Abstract
AbstractI construct infinitely many nondiffeomorphic examples of $5$ -dimensional contact manifolds which are tight, admit no strong fillings and do not have Giroux torsion. I obtain obstruction results for symplectic cobordisms, for which I give a proof not relying on the polyfold abstract perturbation scheme for Symplectic Field Theory (SFT). These results are part of my PhD thesis [23], and are the first applications of higher-dimensional Siefring intersection theory for holomorphic curves and hypersurfaces, as outlined in [23, 24], as a prequel to [30].
Highlights
This paper is a followup to [Mo], where we address the general problem of constructing “interesting” families of contact structures in higher dimensions, together with developing general computational techniques for SFT-type invariants
After carrying out a detailed construction of our examples in any odd dimension, we focus on a family of 5-dimensional particular cases, in order to illustrate the techniques developed for the general case
From our knowledge of the SFT differential of our 5-dimensional examples, we show that they do not have Giroux torsion
Summary
If Y is the unit cotangent bundle of a hyperbolic surface, and (M = Σ × Y, ξk) is the corresponding 5-dimensional contact manifold of Theorem 1.3 with k ≥ 3, (M, ξk) does not have Giroux torsion. One can twist the contact structure of Theorem 1.3 close to the dividing set, by performing the l-fold Lutz–Mori twist along a hypersurface H lying in ∂( k Y × I × S1) This notion was defined in [MNW13], and builds on ideas by Mori in dimension 5 [Mori09]. It consists in gluing copies, along H, of a 2πl-Giroux torsion domain (GTl, ξGT ) := (Y × [0, 2πl] × S1, ker λGT ), the contact manifold obtained by gluing l copies of GT = GT1 together. We have taken special care in that the approach taken provides results that will be fully rigorous after the polyfold machinery of Hofer–Wysocki–Zehnder is complete, and gives several direct results that are already rigorous
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