Algebraic Torsion in higher-dimensional contact manifolds

TL;DR

This paper constructs higher-dimensional contact manifolds with finite algebraic torsion, proving a conjecture relating Giroux torsion to algebraic torsion, and explores implications for symplectic fillings and cobordisms.

Contribution

It introduces examples of higher-dimensional tight contact manifolds with algebraic torsion and establishes a link between Giroux torsion and algebraic torsion in all odd dimensions.

Findings

Giroux torsion implies algebraic 1-torsion in any odd dimension

Constructs infinitely many non-diffeomorphic 5D tight contact manifolds with no strong fillings

Provides obstruction results for symplectic cobordisms without relying on SFT machinery

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.