Higher rho-invariants and the surgery structure set

TL;DR

This paper develops noncommutative eta- and rho-forms for homotopy equivalences, establishing their properties, and connects them with L-theory and C*-algebraic K-theory, providing new insights into higher signatures and signatures invariance.

Contribution

It introduces a new framework for rho-forms on the structure set and links them with index theory, extending previous methods to manifolds with boundary.

Findings

Proved a product formula for rho-forms.

Established rho-forms as well-defined on the structure set.

Provided a unified analytic proof of homotopy invariance of higher signatures.

Abstract

We study noncommutative eta- and rho-forms for homotopy equivalences. We prove a product formula for them and show that the rho-forms are well-defined on the structure set. We also define an index theoretic map from L-theory to C*-algebraic K-theory and show that it is compatible with the rho-forms. Our approach, which is based on methods of Hilsum-Skandalis and Piazza-Schick, also yields a unified analytic proof of the homotopy invariance of the higher signature class and of the L^2-signature for manifolds with boundary.