1,094 research outputs found
Quantitative embedded contact homology
Define a "Liouville domain" to be a compact exact symplectic manifold with
contact-type boundary. We use embedded contact homology to assign to each
four-dimensional Liouville domain (or subset thereof) a sequence of real
numbers, which we call "ECH capacities". The ECH capacities of a Liouville
domain are defined in terms of the "ECH spectrum" of its boundary, which
measures the amount of symplectic action needed to represent certain classes in
embedded contact homology. Using cobordism maps on embedded contact homology
(defined in joint work with Taubes), we show that the ECH capacities are
monotone with respect to symplectic embeddings. We compute the ECH capacities
of ellipsoids, polydisks, certain subsets of the cotangent bundle of T2, and
disjoint unions of examples for which the ECH capacities are known. The
resulting symplectic embedding obstructions are sharp in some interesting
cases, for example for the problem of embedding an ellipsoid into a ball (as
shown by work of McDuff-Schlenk) or embedding a disjoint union of balls into a
ball. We also state and present evidence for a conjecture under which the
asymptotics of the ECH capacities of a Liouville domain recover its symplectic
volume.Comment: 39 pages, v3 has minor correction
Beyond ECH capacities
ECH (embedded contact homology) capacities give obstructions to
symplectically embedding one four-dimensional symplectic manifold with boundary
into another. These obstructions are known to be sharp when the domain and
target are ellipsoids (proved by McDuff), and more generally when the domain is
a "concave toric domain" and the target is a "convex toric domain" (proved by
Cristofaro-Gardiner). However ECH capacities often do not give sharp
obstructions, for example in many cases when the domain is a polydisk. This
paper uses more refined information from ECH to give stronger symplectic
embedding obstructions when the domain is a polydisk, or more generally a
convex toric domain. We use these new obstructions to reprove a result of
Hind-Lisi on symplectic embeddings of a polydisk into a ball, and generalize
this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We
also obtain a new obstruction to symplectically embedding one polydisk into
another, in particular proving the four-dimensional case of a conjecture of
Schlenk.Comment: 41 pages; v4: a couple of minor corrections and clarification
An index inequality for embedded pseudoholomorphic curves in symplectizations
Let be a surface with a symplectic form, let be a
symplectomorphism of , and let be the mapping torus of . We
show that the dimensions of moduli spaces of embedded pseudoholomorphic curves
in , with cylindrical ends asymptotic to periodic orbits of
or multiple covers thereof, are bounded from above by an additive relative
index. We deduce some compactness results for these moduli spaces.
This paper establishes some of the foundations for a program with Michael
Thaddeus, to understand the Seiberg-Witten Floer homology of in terms of
such pseudoholomorphic curves. Analogues of our results should also hold in
three dimensional contact homology.Comment: 60 pages, LaTeX 2
Reidemeister torsion in generalized Morse theory
In two previous papers with Yi-Jen Lee, we defined and computed a notion of
Reidemeister torsion for the Morse theory of closed 1-forms on a finite
dimensional manifold. The present paper gives an a priori proof that this Morse
theory invariant is a topological invariant. It is hoped that this will provide
a model for possible generalizations to Floer theory.Comment: 42 pages, LateX2e; some corrections, improved expositio
Embedded contact homology and its applications
Embedded contact homology (ECH) is a kind of Floer homology for contact
three-manifolds. Taubes has shown that ECH is isomorphic to a version of
Seiberg-Witten Floer homology (and both are conjecturally isomorphic to a
version of Heegaard Floer homology). This isomorphism allows information to be
transferred between topology and contact geometry in three dimensions. In this
article we first give an overview of the definition of embedded contact
homology. We then outline its applications to generalizations of the Weinstein
conjecture, the Arnold chord conjecture, and obstructions to symplectic
embeddings in four dimensions.Comment: expository article to accompany invited talk at 2010 IC
Mean action and the Calabi invariant
Given an area-preserving diffeomorphism of the closed unit disk which is a
rotation near the boundary, one can naturally define an "action" function on
the disk which agrees with the rotation number on the boundary. The Calabi
invariant of the diffeomorphism is the average of the action function over the
disk. Given a periodic orbit of the diffeomorphism, its "mean action" is
defined to be the average of the action function over the orbit. We show that
if the Calabi invariant is less than the boundary rotation number, then the
infimum over periodic orbits of the mean action is less than or equal to the
Calabi invariant. The proof uses a new filtration on embedded contact homology
determined by a transverse knot, which might be of independent interest. (An
analogue of this filtration can be defined for any other version of contact
homology in three dimensions that counts holomorphic curves.)Comment: 34 pages; v2 has minor corrections and clarifications and an
additional reference; v3 has minor corrections following referee's
suggestion
Rounding corners of polygons and the embedded contact homology of T^3
The embedded contact homology (ECH) of a 3-manifold with a contact form is a
variant of Eliashberg-Givental-Hofer's symplectic field theory, which counts
certain embedded J-holomorphic curves in the symplectization. We show that the
ECH of T^3 is computed by a combinatorial chain complex which is generated by
labeled convex polygons in the plane with vertices at lattice points, and whose
differential involves `rounding corners'. We compute the homology of this
combinatorial chain complex. The answer agrees with the Ozsvath--Szabo Floer
homology HF^+(T^3).Comment: This is the version published by Geometry & Topology on 26 March 200
Axiomatic S^1 Morse-Bott theory
In various situations in Floer theory, one extracts homological invariants
from "Morse-Bott" data in which the "critical set" is a union of manifolds, and
the moduli spaces of "flow lines" have evaluation maps taking values in the
critical set. This requires a mix of analytic arguments (establishing
properties of the moduli spaces and evaluation maps) and formal arguments
(defining or computing invariants from the analytic data). The goal of this
paper is to isolate the formal arguments, in the case when the critical set is
a union of circles. Namely, we state axioms for moduli spaces and evaluation
maps (encoding a minimal amount of analytical information that one needs to
verify in any given Floer-theoretic situation), and using these axioms we
define homological invariants. More precisely, we define a (almost) category of
"Morse-Bott systems". We construct a "cascade homology" functor on this
category, based on ideas of Bourgeois and Frauenfelder, which is "homotopy
invariant". This machinery is used in our work on cylindrical contact homology.Comment: 48 pages (v3 has minor clarifications, mainly at the end, following
referee's suggestions
Cylindrical contact homology for dynamically convex contact forms in three dimensions
We show that for dynamically convex contact forms in three dimensions, the
cylindrical contact homology differential d can be defined by directly counting
holomorphic cylinders for a generic almost complex structure, without any
abstract perturbation of the Cauchy-Riemann equation. We also prove that d^2 =
0. Invariance of cylindrical contact homology in this case can be proved using
S^1-dependent almost complex structures, similarly to work of Bourgeois-Oancea;
this will be explained in another paper.Comment: v3: corrected Lemma 2.5(b), to appear in Journal of Symplectic
Geometr
Gluing pseudoholomorphic curves along branched covered cylinders II
This paper and its prequel ("Part I") prove a generalization of the usual
gluing theorem for two index 1 pseudoholomorphic curves U_+ and U_- in the
symplectization of a contact 3-manifold. We assume that for each embedded Reeb
orbit gamma, the total multiplicity of the negative ends of U_+ at covers of
gamma agrees with the total multiplicity of the positive ends of U_- at covers
of gamma. However, unlike in the usual gluing story, here the individual
multiplicities are allowed to differ. In this situation, one can often glue U_+
and U_- to an index 2 curve by inserting genus zero branched covers of
R-invariant cylinders between them. This paper shows that the signed count of
such gluings equals a signed count of zeroes of a certain section of an
obstruction bundle over the moduli space of branched covers of the cylinder.
Part I obtained a combinatorial formula for the latter count and, assuming the
result of the present paper, deduced that the differential d in embedded
contact homology satisfies d^2=0. The present paper completes all of the
analysis that was needed in Part I. The gluing technique explained here is in
principle applicable to more gluing problems. We also prove some lemmas
concerning the generic behavior of pseudoholomorphic curves in
symplectizations, which may be of independent interest.Comment: 123 pages; some corrections following referee's suggestions, to
appear in Journal of Symplectic Geometr
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