Enlargeable Foliations and the Monodromy Groupoid: Infinite Covers

TL;DR

This paper proves that the foliated Rosenberg index is nonzero for certain enlargeable, spin foliations, extending previous results to noncompact cases using advanced index theory and $KK$-theory techniques.

Contribution

It generalizes the nonvanishing index result to noncompactly enlargeable, spin foliations by employing relative index theorem and $KK$-equivalence methods.

Findings

Noncompactly enlargeable, spin foliations have nonzero foliated Rosenberg index.

The use of $KK$-theory reduces infinite-dimensional problems to finite dimensions.

Extension of previous results to broader classes of foliations.

Abstract

In this paper, we prove that the foliated Rosenberg index of a possibly noncompactly enlargeable, spin foliation is nonzero. It generalizes our previous result. The difficulty brought by the noncompactness is reflected in the infinite dimensionality of some vector bundles which, fortunately, can be reduced to finite dimensional vector bundles by the idea of relative index theorem and $KK$-equivalence between the $C^\ast$-algebra of compact operators and $\mathbb{C}$.