Spectral sections, twisted rho invariants and positive scalar curvature

Abstract

We had previously defined in [10], the rho invariant ρspin(Y; ε;H; g) for the twisted Dirac operator δεH on a closed odd dimensional Riemannian spin manifold (Y; g), acting on sections of a flat hermitian vector bundle ε over Y, where H = Σij+1H2j+1 is an odddegree differential form on Y and H2j+1 is a real-valued differential form of degree 2j + 1. Here we show that it is a conformal invariant of the pair (H; g). In this paper we express the defect integer ρspin(Y; ε;H; g)-ρspin(Y; ε; g) in terms of spectral flows and prove that ρspin(Y; ε;H; g) ⊂ Q, whenever g is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum-Connes conjecture holds for π1(Y) (which is assumed to be torsion-free), then we show that ρspin(Y; ε;H; rg) D 0 for all r ≤ 0, significantly generalizing results in [10]. These results are proved using the Bismut-Weitzenböck formula, a scaling trick, the technique of noncommutative spectral sections, and the Higson-Roe approach [22].