Contact homology of left-handed stabilizations and plumbing of open books

TL;DR

This paper constructs contact structures with vanishing contact homology on higher-dimensional closed contact manifolds using open book decompositions with cotangent bundle pages and left-handed Dehn twists, extending to connected sums.

Contribution

It introduces a method to produce contact structures with zero contact homology via open books with cotangent bundle pages and plumbing operations, expanding understanding of contact topology.

Findings

Existence of contact structures with vanishing contact homology on closed manifolds.

Construction uses open books with cotangent bundle pages and left-handed Dehn twists.

Extension of results to connected sums and plumbing operations.

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.