Naturality of Heegaard Floer invariants under positive rational contact surgery

TL;DR

This paper proves that the Heegaard Floer contact invariant behaves naturally under positive rational contact surgery, providing criteria for its nonvanishing and methods for direct calculation.

Contribution

It establishes the naturality of the contact invariant under rational surgeries and characterizes the induced spin-c structures, extending previous results to a broader setting.

Findings

Heegaard Floer contact invariant maps under positive rational contact surgery.

Necessary and sufficient conditions for nonvanishing of the invariant after surgery.

Method for direct calculation using the rational surgery mapping cone.

Abstract

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. In addition we characterize the 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 when Y is 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 Ozsv\'ath and Szab\'o. The proof involves a construction called reducible open book surgery, which reduces in special cases to the capping-off construction studied by Baldwin.