Algebraic Torsion in higher-dimensional contact manifolds

Abstract

We construct examples in any odd dimension of contact manifolds with finite and non-zero algebraic torsion (in the sense of Latschev-Wendl), which are therefore tight and do not admit strong symplectic fillings. We prove that Giroux torsion implies algebraic $1$-torsion in any odd dimension, which proves a conjecture by Massot-Niederkrueger-Wendl. We construct infinitely many non-diffeomorphic examples of $5$-dimensional contact manifolds which are tight, admit no strong fillings, and do not have Giroux torsion. We obtain obstruction results for symplectic cobordisms, for which we give a proof not relying on SFT machinery. We give a tentative definition of a higher-dimensional spinal open book decomposition, based on the $3$-dimensional one of Lisi-van Horn Morris-Wendl. An appendix written in co-authorship with Richard Siefring gives a basic outline of the intersection theory for punctured holomorphic curves and hypersurfaces, which generalizes his $3$-dimensional results of Siefring to higher dimensions. From the intersection theory we obtain an application to codimension-$2$ holomorphic foliations, which we use to restrict the behaviour of holomorphic curves in our examples.

187 pages, 26 figures, PhD Thesis

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• Lectures on Symplectic Field Theory
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Cited by in corpus (7)

• On symplectic fillings of spinal open book decompositions I: Geometric constructions
• Holomorphic curves in the presence of holomorphic hypersurface foliations
• Automatic transversality in contact homology II: filtrations and computations
• Symplectic embeddings into disk cotangent bundles
• Embedded contact homology of prequantization bundles
• S^1-equivariant contact homology for hypertight contact forms
• On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification