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October 2018 Naturality of Heegaard Floer invariants under positive rational contact surgery
Thomas E. Mark, Bülent Tosun
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J. Differential Geom. 110(2): 281-344 (October 2018). DOI: 10.4310/jdg/1538791245


For a nullhomologous Legendrian knot in a closed contact $3$-manifold $Y$ we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of $Y$ is mapped by a surgery cobordism to the contact invariant of the result of contact surgery, and we characterize the $\mathrm{spin}^c$ structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery on a Legendrian knot in the standard $3$-sphere, generalizing previous results of Lisca–Stipsicz and Golla. In fact, our methods allow direct calculation of the contact invariant in terms of the rational surgery mapping cone of Ozsváth and Szabó. The proof involves a construction called reducible open book surgery, which reduces in special cases to the capping-off construction studied by Baldwin.


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Thomas E. Mark. Bülent Tosun. "Naturality of Heegaard Floer invariants under positive rational contact surgery." J. Differential Geom. 110 (2) 281 - 344, October 2018.


Received: 23 September 2015; Published: October 2018
First available in Project Euclid: 6 October 2018

zbMATH: 06958642
MathSciNet: MR3861812
Digital Object Identifier: 10.4310/jdg/1538791245

Rights: Copyright © 2018 Lehigh University


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Vol.110 • No. 2 • October 2018
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