Enlargeable foliations and the monodromy groupoid

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.