Résumé
We had previously defined in [10], the rho invariant ρ spin pY, E, H, gq for the twisted Dirac operator g E H on a closed odd dimensional Riemannian spin manifold pY, gq, acting on sections of a flat hermitian vector bundle E over Y , where H ° i j 1 H 2j 1 is an odd-degree differential form on Y and H 2j 1 is a real-valued differential form of degree 2j 1. Here we show that it is a conformal invariant of the pair pH, gq. In this paper we express the defect integer ρ spin pY, E, H, gq¡ρ spin pY, E, gq in terms of spectral flows and prove that ρ spin pY, E, H, gq Q, whenever g is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum-Connes conjecture holds for π 1 pY q (which is assumed to be torsion-free), then we show that ρ spin pY, E, H, rgq 0 for all r 4 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].