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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.06781 |
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| _version_ | 1866912591044935680 |
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| author | Ba, Te Zhou, Ze |
| author_facet | Ba, Te Zhou, Ze |
| contents | We show that a smooth $d$-manifold $M$ is diffeomorphic to $\mathbb R^d$ if it admits a Lyapunov-Reeb function, i.e., a smooth map $f:M\to\mathbb R$ that is proper, lower-bounded, and has a unique critical point. By constructing such functions, we prove that the moduli spaces of self-avoiding polygonal linkages and configurations are diffeomorphic to Euclidean spaces. This resolves the Refined Carpenter's Rule Problem and confirms a conjecture proposed by González and Sedano-Mendoza. Furthermore, we describe foliation structures of these moduli spaces via level sets of Lyapunov-Reeb functions and develop algorithms for related problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06781 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Morse theory and moduli spaces of self-avoiding polygonal linkages Ba, Te Zhou, Ze Combinatorics Dynamical Systems Geometric Topology 52C25, 57R70, 68U15 We show that a smooth $d$-manifold $M$ is diffeomorphic to $\mathbb R^d$ if it admits a Lyapunov-Reeb function, i.e., a smooth map $f:M\to\mathbb R$ that is proper, lower-bounded, and has a unique critical point. By constructing such functions, we prove that the moduli spaces of self-avoiding polygonal linkages and configurations are diffeomorphic to Euclidean spaces. This resolves the Refined Carpenter's Rule Problem and confirms a conjecture proposed by González and Sedano-Mendoza. Furthermore, we describe foliation structures of these moduli spaces via level sets of Lyapunov-Reeb functions and develop algorithms for related problems. |
| title | Morse theory and moduli spaces of self-avoiding polygonal linkages |
| topic | Combinatorics Dynamical Systems Geometric Topology 52C25, 57R70, 68U15 |
| url | https://arxiv.org/abs/2506.06781 |