Morse theory on Lie groupoids

Cristian Ortiz; Fabricio Valencia

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arXiv · 2207.07594

Morse theory on Lie groupoids

Cristian Ortiz·Fabricio Valencia

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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.

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BibTeX RIS

Cristian Ortiz, Fabricio Valencia. 2024-06-02. Morse theory on Lie groupoids. https://arxiv.org/abs/2207.07594

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