Enlargeable foliations and the monodromy groupoid

TL;DR

This paper extends the concept of the Rosenberg index, originally used to obstruct positive scalar curvature on spin manifolds, to the setting of spin foliations, linking enlargeability to nonvanishing index elements.

Contribution

It introduces a foliation version of the Rosenberg index and proves its nonvanishing under compact enlargeability of the foliation.

Findings

Foliation Rosenberg index is nonzero for compactly enlargeable spin foliations.

Generalizes scalar curvature obstructions to foliated manifolds.

Establishes a new link between enlargeability and index theory in foliation geometry.

Abstract

Let $M$ be a spin manifold, the Dirac operator with coefficient in the universal flat Hilbert $C^\ast \pi_1(M)$-module determines a "Rosenberg index element" which, according to B.Hanke and T.Schick, subsumes the enlargeablility obstruction of positive scalar curvature on $M$. In this note, we generalize this result to the case of spin foliation. More precisely, given a foliation $(M,F)$ with $F$ spin, we shall define a foliation version of "Rosenberg index element" and prove that it is nonzero at the presence of compactly enlargeability of $(M,F)$.