Spectral sections, twisted rho invariants and positive scalar curvature

Abstract

We had previously defined in [10], the rho invariant \rho_{spin}(Y,\Epsilon,H, g) for the twisted Dirac operator \mathrlap{/}{\partial}^{\mathcal E}_H on a closed odd dimensional Riemannian spin manifold (Y, g) , acting on sections of a flat hermitian vector bundle \Epsilon over Y , where H = \sum 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 (H, g) . In this paper we express the defect integer \rho_{spin}(Y,\Epsilon,H, g) - \rho_{spin}(Y,\Epsilon, g) in terms of spectral flows and prove that \rho_{spin}(Y,\Epsilon,H, g) \in \mathbb Q , whenever g is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum–Connes conjecture holds for \pi_1(Y) (which is assumed to be torsion-free), then we show that \rho_{spin}(Y,\Epsilon,H, rg) =0 for all r \gg 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].