Reeb graph invariants of Morse functions and $3$-manifold groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914701294698496 |
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| author | Michalak, Łukasz Patryk |
| author_facet | Michalak, Łukasz Patryk |
| contents | In this work we are focused on the existence of Morse functions on a closed manifold $M$ which are far from being ordered, i.e. whose Reeb graphs have positive first Betti number, especially the maximal possible, equals $\operatorname{corank}(π_1(M))$. In the case of $3$-manifolds we describe the minimal number of critical points needed to construct such functions, which is related with the number of vertices of degree $2$ in Reeb graphs. We define a new invariant of $3$-manifold groups and their presentations, and using Heegaard splittings we show its utility in determining occurrence of disordered Morse functions. In particular, the Freiheitssatz, a result for one-relator groups, allows us to calculate this invariant in the case of orientable circle-bundles over a surface, which provides an interesting example of the behaviour of Morse functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02291 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reeb graph invariants of Morse functions and $3$-manifold groups Michalak, Łukasz Patryk Geometric Topology Group Theory Primary: 57M15, Secondary: 57K31, 20F05 In this work we are focused on the existence of Morse functions on a closed manifold $M$ which are far from being ordered, i.e. whose Reeb graphs have positive first Betti number, especially the maximal possible, equals $\operatorname{corank}(π_1(M))$. In the case of $3$-manifolds we describe the minimal number of critical points needed to construct such functions, which is related with the number of vertices of degree $2$ in Reeb graphs. We define a new invariant of $3$-manifold groups and their presentations, and using Heegaard splittings we show its utility in determining occurrence of disordered Morse functions. In particular, the Freiheitssatz, a result for one-relator groups, allows us to calculate this invariant in the case of orientable circle-bundles over a surface, which provides an interesting example of the behaviour of Morse functions. |
| title | Reeb graph invariants of Morse functions and $3$-manifold groups |
| topic | Geometric Topology Group Theory Primary: 57M15, Secondary: 57K31, 20F05 |
| url | https://arxiv.org/abs/2403.02291 |