Defining and classifying TQFTs via surgery
We give a presentation of the n-dimensional oriented cobordism category Cobn with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor F from the category of smooth oriented manifolds and diffeomorphisms to an arbitra...
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| Format: | Journal article |
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European Mathematical Society
2018
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| _version_ | 1797055291742748672 |
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| author | Juhasz, A |
| author_facet | Juhasz, A |
| author_sort | Juhasz, A |
| collection | OXFORD |
| description | We give a presentation of the n-dimensional oriented cobordism category Cobn with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor F from the category of smooth oriented manifolds and diffeomorphisms to an arbitrary category C, and morphisms induced by surgeries along framed spheres, we obtain a necessary and sufficient set of relations these have to satisfy to extend to a functor from Cobn to C. If C is symmetric and monoidal, then we also characterize when the extension is a TQFT. This framework is well-suited to defining natural cobordism maps in Heegaard Floer homology. It also allows us to give a short proof of the classical correspondence between (1+1)-dimensional TQFTs and commutative Frobenius algebras. Finally, we use it to classify (2+1)-dimensional TQFTs in terms of J-algebras, a new algebraic structure that consists of a split graded involutive nearly Frobenius algebra endowed with a certain mapping class group representation. This solves a long-standing open problem. As a corollary, we obtain a structure theorem for (2+1)-dimensional TQFTs that assign a vector space of the same dimension to every connected surface. We also note that there are 22 ω nonequivalent lax monoidal TQFTs over C that do not extend to (1+1+1)-dimensional ones. |
| first_indexed | 2024-03-06T19:08:35Z |
| format | Journal article |
| id | oxford-uuid:16021b4e-6b4d-4109-9fda-34d4461855be |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:08:35Z |
| publishDate | 2018 |
| publisher | European Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:16021b4e-6b4d-4109-9fda-34d4461855be2022-03-26T10:28:46ZDefining and classifying TQFTs via surgeryJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:16021b4e-6b4d-4109-9fda-34d4461855beSymplectic Elements at OxfordEuropean Mathematical Society2018Juhasz, AWe give a presentation of the n-dimensional oriented cobordism category Cobn with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor F from the category of smooth oriented manifolds and diffeomorphisms to an arbitrary category C, and morphisms induced by surgeries along framed spheres, we obtain a necessary and sufficient set of relations these have to satisfy to extend to a functor from Cobn to C. If C is symmetric and monoidal, then we also characterize when the extension is a TQFT. This framework is well-suited to defining natural cobordism maps in Heegaard Floer homology. It also allows us to give a short proof of the classical correspondence between (1+1)-dimensional TQFTs and commutative Frobenius algebras. Finally, we use it to classify (2+1)-dimensional TQFTs in terms of J-algebras, a new algebraic structure that consists of a split graded involutive nearly Frobenius algebra endowed with a certain mapping class group representation. This solves a long-standing open problem. As a corollary, we obtain a structure theorem for (2+1)-dimensional TQFTs that assign a vector space of the same dimension to every connected surface. We also note that there are 22 ω nonequivalent lax monoidal TQFTs over C that do not extend to (1+1+1)-dimensional ones. |
| spellingShingle | Juhasz, A Defining and classifying TQFTs via surgery |
| title | Defining and classifying TQFTs via surgery |
| title_full | Defining and classifying TQFTs via surgery |
| title_fullStr | Defining and classifying TQFTs via surgery |
| title_full_unstemmed | Defining and classifying TQFTs via surgery |
| title_short | Defining and classifying TQFTs via surgery |
| title_sort | defining and classifying tqfts via surgery |
| work_keys_str_mv | AT juhasza definingandclassifyingtqftsviasurgery |