Homotopy invariance of relative eta-invariants and 𝐶*-algebra 𝐾-theory

Abstract

We prove a close cousin of a theorem of Weinberger about the homotopy invariance of certain relative eta-invariants by placing the problem in operator K K -theory. The main idea is to use a homotopy equivalence h : M → M ′ h:M \to M’ to construct a loop of invertible operators whose “winding number" is related to eta-invariants. The Baum-Connes conjecture and a technique motivated by the Atiyah-Singer index theorem provides us with the invariance of this winding number under twistings by finite-dimensional unitary representations of π 1 ( M ) \pi _{1}(M) .