Bordered Floer homology and contact structures

Alishahi, Akram and Földvári, Viktória and Hendricks, Kristen and Licata, Joan and Petkova, Ina and Vértesi, Vera (2023) Bordered Floer homology and contact structures. FORUM OF MATHEMATICS SIGMA. ISSN 2050-5094

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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, ξ, F) whose convex boundary is equipped with a signed singular foliation 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 cD and cA 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 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-Matíc.

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