Spectral sections, twisted rho invariants and positive scalar curvature

We had previously defined in [10], the rho invariant rho(spin) (Y, epsilon, H, g) for the twisted Dirac operator (epsilon)(H) on a closed odd dimensional Riemannian spin manifold (Y, g), acting on sections of a fiat hermitian vector bundle epsilon over Y, where H =Sigma i(j +1) H2j +1 is an odddegree differential form on Y and H2 j+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 pspi (Y, epsilon, H, g) rho(spin)(Y,epsilon, g) in terms of spectral flows and prove that p(spin) (Y, epsilon, H, g) is an element of Q, whenever g is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum Connes conjecture holds for in pi(1)(Y) (which is assumed to be torsion-free), then we show that rho(spin)(Y, epsilon, H, r g) = 0 for all r >> 0, significantly generalizing results in [10]. These results are proved using the Bismut Weitzenbock formula, a scaling trick, the technique of noncommutative spectral sections, and the Higson Roe approach [22].