem (Q2829322)

This survey article presents a short introduction of a series of papers of Sullivan and Teleman about the signature index theorem on Lipschitz manifolds and how these constructions imply Novikov's theorem on the topological invariance of rational Pontryagin classes. The importance of the Lipschitz manifolds is that any topological manifold of dimension not equal to 4 has a Lipschitz structure and for any two Lipschitz structures, there exists a Lipschitz homeomorphism between them which is isotopic to the identity. Since differential forms, Riemannian metrics and Sobolev spaces are well constructed on Lipschitz manifolds, we could do analysis on topological manifolds taking advantage of the Lipschitz structures.NEWLINENEWLINEIn this survey, the basics of signature operators and Hodge theory on Lipschitz Riemannian manifolds are discussed in some detail. And another approach of the results of Teleman and Novikov in the frame work of the KK-theory by Hilsum is briefly reviewed.NEWLINENEWLINEFor the entire collection see [Zbl 1323.00070].