Morse theory on Lie groupoids (Q6564146)

Classical Morse theory has been generalized and adapted to different settings in various ways. The aim of this paper is to study Morse theory on Lie groupoids and their differentiable stacks. The authors begin by introducing Morse Lie groupoid morphisms and by studying their main properties. In particular, they show that the property of being a Morse Lie groupoid morphism is Morita invariant. They also consider several examples. They then go on to state a version of the Morse lemma in the Lie groupoid setting. The lemma is used to describe topological behaviour of the critical subgroupoid levels of a Morse Lie groupoid morphism around its nondegenerate critical orbits.\N\NA stack can be thought of as a generalization of a manifold which makes it possible to study higher symmetries and singular geometric features. In Section 7, the authors adapt some of the Morse theory results for Lie groupoids to the setting of differentiable stacks. They obtain Morse type inequalities for certain separated differentiable stacks. In the last two sections, they study Morse-Smale dynamics and the Morse double complex in the framework of Lie groupoids.