Higher <i>ρ</i>-invariants and the surgery structure set

Abstract

We study noncommutative η- and ρ-forms for homotopy equivalences. We prove a product formula for them and show that the ρ-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 ρ-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 L2-signature for manifolds with boundary.