Periodic Floer homology and embedded contact homology
The Seiberg-Witten invariants are currently one of the most powerful
tools available for understanding the topology of smooth 4-manifolds.
Taubes' famous ``Seiberg-Witten = Gromov'' theorem asserts that the
Seiberg-Witten invariants of a closed symplectic 4-manifold X are
equal to a certain count of (mostly) embedded pseudoholomorphic
curves in X. Seiberg-Witten theory also associates ``Floer
homology'' groups to a closed 3-manifold Y, by considering solutions
to the Seiberg-Witten equations on the 4-manifold
R x Y. The goal of this project is to develop an equivalent
version of Floer homology that counts appropriate (mostly) embedded
pseudoholomorphic curves in R x Y. That is, we would like to
extend Taubes' correspondence to the noncompact 4-manifold R x
Y.
If Y is a 3-manifold with a contact form, then the invariant we define
is called "embedded contact homology" (ECH). This is the homology of
a chain complex which is generated by certain unions of Reeb orbits
(i.e. periodic orbits of the Reeb vector field associated to the
contact form), and whose differential counts pseudoholomorphic curves
in R x Y satisfying a certain index condition. ECH is
reminiscent of the symplectic field theory (SFT) of
Eliashberg-Givental-Hofer. However, the ECH index condition on the
pseudoholomorphic curves substantially constrains their topology, and
in particular forces them to be embedded (except that they may contain
multiply covered R-invariant cylinders). Also, the precise
definitions of the generators and relative index of ECH deviate from
those of SFT, causing the two theories to behave quite differently.
The main conjecture is that the ECH of Y agrees with a version of the
Seiberg-Witten (or the conjecturally isomorphic Ozsvath-Szabo) Floer
homology of Y with the opposite orientation. For the precise
statement of the conjecture see the introduction to paper [3] below.
This conjecture implies that ECH is a topological invariant. In
contrast, at least some versions of SFT are highly sensitive to the
contact structure. (However the ECH of a contact 3-manifold also
contains a canonical element, represented by the empty set of Reeb
orbits, which can distinguish some contact structures and which is
conjectured to agree with analogous invariants of contact structures
in the Seiberg-Witten and Ozsvath-Szabo theories.) The examples that
we have computed so far agree perfectly with the predictions of this
conjecture. (Update: the conjectured isomorphism between ECH and
Seiberg-Witten Floer homology has been proved by Taubes in 2008.)
The above conjecture has implications for contact dynamics. In
particular, it implies a version of the Weinstein conjecture,
asserting that for any contact form on a closed 3-manifold, the
associated Reeb vector field has a periodic orbit. The reason is that
ECH is the homology of a chain complex generated by certain unions of
Reeb orbits, while the corresponding SWF homology is known by results
of Kronheimer-Mrowka to be infinitely generated. If there were no
Reeb orbits, then the ECH would have just one generator, namely the
empty set of Reeb orbits. (The Weinstein conjecture was
proved by Taubes in 2006, and his proof can be regarded as a step towards
showing that ECH is isomorphic to SWF.)
When Y is the mapping torus of a surface symplectomorphism f, there is
a similar invariant called "periodic Floer homology" (PFH). This is a
generalization of the symplectic Floer homology of f, which is
generated not by fixed points of f but rather by unions of periodic
orbits of f. This also has a conjectural relation with Seiberg-Witten
and Ozsvath-Szabo Floer homology, which is confirmed for those
examples where both PFH and at least one of the other two theories
have been computed.
Below are some papers about this story. If you prefer watching movies to
reading papers, you can view the introductory lectures on
ECH that I gave at MSRI (and to get the full story, watch the other lectures at that conference):
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[1] An index inequality for embedded pseudoholomorphic curves in
symplectizations
Journal of the European Math. Soc.
4 (2002), 313-361. PDF
PS
This paper defines the index, or grading, in PFH. When Y is the
mapping torus of a surface symplectomorphism, this index associates to
each pseudoholomorphic curve C in R x Y an integer I(C), which
depends only on the relative homology class of C. One can think of
I(C) as a kind of measure of the complexity of C, which is quite
different from usual measures of complexity such as genus or number of
ends. In particular, we show that for a generic almost complex
structure, there are no curves in R x Y with I<0; a curve has
I=0 if and only if it is a cover of a union of R-invariant cylinders;
I=1 curves are (mostly) embedded, have Fredholm index 1, and satisfy
additional topological constraints; and there are good compactness
results for moduli spaces of I=1 and I=2 curves. An analogous story
holds for ECH; see paper [3] below. The differential in PFH or ECH
counts curves with I=1. The index I is the key, nontrivial part of
the definition of PFH and ECH.
-
[2] The periodic Floer homology of a Dehn twist (with M. Sullivan),
Algebraic and Geometric
Topology 5 (2005), 301-354. download.
This paper reviews the definition of PFH and computes the PFH of some
Dehn twists. Some of the geometric techniques introduced here play an
important role in paper [3] below.
-
[3] Rounding corners of polygons and the
embedded contact homology of T^3 (with M. Sullivan),
Geometry and Topology 10 (2006), 169-266.
download
This paper gives an introduction to ECH, and computes the ECH of T^3.
The latter is described 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".
Much of the paper develops algebraic techniques to compute the
homology of this combinatorial chain complex. The answer agrees with
the Ozsvath-Szabo and Seiberg-Witten Floer homologies of T^3. My
student Eli Lebow has extended the methods of this paper to compute
the ECH of (most) T^2-bundles over S^1.
- [4] Gluing pseudoholomorphic curves along
branched covered cylinders I (with C. H. Taubes.)
Journal of Symplectic Geometry 5 (2007), 43-137.
ps
This paper and its sequel [5] contain the
gluing analysis to prove that d^2=0 in embedded contact homology and
periodic Floer homology. This requires an extension of the standard
gluing machinery, in order to glue together two J-holomorphic curves
in
R x Y whose ends are at the same Reeb orbits, but with
different covering multiplicities. To glue two such curves together,
we need to insert a branched cover of an
R-invariant cylinder between them. The number of gluings is
given by a count of zeroes of a certain section of an obstruction
bundle over a noncompact moduli space of branched covered cylinders.
We obtain a general combinatorial formula for the number of gluings as
a certain sum over labeled forests. In the case of interest for ECH
and PFH, the number of gluings comes out to be 1. This is just what
is needed to show that d^2 counts the number of boundary points of a
compact 1-manifold, so that d^2=0. Paper [4] explains the more algebraic
aspects of this story, using an analytic result from [5].
- [5] Gluing pseudoholomorphic curves along branched covered
cylinders II (with C. H. Taubes.)
arxiv:0705.2074, submitted. 120 pages. pdf
This paper completes all the analysis that was needed in [4].
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.
- [6] The embedded contact homology index revisited
arXiv:0805.1240, 45 pages. pdf
![[new]](new.gif)
In this paper we refine the relative grading on ECH to an absolute
grading, which associates to each union of Reeb orbits a homotopy
class of oriented 2-plane fields. This is obtained by
modifying the contact plane field in a canonical (up to homotopy)
manner in a neighborhood of each Reeb orbit in the union.
We also simplify the proof of the ECH index inequality, and extend it to
symplectic cobordisms between three-manifolds with Hamiltonian
structures. Included are new inequalities on the ECH index
of unions and multiple covers of holomorphic curves in cobordisms.
Finally, we define a new relative filtration on ECH, or any other
kind of contact homology of a contact 3-manifold, which is similar
to the ECH index and related to the Euler characteristic of
holomorphic curves. This does not give new topological invariants
except possibly in special situations, but it is a useful
computational tool.
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