Morse theory on Lie groupoids - Quintero, Fabricio Valencia (2024)
We extend classical Morse theory to the realm of Lie groupoids and their differentiable stacks. This allows us to obtain both topological and geometrical information of the singular objects represented by the corresponding orbit spaces, offering a unified approach to study equivariant Morse theory as well as Morse theory for orbifolds. We show a groupoid version of the Morse lemma, describe the topological behavior of Lie groupoids around nondegenerate critical orbits, study MorseSmale dynamics, and recover the BottShulmanStasheff cohomology of a Lie groupoid by using Morse theory techniques. We define Morse stacky functions, thus proving analogues of the previous results in the context of differentiable stacks. The latter enables us to get Morse-like inequalities for compact orbit spaces of proper Lie groupoids. In order to develop a 2-equivariant Morse theory over Lie groupoids we introduce a natural notion of isometric Lie 2-group action on Riemannian groupoids. The global and infinitesimal counterparts of such a notion are explored in detail. We also study the existence of closed stacky geodesics on Riemannian stacks and describe constructions which explain how to obtain the equivariant cohomology of toric symplectic stacks.