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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  L  2 ‐signature for manifolds with boundary.

Higher ρ ‐invariants and the surgery structure set