Morse theory on Lie groupoids

TL;DR

This paper extends Morse theory to Lie groupoids and differentiable stacks, introducing Morse Lie groupoid morphisms, establishing their properties, and connecting them to cohomology theories.

Contribution

It introduces Morse Lie groupoid morphisms, proves their Morita invariance, and develops a Morse theory framework for differentiable stacks and Lie groupoids.

Findings

Morse Lie groupoid morphisms are Morita invariant.

A groupoid version of the Morse lemma is established.

A Morse double complex relates to Bott-Shulman-Stasheff cohomology.

Abstract

In this paper we introduce Morse Lie groupoid morphisms and study their main properties. We show that this notion is Morita invariant which gives rise to a well defined notion of Morse function on differentiable stacks. We show a groupoid version of the Morse lemma which is used to describe the topological behavior of the critical subgroupoid levels of a Morse Lie groupoid morphism around its nondegenerate critical orbits. We also prove Morse type inequalities for certain separated differentiable stacks and construct a Morse double complex whose total cohomology is isomorphic to the Bott-Shulman-Stasheff cohomology of the underlying Lie groupoid. We provide several examples and applications.