17 research outputs found
Homological algebra related to surfaces with boundary
In this article we describe an algebraic framework which can be used in three
related but different contexts: string topology, symplectic field theory, and
Lagrangian Floer theory of higher genus. It turns out that the relevant
algebraic structure for all three contexts is a homotopy version of involutive
bi-Lie algebras, which we call IBL-algebras.Comment: 127 pages, 22 figures. Some references added in version 2. Fixed a
tex problem in version
Compactness for holomorphic curves with switching Lagrangian boundary conditions
We prove a compactness result for holomorphic curves with boundary on an
immersed Lagrangian submanifold with clean self-intersection. As a consequence,
we show that the number of intersections of such holomorphic curves with the
self-intersection locus is uniformly bounded in terms of the Hofer energy.Comment: 28 pages, no figure
The Gromov width of 4-dimensional tori
We show that every 4-dimensional torus with a linear symplectic form can be
fully filled by one symplectic ball. If such a torus is not symplectomorphic to
a product of 2-dimensional tori with equal sized factors, then it can also be
fully filled by any finite collection of balls provided only that their total
volume is less than that of the 4-torus with its given linear symplectic form.Comment: improved exposition, proof of Proposition 3.9 clarified, discussion
of ellipsoid embeddings remove
Algebraic Torsion in Contact Manifolds
We extract a nonnegative integer-valued invariant, 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 zero 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 one (though the converse is not
true). We also construct examples for each nonnegative k 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.Comment: 53 pages, 4 figures, with an appendix by Michael Hutchings; v.3 is a
final update to agree with the published paper, and also corrects a minor
error that appeared in the published version of the appendi