The blow-down map in Lie algebroid cohomology
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929399495917568 |
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| author | Schüßler, Andreas |
| author_facet | Schüßler, Andreas |
| contents | We study the blow-down map in cohomology in the context of real projective blowups of Lie algebroids. Using the blow-down map in cohomology we compute the Lie algebroid cohomology of the blowup of transversals of arbitrary codimension, generalising the Mazzeo-Melrose theorem on b-cohomology. To prove the result we develop a Gysin sequence for Lie algebroids. As another example we use the developed tools to compute the Lie algebroid cohomology of the action Lie algebroid $\mathfrak{so}(3)\ltimes \mathbb{R}^3$, a result known in Poisson geometry literature. Moreover, we use similar techniques to compute the de Rham cohomology of real projective blowups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_17668 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The blow-down map in Lie algebroid cohomology Schüßler, Andreas Differential Geometry Symplectic Geometry 53D17, 17B56 We study the blow-down map in cohomology in the context of real projective blowups of Lie algebroids. Using the blow-down map in cohomology we compute the Lie algebroid cohomology of the blowup of transversals of arbitrary codimension, generalising the Mazzeo-Melrose theorem on b-cohomology. To prove the result we develop a Gysin sequence for Lie algebroids. As another example we use the developed tools to compute the Lie algebroid cohomology of the action Lie algebroid $\mathfrak{so}(3)\ltimes \mathbb{R}^3$, a result known in Poisson geometry literature. Moreover, we use similar techniques to compute the de Rham cohomology of real projective blowups. |
| title | The blow-down map in Lie algebroid cohomology |
| topic | Differential Geometry Symplectic Geometry 53D17, 17B56 |
| url | https://arxiv.org/abs/2406.17668 |