[Paper Review] Capping off open books and the Ozsvath-Szabo contact invariant

John A. Baldwin|arXiv (Cornell University)|Jan 24, 2009

Geometric and Algebraic Topology21 references4 citations

TL;DR

This paper establishes a $U$-equivariant map in Heegaard Floer homology between the $HF^+$ of the negative of a capping-off 3-manifold and the original, showing that the Ozsváth-Szabó contact invariant $c^+$ is preserved under capping off boundary components of open books. The key result is a cobordism-induced map sending $c^+(S_{g,r-1}, ilde\phi)$ to $c^+(S_{g,r},\phi)$, which enables new obstructions to tightness and fillability in genus one open books with multiple boundary components.

ABSTRACT

If (S,h) is an open book with disconnected binding then we can form a new open book (S',h') by capping off one of the boundary components of S with a disk. We define a U-equivariant map on Heegaard Floer homology which sends c^+(S',h') to c^+(S,h), and we discuss various applications. In particular, we determine the support genera of almost all contact structures compatible with genus one, one boundary component open books. In addition, we compute the 3-dimensional invariant associated to any contact structure with non-vanishing contact invariant which is compatible with a genus one open book with periodic monodromy.

Motivation & Objective

• To understand the geometric effect of capping off a boundary component in an open book decomposition on the Ozsváth-Szabó contact invariant.
• To establish a canonical map in Heegaard Floer homology that relates the $HF^+$ of the capping-off cobordism to the original manifold.
• To use this map to derive new obstructions to tightness and symplectic fillability for contact structures supported by genus one open books with multiple boundary components.
• To determine the support genus of contact structures compatible with genus one, one-boundary-component open books.
• To compute the $d_3$ invariant for all tight contact structures supported by genus one open books with periodic monodromy.

Proposed method

• Construct a 2-handle cobordism $W$ from $-M_{S_{g,r-1}, ilde\phi}$ to $-M_{S_{g,r},\phi}$ by attaching a 0-framed 2-handle along the binding component corresponding to the capped-off boundary.
• Define a $\text{Spin}^c$ structure $\mathfrak{s}_0$ on $W$ and use the induced map $F^{+}_{W,\mathfrak{s}_0}: HF^{+}(-M_{S_{g,r-1},\tilde\phi}) \to HF^{+}(-M_{S_{g,r},\phi})$ in Heegaard Floer homology.
• Prove that this map sends the contact invariant $c^+(S_{g,r-1},\tilde\phi)$ to $c^+(S_{g,r},\phi)$, establishing a $U$-equivariant relation.
• Use the map to derive consequences for the vanishing of $c^+$ under capping, particularly in genus one open books.
• Apply the result to compute $d_3(\xi)$ for all tight contact structures supported by genus one open books with periodic monodromy.
• Leverage the map and known results on pseudo-Anosov diffeomorphisms to derive conditions under which $c^+(\xi)$ lies in the image of $U^d$ for all $d \in \mathbb{N}$.

Experimental results

Research questions

• RQ1 Does capping off a boundary component preserve the non-vanishing of the Ozsváth-Szabó contact invariant in Heegaard Floer homology?
• RQ2 Can the $HF^+$-map induced by capping-off cobordisms provide new obstructions to tightness or symplectic fillability?
• RQ3 What is the support genus of contact structures compatible with genus one, one-boundary-component open books?
• RQ4 How does the $d_3$ invariant behave for tight contact structures supported by genus one open books with periodic monodromy?
• RQ5 Under what conditions on the fractional Dehn twist coefficients does $c^+(\xi)$ lie in the image of $U^d$ for all $d \in \mathbb{N}$?

Key findings

• There exists a $U$-equivariant map $F^{+}_{W,\mathfrak{s}_0}: HF^{+}(-M_{S_{g,r-1},\tilde\phi}) \to HF^{+}(-M_{S_{g,r},\phi})$ sending $c^+(S_{g,r-1},\tilde\phi)$ to $c^+(S_{g,r},\phi)$, establishing a canonical relation under capping off.
• If $c^+(S_{g,r-1},\tilde\phi) = 0$, then $c^+(S_{g,r},\phi) = 0$, providing a vanishing obstruction for the contact invariant.
• For genus one open books with $r$ boundary components and pseudo-Anosov monodromy having exactly two singularities per boundary, if any fractional Dehn twist coefficient is less than 1, then $c^+(\xi)$ lies in the image of $U^d$ for all $d \in \mathbb{N}$.
• The $d_3$ invariant is computed for all tight contact structures supported by genus one open books with periodic monodromy.
• The paper provides a classification of support genera for contact structures compatible with genus one, one-boundary-component open books.
• A conjecture is proposed linking the non-vanishing of $c^+(\xi)$ outside the image of $U^d$ to strong symplectic fillability with $b_2^+ > 0$, under specific foliation and monodromy conditions.