[Paper Review] Contact homology of left-handed stabilizations and plumbing of open books

Frédéric Bourgeois, Otto van Koert|arXiv (Cornell University)|Mar 4, 2008

Geometric and Algebraic Topology25 references22 citations

TL;DR

This paper constructs contact structures with vanishing contact homology on closed contact manifolds of dimension greater than 1 by using open book decompositions with left-handed Dehn twists. The key mechanism is identifying a closed Reeb orbit that bounds a unique finite-energy holomorphic plane, making the unit element of the contact homology algebra exact, thus forcing contact homology to vanish. This construction generalizes to connected sums via plumbing of open books, supporting a conjecture that left-handed stabilizations yield vanishing contact homology in all dimensions.

ABSTRACT

We show that on any closed contact manifold of dimension greater than 1 a contact structure with vanishing contact homology can be constructed. The basic idea for the construction comes from Giroux. We use a special open book decomposition for spheres. The page is the cotangent bundle of a sphere and the monodromy is given by a left-handed Dehn twist. In the resulting contact manifold we exhibit a closed Reeb orbit that bounds a single finite energy plane. As a result, the unit element of the contact homology algebra is exact and so the contact homology vanishes. This result can be extended to other contact manifolds by using connected sums. The latter is related to the plumbing- or 2-Murasugi sum of the contact open books. We shall give a possible description of this construction and some conjectures about the plumbing operation.

Motivation & Objective

• To demonstrate that contact homology vanishes for certain contact structures constructed via left-handed Dehn twists in open book decompositions.
• To extend this result to connected sums of contact manifolds using plumbing (2-Murasugi sum) of open books.
• To provide evidence for a conjecture that left-handed stabilizations of contact open books yield vanishing contact homology in all dimensions ≥3.
• To explore the relationship between plumbing of open books and contact connected sums, particularly in higher dimensions.
• To support the idea that vanishing contact homology is a distinguishing feature of overtwisted-like structures in higher-dimensional contact topology.

Proposed method

• Construct an open book for the sphere $S^{2n-1}$ with page $T^*S^{n-1}$ and monodromy given by a single left-handed Dehn-Seidel twist.
• Identify a closed Reeb orbit in the resulting contact manifold that bounds a unique finite-energy holomorphic plane.
• Use the uniqueness of this holomorphic plane to show that the unit element in the contact homology algebra is exact, implying the entire homology vanishes.
• Apply the plumbing construction (2-Murasugi sum) to combine two contact open books by gluing their pages along Lagrangian balls with Legendrian boundaries.
• Use Weinstein neighborhood theorem to identify coordinates on the plumbing neighborhood and define the new monodromy as the composition $\tilde{\psi}_2 \circ \tilde{\psi}_1$.
• Conjecture that the resulting open book supports the contact connected sum, generalizing known 3-dimensional results to higher dimensions.

Experimental results

Research questions

• RQ1 Can contact homology vanish for contact structures obtained via left-handed Dehn twists in higher-dimensional open books?
• RQ2 Does the plumbing construction of open books (2-Murasugi sum) yield a contact structure contactomorphic to the connected sum of the original contact manifolds in higher dimensions?
• RQ3 Is the unit element in contact homology exact when a Reeb orbit bounds a unique finite-energy holomorphic plane?
• RQ4 Can left-handed stabilizations of contact open books be shown to have vanishing contact homology in dimensions greater than 3?
• RQ5 Does the plumbing of pages with Lagrangian balls and composition of monodromies preserve contact compatibility and support the connected sum structure?

Key findings

• Contact homology vanishes for the contact structure on $S^{2n-1}$ obtained from an open book with page $T^*S^{n-1}$ and monodromy a single left-handed Dehn twist.
• The vanishing occurs because a closed Reeb orbit bounds a unique finite-energy holomorphic plane, rendering the unit element of the contact homology algebra exact.
• The construction extends to connected sums via plumbing of open books, preserving the vanishing of contact homology.
• The plumbing operation is conjectured to yield a contact open book supporting the contact connected sum, generalizing a known 3-dimensional result by Torisu.
• The result supports the conjecture that left-handed stabilizations of contact open books have vanishing contact homology in all dimensions ≥3, particularly when the Lagrangian ball is boundary-parallel.