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 like in the computation for the overtwisted case. 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 contact open books. We shall give a possible description of this construction and some conjectures about the plumbing operation.
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