Stated skein modules of 3-manifolds and TQFT
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| Format: | Preprint |
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2022
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| _version_ | 1866912326867746816 |
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| author | Costantino, Francesco Le, Thang T. Q. |
| author_facet | Costantino, Francesco Le, Thang T. Q. |
| contents | We study the behaviour of the Kauffman bracket skein modules of 3-manifolds under gluing along surfaces. For this purpose we extend the notion of Kauffman bracket skein modules to $3$-manifolds with marking consisting of open intervals and circles in the boundary. The new module is called the stated skein module.
The first main results concern non-injectivity of certain natural maps defined when forming connected sums along a sphere or along a closed disk. These maps are injective for surfaces, or for generic quantum parameter, but we show that in general they are not injective when the quantum parameter is a root of 1. The result applies to the classical skein modules as well. A particular interesting result is that when the quantum parameter is a root of 1, the empty skein is zero in a connected sum where each constituent manifold has non-empty marking.
We also prove various non injectivity results for the Chebyshev-Frobenius map and the natural map induced by the deletion of marked balls.
We then consider the general case of gluing along a surface, showing that the stated skein module can be interpreted as a monoidal symmetric functor from a category of "decorated cobordisms" to a Morita category of algebras and their bimodules. We apply this result to deduce several properties of stated skein modules as a Van-Kampen like theorem as well as a computation through Heegaard decompositions and a relation to Hochshild homology for trivial circle bundles over surfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_10906 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stated skein modules of 3-manifolds and TQFT Costantino, Francesco Le, Thang T. Q. Geometric Topology Quantum Algebra Primary 57N10, Secondary: 57M25 We study the behaviour of the Kauffman bracket skein modules of 3-manifolds under gluing along surfaces. For this purpose we extend the notion of Kauffman bracket skein modules to $3$-manifolds with marking consisting of open intervals and circles in the boundary. The new module is called the stated skein module. The first main results concern non-injectivity of certain natural maps defined when forming connected sums along a sphere or along a closed disk. These maps are injective for surfaces, or for generic quantum parameter, but we show that in general they are not injective when the quantum parameter is a root of 1. The result applies to the classical skein modules as well. A particular interesting result is that when the quantum parameter is a root of 1, the empty skein is zero in a connected sum where each constituent manifold has non-empty marking. We also prove various non injectivity results for the Chebyshev-Frobenius map and the natural map induced by the deletion of marked balls. We then consider the general case of gluing along a surface, showing that the stated skein module can be interpreted as a monoidal symmetric functor from a category of "decorated cobordisms" to a Morita category of algebras and their bimodules. We apply this result to deduce several properties of stated skein modules as a Van-Kampen like theorem as well as a computation through Heegaard decompositions and a relation to Hochshild homology for trivial circle bundles over surfaces. |
| title | Stated skein modules of 3-manifolds and TQFT |
| topic | Geometric Topology Quantum Algebra Primary 57N10, Secondary: 57M25 |
| url | https://arxiv.org/abs/2206.10906 |