Higher ρ -invariants and the surgery structure set

Abstract

We study noncommutative ηand 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.

Mapping the surgery exact sequence for topological manifolds to analysis

“…In this section we are going to check that the construction we made for and Γ are well defined on the structure set S T OP (N ). For this purpose we will use the results presented in [32,18,17], where the authors have developed the theory in the smooth setting. Their methods are rather abstract and they also hold in the Lipschitz context.…”

“…First of all we need a generalization of Theorem 3.2 for manifolds with cylindrical ends. This result is given by [32,Proposition 8.1], where a perturbation of the signature operator is associated to the homotopy equivalence F . Such a perturbation makes the operator invertible, as in the usual case.…”

Relative geometric assembly and mapping cones, part I: the geometric model and applications

“…The fact that (u, u ∂ ) is of product type near the boundary allows us to use [44,Theorem 8.4] which implies that ind AP S (D W ∪−W sign , (α ∪ α )φ * A) = 0. We conclude that x = 0.…”

Adiabatic groupoid and secondary invariants in K-theory

“…We want to construct an operator in L(E 1 , E 2 ) that satisfies the hypotheses of [23, Lemma 2.1]. The following material is from [23] and [50,Section 2].…”