Algebraic Torsion in Contact Manifolds

By J. Latschev, Chris Wendl, Michael Hutchings. 2010

60 citations recorded by the source corpus.

We extract an invariant taking values in $${\mathbb{N}\cup\{\infty\}}$$ , which we call the order of algebraic torsion, from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order 0 if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order 1 (though the converse is not true). We also construct examples for each $${k \in \mathbb{N}}$$ of contact 3-manifolds that have algebraic torsion of order k but not k − 1, and derive consequences for contact surgeries on such manifolds.The appendix by Michael Hutchings gives an alternative proof of our cobordism obstructions in dimension three using a refinement of the contact invariant in Embedded Contact Homology.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/60be851bda3484621e0d16657a1736fde2054c90

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. J. Pardon
2. Patrick Massot
3. Klaus Niederkruger
4. Chris Wendl

In Geometry and Topology

1. J. Pardon
2. Klaus Niederkruger
3. Chris Wendl
4. Patrick Massot
5. Michael Hutchings

In Mathematical Physics

1. J. Pardon
2. Klaus Niederkruger
3. Chris Wendl
4. Patrick Massot
5. Michael Hutchings

Citations in the corpus

• A hierarchy of local symplectic filling obstructions for contact $3$-manifolds (2010, background): “ can be embedded in (M,ξ), and can take arbitrarily large values while PT(M,ξ) ≤ 1. Likewise, (M,ξ) has Giroux torsion zero whenever PT(M,ξ) ≥ 2. The theorem above follows from some results proved in [LW11] using notions from Symplectic Field Theory, which also lie in the background of our choice of”
• Weak and strong fillability of higher dimensional contact manifolds (2011, background): “Corollary 7 If (V , ξ) has fully twisted algebraic torsion in the sense of [39], then it is not weakly fillable.”
• Contact homology and virtual fundamental cycles (2015, methodology): “py, though it is not clear if this is any improvement. •(Symplectic field theory) To construct the symplectic field theory “homology groups” as described in Cieliebak–Latschev [CL09] and Latschev–Wendl [LW11] (specializing work of Eliashberg–Givental–Hofer [EGH00]) using the methods of this paper, it ”