Spectral sections, twisted rho invariants and positive scalar curvature

Abstract

We had previously defined in [10], the rho invariant ${\rho }_{spin}(Y,\mathrm{E},H,g)$ for the twisted Dirac operator $/{\partial }_H^{\mathcal{E}}$ on a closed odd dimensional Riemannian spin manifold $(Y,g)$, acting on sections of a flat hermitian vector bundle $\mathrm{E}$ 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 ${\rho }_{spin}(Y,\mathrm{E},H,g)-{\rho }_{spin}(Y,\mathrm{E},g)$ in terms of spectral flows and prove that ${\rho }_{spin}(Y,\mathrm{E},H,g)\in \mathbb{Q}$, 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 ${\rho }_{spin}(Y,\mathrm{E},H,rg)=0$ 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].