Enlargeable foliations and the monodromy groupoid

Abstract

Let $M$ be a closed spin manifold, the Dirac operator with coefficient in the universal flat Hilbert $C^{\ast }{\pi }_1$-module determines a Rosenberg index element which, according to B. Hanke and T. Schick, subsumes the enlargeability obstruction of positive scalar curvature on $M$. In this paper, we generalize this result to the case of spin foliation. More precisely, given a foliation $(M,F)$ with $F$ spin, we will define a foliation version of Rosenberg index element and prove that it is nonzero in the presence of enlargeability of $(M,F)$. As a consequence, the foliation version of Rosenberg index element subsumes the enlargeability obstruction to the existence of leafwise positive scalar curvature metric.