Dehn twists in Heegaard Floer homology
The main results of this paper are two long exact sequences in the hat-version of Heegaard Floer homology in \(\mathbb{Z}_2\)-coefficients, together with their application in contact geometry. A Heegaard splitting represents a \(3\)-manifold \(Y\) as glued from two handlebodies of genus \(g\) along their boundary \(\Sigma\). A Heegaard diagram describes the gluing map using two collections of \(g\) embedded curves \((\alpha, \beta)\). The collections \(\alpha\) and \(\beta\) define Lagrangian tori \(T_\alpha\) and \(T_\beta\) in the \(g\)-fold symmetric product \(Sym^g \Sigma\). When the Heegaard diagram is admissible, the hat-version of Heegaard Floer homology \(\widehat{HF}(Y)\) is defined from a variant of the Lagrangian Floer homology for the pair \(T_\alpha\) and \(T_\beta\) [\textit{P.~Ozsváth} and \textit{Z.~Szabó}, Ann. Math. (2) 159, 1027--1158 (2004; Zbl 1073.57009)]. Let \(K\) be a knot in \(Y\), then a Heegaard splitting can be chosen such that \(K\) is an embedded curve on \(\Sigma\). Using a Heegaard diagram subordinate to the knot \(K\), a similar variant of the Lagrangian Floer homology defines the hat-version of knot Floer homology \(\widehat{HFK}(Y, K)\). In \S 2, the author shows that the construction of \(\widehat{HFK}(Y, K)\) [\textit{P.~Ozsváth} and \textit{Z.~Szabó}, Adv. Math. 186, 58--116 (2004; Zbl 1062.57019)] extends to arbitrary knots \(K\). Let \(Y_0(K)\) (respectively \(Y_{\pm 1}(K)\)) be the result of \(0\)-surgery (respectively \((\pm 1)\)-surgery) along the knot \(K\) and let \(\mu\) be the meridian of \(K\) in \(Y\). The main result of the present paper is the exact sequence proven in \S 4: \[ \ldots \to \widehat{HFK}(Y, K) \to \widehat{HF}(Y_{-1}(K)) \to \widehat{HFK}(Y_0(K), \mu) \to \ldots \] and a similar one with \(\widehat{HF}(Y_{+1}(K))\) and opposite arrows instead. The Dehn twists in the title is due to the fact that \(Y_{\pm 1}(K)\) can be obtained by performing a \((\mp)\)-Dehn surgery along \(K\) on the gluing map that defines \(Y\). The exact sequences are shown by analysing directly how the Heegaard Floer complex behaves under the Dehn twists, using \(\delta\)-adapted Heegaard diagrams (Definition 4.1). In \S 5, the author shows that the map \(\Gamma_1\) in the exact sequences that relates \(\widehat{HFK}(Y, K)\) and \(\widehat{HF}(Y_{\pm 1}(K))\) is topological, i.e. it depends only on the cobordism associated to the surgery. The rest of the paper (\S 6 and \S 7) discusses the implication of the exact sequence when \(K = L\) is a Legendrian knot in a contact manifold \((Y, \xi)\). In this case, one performs \((\pm 1)\)-surgery along \(L\) in the contact category and uses the cohomological version of the exact sequences (which corresponds to changing the orientation of \(Y\), and the directions of the maps). Two invariants are discussed. The first one is the \textit{contact element} \(c(Y, \xi) \in \widehat{HF}(-Y)\) [\textit{K.~Honda, W.~H.~Kazez} and \textit{G.~Matić}, J. Differ. Geom. 83, No.~2 289--311 (2009; Zbl 1186.53098)] and the other is the Legendrian invariant \(\widehat{\mathcal L}(L) \in \widehat{HFK}(-Y, L)\) of \(L\) [\textit{P.~Lisca, P.~Ozváth, A.~I.~Stipsicz} and \textit{Z.~Szabó}, J. Eur. Math. Soc. (JEMS) 11, No.~6, 1307--1363 (2009; Zbl 1232.57017)]. Let \((Y_L^\pm, \xi_L^\pm)\) denote the result of the \((\pm 1)\)-contact surgery, then the main result in this part of the paper is \[ \Gamma_{-W}(\widehat{\mathcal L}(L)) = c(Y^+_L, \xi_L^+) \text{ and } \Gamma_{-W}(c(Y_L^-, \xi_L^-)) = 0, \] where \(W\) is the cobordism associated to the contact surgeries and \(\Gamma_{-W}\) corresponds to \(\Gamma_1\) above. The paper goes on to prove relations among other contact invariants, as well as new and known vanishing results for \(c(Y^+_L, \xi_L^+)\) for various types of \((Y, \xi, L)\).
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