The index of a transverse Dirac-type operator: the case of abelian Molino sheaf

Abstract

Abstract. We give a local formula for the index of a transverse Dirac-type operator on a compact manifold with a Riemannian foliation, under the assumption that the Molino sheaf is a sheaf of abelian Lie algebras.

Formality and Kontsevich–Duflo type theorems for Lie pairs

“…In a separate publication, we will study Tsygan's formality for Lie pairs and its application to the index theorem. When the Lie pair arises from a regular foliation, it would be interesting to explore the connection with the work of Gorokhovsky-Lott [31] and Pflaum-Posthuma-Tang [57] on the transverse index theorem.…”

Spectral triples on proper étale groupoids

“…To this end, we develop an explicit model for a spectral triple on the smooth convolution algebra C 8 c pΘq of an effective properétale Lie groupoid. Similar model has been studied in [5]. I wish to thank Andrzej Sitarz and the referee for valuable comments leading to several improvements.…”

“…Since X is complete, ð has a self-adjoint closure on L 2 pF Σ q which makes ð finitely summable. The action (8) defines a faithful˚-representation which preserves the domain of ð, [5]. It remains to verify that the commutators rð, f s extend to bounded operators on L 2 pF Σ q.…”

Resolutions of Proper Riemannian Lie Groupoids

“…Example 7.7. Another example of a proper Lie groupoid is constructed in [11]. Here one starts with a regular Riemannian foliation (M, F, g) and considers the action of the holonomy groupoid Hol(F) on the normal bundle N F to the foliation.…”