Contact Structures on Plumbed 3-Manifolds

Abstract

In this paper, we show that the Ozsváth-Szabó contact invariant $c^{+}(\xi )\in H{F}^{+}(-Y)$ of a contact 3-manifold $(Y,\xi )$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$, and $\xi$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsváth and Szabó to determine the Heegaard-Floer homology of such 3-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsváth-Stipsicz-Szabó obstruction to admitting a planar open book. Then we define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. We do a sample calculation showing that the invariant can get infinitely many distinct values.

Added some examples and comments

References in corpus (8)

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Cited by in corpus (2)

• Contact Structures on AR-singularity links
• Indefinite Stein fillings and Pin(2)-monopole Floer homology