Discrete Morse theory on $ΩS^2$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916327907655680 |
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| author | Johnson, Lacey Knudson, Kevin |
| author_facet | Johnson, Lacey Knudson, Kevin |
| contents | A classical result in Morse theory is the determination of the homotopy type of the loop space of a manifold. In this paper, we study this result through the lens of discrete Morse theory. This requires a suitable simplicial model for the loop space. Here, we use Milnor's $\textrm{F}^+\textrm{K}$ construction to model the loop space of the sphere $S^2$, describe a discrete gradient on it, and identify a collection of critical cells. We also compute the action of the boundary operator in the Morse complex on these critical cells, showing that they are potential homology generators. A careful analysis allows us to recover the calculation of the first homology of $ΩS^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12156 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete Morse theory on $ΩS^2$ Johnson, Lacey Knudson, Kevin Algebraic Topology 57Q70, 55P35 A classical result in Morse theory is the determination of the homotopy type of the loop space of a manifold. In this paper, we study this result through the lens of discrete Morse theory. This requires a suitable simplicial model for the loop space. Here, we use Milnor's $\textrm{F}^+\textrm{K}$ construction to model the loop space of the sphere $S^2$, describe a discrete gradient on it, and identify a collection of critical cells. We also compute the action of the boundary operator in the Morse complex on these critical cells, showing that they are potential homology generators. A careful analysis allows us to recover the calculation of the first homology of $ΩS^2$. |
| title | Discrete Morse theory on $ΩS^2$ |
| topic | Algebraic Topology 57Q70, 55P35 |
| url | https://arxiv.org/abs/2407.12156 |