Relating categorical dimensions in topology and symplectic geometry
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866918161784242176 |
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| author | Hanlon, Andrew Hicks, Jeff Lazarev, Oleg |
| author_facet | Hanlon, Andrew Hicks, Jeff Lazarev, Oleg |
| contents | We study several notions of dimension for (pre-)triangulated categories naturally arising from topology and symplectic geometry. We prove new bounds on these dimensions and raise several questions for further investigation. For instance, we relate the Rouquier dimension of the wrapped Fukaya category of either the cotangent bundle of a smooth manifold $M$ or more generally a Weinstein domain $X$ to quantities of geometric interest. These quantities include the minimum number of critical values of a Morse function on $M$, the Lusternik-Schnirelmann category of $M$, the number of distinct action values of a Hamiltonian diffeomorphism of $X$, and the smallest $n$ such that $X$ admits a Weinstein embedding into $\mathbb{R}^{2n+1}$. Along the way, we introduce a notion of the Lusternik-Schnirelmann category for dg-categories and construct exact Lagrangian cobordisms for restriction to a Liouville subdomain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_13677 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relating categorical dimensions in topology and symplectic geometry Hanlon, Andrew Hicks, Jeff Lazarev, Oleg Symplectic Geometry Algebraic Topology Category Theory 53D40, 18G80, 55M10 We study several notions of dimension for (pre-)triangulated categories naturally arising from topology and symplectic geometry. We prove new bounds on these dimensions and raise several questions for further investigation. For instance, we relate the Rouquier dimension of the wrapped Fukaya category of either the cotangent bundle of a smooth manifold $M$ or more generally a Weinstein domain $X$ to quantities of geometric interest. These quantities include the minimum number of critical values of a Morse function on $M$, the Lusternik-Schnirelmann category of $M$, the number of distinct action values of a Hamiltonian diffeomorphism of $X$, and the smallest $n$ such that $X$ admits a Weinstein embedding into $\mathbb{R}^{2n+1}$. Along the way, we introduce a notion of the Lusternik-Schnirelmann category for dg-categories and construct exact Lagrangian cobordisms for restriction to a Liouville subdomain. |
| title | Relating categorical dimensions in topology and symplectic geometry |
| topic | Symplectic Geometry Algebraic Topology Category Theory 53D40, 18G80, 55M10 |
| url | https://arxiv.org/abs/2308.13677 |