Veering triangulations and transverse foliations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910680391614464 |
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| author | Zung, Jonathan |
| author_facet | Zung, Jonathan |
| contents | We present a combinatorial approach to the existence of foliations and contact structures transverse to a given pseudo-Anosov flow. Let $φ$ be a transitive pseudo-Anosov flow on a closed oriented 3-manifold. Our main technical result is that every codimension 1 foliation transverse to $φ$ is carried by a single branched surface coming from a veering triangulation. Combined with recent breakthrough work of Massoni, this reduces the existence problem for transverse foliations to something like the feasibility of a system of inequalities (rather than equations!) over $Homeo_+([0,1])$. As a proof of concept, we show that for the hyperbolic, fibered, non-L-space knot $10_{145}$, the natural pseudo-Anosov flow on the slope $s$ Dehn surgery admits a transverse foliation for $s\in (-\infty, 3)$, but does not admit such a foliation for $s\in [5,\infty)$. The negative result is part of a more general Milnor--Wood type phenomenon which puts limitations on some well known methods for constructing taut foliations on Dehn surgeries. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_00227 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Veering triangulations and transverse foliations Zung, Jonathan Geometric Topology 57R30 We present a combinatorial approach to the existence of foliations and contact structures transverse to a given pseudo-Anosov flow. Let $φ$ be a transitive pseudo-Anosov flow on a closed oriented 3-manifold. Our main technical result is that every codimension 1 foliation transverse to $φ$ is carried by a single branched surface coming from a veering triangulation. Combined with recent breakthrough work of Massoni, this reduces the existence problem for transverse foliations to something like the feasibility of a system of inequalities (rather than equations!) over $Homeo_+([0,1])$. As a proof of concept, we show that for the hyperbolic, fibered, non-L-space knot $10_{145}$, the natural pseudo-Anosov flow on the slope $s$ Dehn surgery admits a transverse foliation for $s\in (-\infty, 3)$, but does not admit such a foliation for $s\in [5,\infty)$. The negative result is part of a more general Milnor--Wood type phenomenon which puts limitations on some well known methods for constructing taut foliations on Dehn surgeries. |
| title | Veering triangulations and transverse foliations |
| topic | Geometric Topology 57R30 |
| url | https://arxiv.org/abs/2411.00227 |