Higher rho-invariants and the surgery structure set

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.

42 pages; exposition improved; version accepted by Journal of Topology

References in corpus (3)

• Combable groups have group cohomology of polynomial growth
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• Homological index formulas for elliptic operators over C*-algebras

Cited by in corpus (11)

• Rho-classes, index theory and Stolz' positive scalar curvature sequence
• The surgery exact sequence, K-theory and the signature operator
• Boundaries, spectral triples and K-homology
• Adiabatic groupoids and secondary invariants in K-theory
• Relative geometric assembly and mapping cones, Part I: The geometric model and applications
• Realizing the analytic surgery group of Higson and Roe geometrically, Part II: Relative eta-invariants
• Higher genera for proper actions of Lie groups, Part 2: the case of manifolds with boundary
• C*-Algebraic Higher Signatures and an Invariance Theorem in Codimension Two
• Additive higher rho invariant for structure group in differential point of view
• Relative geometric assembly and mapping cones, Part II: Chern characters and the Novikov property
• On localized signature and higher rho invariant of fibered manifolds