Combed Trisection Diagrams and Non-Semisimple 4-Manifold Invariants
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916493348831232 |
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| author | Chaidez, Julian Cotler, Jordan Cui, Shawn X. |
| author_facet | Chaidez, Julian Cotler, Jordan Cui, Shawn X. |
| contents | Given a triple $H$ of (possibly non-semisimple) Hopf algebras equipped with pairings satisfying a set of properties, we describe a construction of an associated smooth, scalar invariant $τ_H(X,π)$ of a simply connected, compact, oriented $4$-manifold $X$ and an open book $π$ on its boundary. This invariant generalizes an earlier semisimple version and is calculated using a trisection diagram $T$ for $X$ and a certain type of combing of the trisection surface. We explain a general calculation of this invariant for a family of exotic 4-manifolds with boundary called Stein nuclei, introduced by Yasui. After investigating many low-dimensional Hopf algebras up to dimension 11, we have not been able to find non-semisimple Hopf triples that satisfy the criteria for our invariant. Nonetheless, appropriate Hopf triples may exist outside the scope of our explorations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08461 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Combed Trisection Diagrams and Non-Semisimple 4-Manifold Invariants Chaidez, Julian Cotler, Jordan Cui, Shawn X. Quantum Algebra Mathematical Physics Geometric Topology Given a triple $H$ of (possibly non-semisimple) Hopf algebras equipped with pairings satisfying a set of properties, we describe a construction of an associated smooth, scalar invariant $τ_H(X,π)$ of a simply connected, compact, oriented $4$-manifold $X$ and an open book $π$ on its boundary. This invariant generalizes an earlier semisimple version and is calculated using a trisection diagram $T$ for $X$ and a certain type of combing of the trisection surface. We explain a general calculation of this invariant for a family of exotic 4-manifolds with boundary called Stein nuclei, introduced by Yasui. After investigating many low-dimensional Hopf algebras up to dimension 11, we have not been able to find non-semisimple Hopf triples that satisfy the criteria for our invariant. Nonetheless, appropriate Hopf triples may exist outside the scope of our explorations. |
| title | Combed Trisection Diagrams and Non-Semisimple 4-Manifold Invariants |
| topic | Quantum Algebra Mathematical Physics Geometric Topology |
| url | https://arxiv.org/abs/2309.08461 |