Morse theory on Lie groupoids
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910466083651584 |
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| author | Ortiz, Cristian Valencia, Fabricio |
| author_facet | Ortiz, Cristian Valencia, Fabricio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_07594 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Morse theory on Lie groupoids Ortiz, Cristian Valencia, Fabricio Differential Geometry 22A22, 58H05, 57R70, 37D15 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. |
| title | Morse theory on Lie groupoids |
| topic | Differential Geometry 22A22, 58H05, 57R70, 37D15 |
| url | https://arxiv.org/abs/2207.07594 |