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We define larger variants of the vector spaces one obtains by decategorifying bordered (sutured) Heegaard Floer invariants of surfaces. We also define bimodule structures on these larger spaces that are similar to, but more elaborate than, the bimodule structures that arise from decategorifying the higher actions in bordered Heegaard Floer theory introduced by Rouquier and the author. In particular, these new bimodule structures involve actions of both odd generatorsEandFof\mathfrak{gl}(1|1), whereas the previous ones only involved actions ofE. Over\mathbb{F}_2, we show that the new bimodules satisfy the necessary gluing properties to give a 1+1 open-closed TQFT valued in graded algebras and bimodules up to isomorphism; in particular, unlike in previous related work, we have a gluing theorem when gluing surfaces along circles as well as intervals. Over the integers, we show that a similar construction gives two partially defined open-closed TQFTs with two different domains of definition depending on how parities are chosen for the bimodules. We formulate conjectures relating these open-closed TQFTs with the\mathfrak{psl}(1|1)Chern–Simons TQFT recently studied by Mikhaylov and Geer–Young.
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Bordered Floer homology and contact structures
Abstract We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $$(M, \xi , \mathcal {F})$$ whose convex boundary is equipped with a signed singular foliation $$\mathcal {F}$$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $$c_D$$ and $$c_A$$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $$\mathcal {F}$$ in favor of the dividing set and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda–Kazez–Matić.
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- PAR ID:
- 10413989
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 11
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fms.2023.19
