Spectral triples on proper étale groupoids

Abstract

A proper etale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel-Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of the tangent bundle and the obstruction to lift the tangent bundle to a spinor bundle. In the case of an orientable and spin Lie groupoid, an invariant spinor bundle and an invariant Dirac operator will be constructed. This data gives rise to a spectral triple. The algebraic orientability axiom in noncommutative geometry is reformulated to make it compatible with the geometric model.