Morse theory on Lie groupoids

Abstract

In this paper we introduce Morse Lie groupoid morphisms and we study their main properties. We show that Morse Lie groupoid morphisms are Morita invariant, giving rise to a good notion of Morse function on a differentiable stack. We show a groupoid version of the Morse-Bott lemma, which allows us to speak about the index of a non-degenerate critical subgroupoid and hence to describe critical sub-levels by an attaching procedure in the category of topological groupoids. Motivated by the Morse-Bott complex defined by Austin and Braam, we introduce the groupoid Morse double complex, showing that its total cohomology is isomorphic to the Bott-Shulman-Stasheff cohomology of the underlying Lie groupoid. Then we study Morse Lie groupoid morphisms which are invariant under the action of a Lie 2-group yielding an equivariant version of the groupoid Morse double complex. We show that in this case, the associated cohomology is isomorphic to the corresponding equivariant cohomology of the Lie 2-group action. As an application, we show that the equivariant cohomology of toric symplectic stacks can be computed by means of groupoid Morse theory.Comment: 41 pages. Comments are welcom