Cobordisms of sutured manifolds and the functoriality of link Floer homology
It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induce...
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| Format: | Journal article |
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Elsevier
2016
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| _version_ | 1797051695783477248 |
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| author | Juhász, A |
| author_facet | Juhász, A |
| author_sort | Juhász, A |
| collection | OXFORD |
| description | It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured Floer homology. This map is a common generalization of the hat version of the closed 3-manifold cobordism map in Heegaard Floer theory, and the contact gluing map defined by Honda, Kazez, and Mati´c. We show that sutured Floer homology, together with the above cobordism maps, forms a type of TQFT in the sense of Atiyah. Applied to the sutured manifold cobordism complementary to a decorated link cobordism, our theory gives rise to the desired map on link Floer homology. Hence, link Floer homology is a categorification of the multi-variable Alexander polynomial. We outline an alternative definition of the contact gluing map using only the contact element and handle maps. Finally, we show that a Weinstein sutured manifold cobordism preserves the contact element. |
| first_indexed | 2024-03-06T18:23:11Z |
| format | Journal article |
| id | oxford-uuid:0703d878-43c1-4763-91f5-d0ba1b083b1f |
| institution | University of Oxford |
| last_indexed | 2024-03-06T18:23:11Z |
| publishDate | 2016 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:0703d878-43c1-4763-91f5-d0ba1b083b1f2022-03-26T09:05:23ZCobordisms of sutured manifolds and the functoriality of link Floer homologyJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:0703d878-43c1-4763-91f5-d0ba1b083b1fSymplectic Elements at OxfordElsevier2016Juhász, AIt has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured Floer homology. This map is a common generalization of the hat version of the closed 3-manifold cobordism map in Heegaard Floer theory, and the contact gluing map defined by Honda, Kazez, and Mati´c. We show that sutured Floer homology, together with the above cobordism maps, forms a type of TQFT in the sense of Atiyah. Applied to the sutured manifold cobordism complementary to a decorated link cobordism, our theory gives rise to the desired map on link Floer homology. Hence, link Floer homology is a categorification of the multi-variable Alexander polynomial. We outline an alternative definition of the contact gluing map using only the contact element and handle maps. Finally, we show that a Weinstein sutured manifold cobordism preserves the contact element. |
| spellingShingle | Juhász, A Cobordisms of sutured manifolds and the functoriality of link Floer homology |
| title | Cobordisms of sutured manifolds and the functoriality of link Floer homology |
| title_full | Cobordisms of sutured manifolds and the functoriality of link Floer homology |
| title_fullStr | Cobordisms of sutured manifolds and the functoriality of link Floer homology |
| title_full_unstemmed | Cobordisms of sutured manifolds and the functoriality of link Floer homology |
| title_short | Cobordisms of sutured manifolds and the functoriality of link Floer homology |
| title_sort | cobordisms of sutured manifolds and the functoriality of link floer homology |
| work_keys_str_mv | AT juhasza cobordismsofsuturedmanifoldsandthefunctorialityoflinkfloerhomology |