1. science
2. mathematics
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UPPSALA DISSERTATIONS IN MATHEMATICS
92
Orienting Moduli Spaces of Flow
Trees for Symplectic Field Theory
Cecilia Karlsson
Department of Mathematics
Uppsala University
UPPSALA 2016
Dissertation presented at Uppsala University to be publicly examined in Polhemsalen,
Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 19 February 2016 at 13:15
for the degree of Doctor of Philosophy. The examination will be conducted in English.
Faculty examiner: Vincent Colin (Université de Nantes, Faculté des sciences, Département de
mathématiques).
Abstract
Karlsson, C. 2016. Orienting Moduli Spaces of Flow Trees for Symplectic Field Theory.
Uppsala Dissertations in Mathematics 92. 37 pp. Uppsala: Department of Mathematics.
ISBN 978-91-506-2523-3.
This thesis consists of three scientific papers dealing with invariants of Legendrian and
Lagrangian submanifolds. Besides the scientific papers, the thesis contains an introduction to
contact and symplectic geometry, and a brief outline of Symplectic field theory with focus on
Legendrian contact homology.
In Paper I we give an orientation scheme for moduli spaces of rigid flow trees in Legendrian
contact homology. The flow trees can be seen as the adiabatic limit of sequences of punctured
pseudo-holomorphic disks with boundary on the Lagrangian projection of the Legendrian. So
to equip the trees with orientations corresponds to orienting the determinant line bundle of the
dbar-operator over the space of Lagrangian boundary conditions on the punctured disk. We
define an orientation of this line bundle and prove that it is well-defined in the limit. We also
prove that the chosen orientation scheme gives rise to a combinatorial algorithm for computing
the orientation of the trees, and we give an explicit description of this algorithm.
In Paper II we study exact Lagrangian cobordisms with cylindrical Legendrian ends, induced
by Legendrian isotopies. We prove that the combinatorially defined DGA-morphisms used to
prove invariance of Legendrian contact homology for Legendrian knots over the integers can
be derived analytically. This is proved using the orientation scheme from Paper I together with
a count of abstractly perturbed flow trees of the Lagrangian cobordisms.
In Paper III we prove a flexibility result for closed, immersed Lagrangian submanifolds in
the standard symplectic plane.
Keywords: Contact manifolds, Legendrian submanifolds, Lagrangian immersions, Legendrian
contact homology, Morse flow trees, Determinant line bundles, Orientation of moduli spaces,
Exact Lagrangian cobordisms
Cecilia Karlsson, Department of Mathematics, Algebra and Geometry, Box 480, Uppsala
University, SE-751 06 Uppsala, Sweden.
© Cecilia Karlsson 2016
ISSN 1401-2049
ISBN 978-91-506-2523-3
urn:nbn:se:uu:diva-269551 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-269551)
List of papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I
C. Karlsson, Orientations of Morse flow trees in Legendrian contact
homology. Preprint, 2015.
II C. Karlsson, A note on orientations of exact Lagrangian cobordisms
with cylindrical Legendrian ends. Preprint, 2015.
III C. Karlsson, Area-preserving isotopies of self-transverse immersions of
S1 in R2 . Ark. Mat., 51(1):85–97, 2013.
Reprints were made with permission from the publishers.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2
The Legendrian isotopy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.3
Legendrian contact homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.4
A summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2
Symplectic geometry and pseudo-holomorphic disks . . . . . . . . . . . . . . . . . 10
1.3
Contact geometry and exact Lagrangian cobordisms . . . . . . . . . . . . . . . . . . 12
2
Symplectic field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Legendrian contact homology in 1-jet spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1
Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2
The differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3
Invariance over Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4
Invariance over Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Morse flow trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2
Morse flow trees in Legendrian contact homology . . . . . . .
2.2.3
Correspondence between flow trees and disks . . . . . . . . . . . . . .
2.3
Exact Lagrangian cobordisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
14
15
15
16
16
17
18
19
20
21
3
On the appended papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
24
27
28
4
Att orientera modulirum av flödesträd för symplektisk fältteori,
Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Bakgrund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Kontaktgeometri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Symplektisk geometri och pseudoholomorfa kurvor . . . . . . . . . . . . . . . . . . .
4.4
Det Legendriska isotopiproblemet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
Legendrisk kontakthomologi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Flödesträd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7
Sammanfattning av avhandlingens resultat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
30
30
31
32
32
33
33
Acknowledgements
...................................................................................
35
........................................................................................................
36
5
References
1. Introduction
Ever since Lord Kelvin’s suggestion that atoms are “knots in the ether”, mathematicians have tried to classify geometric objects, up to continuous deformations. The aim is to find efficient invariants for the objects, where the invariants
are required to remain unchanged (invariant) under the allowed deformations.
This study is today a branch of the mathematical field Topology, and has been
generalized in many different directions, for example to higher dimensions but
also by imposing additional geometric constraints.
1.1 Outline
This thesis is primarily about algebraic invariants of Legendrian submanifolds,
with the invariants defined over the integers. In this section we give a summary
of the contents of the thesis, and we refer to Section 1.2 and Section 1.3 for
more detailed definitions of the basic concepts, and to Chapter 2 for a more
comprehensive introduction to Symplectic field theory and in particular Legendrian contact homology. In Chapter 3 we give a summary of the main results
of the thesis.
1.1.1 Background
A Legendrian submanifold is a special kind of submanifold, which lives in a
contact manifold and satisfies certain tangential constraints. More precisely,
a contact manifold W is an odd-dimensional manifold∗ , of dimension 2n + 1
say, equipped with a contact structure ξ . This additional structure is given by
a completely non-integrable distribution of tangent hyperplanes. That ξ is a
distribution of tangent hyperplanes means that ξ assigns to each point p of
W a 2n-dimensional subspace of the tangent space TpW , and these subspaces
vary smoothly over W . That ξ is completely non-integrable means that it is
(locally) given as the kernel of a (locally defined) 1-form α which satisfies
α ∧ (dα)∧n 6= 0. This condition implies that there is no 2n-dimensional submanifold of W that is tangent to ξ , not even locally. Moreover, the maximal
dimension of a submanifold that could possibly be tangent to ξ is equal to n,
and such a maximal submanifold is called a Legendrian submanifold.
Contact geometry goes back to the work of Sophus Lie, who used contact
transformations to study systems of differential equations. The field also has
∗ We
assume that all objects are C∞ , unless otherwise stated.
7
connections to optics and thermodynamics. For example, one can use Legendrian submanifolds to describe wavefront propagation. This connection can
also be described by the following citation from one of the founders of modern
contact geometry, Arnold, who in [Arn90] writes the following.
Every mathematician knows that it is impossible to understand any elementary
course in thermodynamics. The reason is that thermodynamics is based on a
rather complicated mathematical theory, on contact geometry.
Contact geometry is often considered as the odd-dimensional counterpart of
symplectic geometry. The latter is a branch of mathematics having its origin in
the study of phase spaces in classical mechanics. Briefly, a symplectic manifold is an even-dimensional manifold V together with a closed, non-degenerate
2-form, which is called the symplectic form. The analogy of Legendrian submanifolds is given by Lagrangian submanifolds, which are submanifolds of
half the dimension of V , and which annihilate the symplectic form.
One can use symplectic geometry to study the geometry of contact manifolds, and vice versa. For example, by taking the product of R and a contact
manifold W , one gets a symplectic manifold which is called the symplectization of W . One can also contactize some symplectic manifolds to get contact
manifolds.
1.1.2 The Legendrian isotopy problem
A Legendrian isotopy is a smooth 1-parameter family of Legendrian submanifolds. The problem of classifying all Legendrian submanifolds of a contact
manifold, up to Legendrian isotopy, is one of the major problems in contact
geometry. This motivates finding effective Legendrian invariants, that is, invariants that are preserved under Legendrian isotopies.
Clearly, if two Legendrian submanifolds are Legendrian isotopic, then they
are also smoothly isotopic. However, by examining the tangency constraints
coming from the Legendrian condition, two classical Legendrian invariants
have been defined. These are called the Thurston-Bennequin invariant and
the rotation number. See, e.g., [Gei08] for a definition in the case when the
dimension of the Legendrians equals 1, and [EES05a] for a generalization to
higher dimensions. With these invariants it is easy to give examples of Legendrian submanifolds that are smoothly isotopic but which fail to be Legendrian
isotopic.
By using pseudo-holomorphic curve techniques, introduced by Gromov in
the paper [Gro85] and developed further by Floer in for example [Flo88], one
has been able to define more sophisticated Legendrian invariants. By using
these invariants it has been proven that there exist smoothly isotopic Legendrian submanifolds, which moreover have the same classical Legendrian invariants, but which are not Legendrian isotopic. See, e.g., [Che02] for examples of such 1-dimensional Legendrians, and [EES05a] for higher-dimensional
8
examples. The results in these papers use an invariant called Legendrian contact homology, which is a homology theory that fits into the package of Symplectic Field Theory (SFT), introduced by Eliashberg, Givental and Hofer in
the paper [EGH00].
1.1.3 Legendrian contact homology
Legendrian contact homology is the homology of a differential graded algebra (DGA) associated to a Legendrian Λ. One should note that Legendrian
contact homology has not been worked out in full detail for all contact manifolds, but in the special case when the contact manifold is given by the 1-jet
space J 1 (M) = T ∗ M × R of a manifold M, then the analytical details were
established in [EES07]. In the special case M = R, this was also done by
Eliashberg in [Eli98] and independently by Chekanov in [Che02]. To handle
the similar issues in the general setting of SFT, the theory of Polyfolds developed by Hofer, Wysocki and Zehnder, see e.g. [Hof], and also Pardon’s work
[Para, Parb] on virtual fundamental cycles, seem to be fruitful.
In the case of a 1-jet space, the DGA of Λ can be defined by considering the
projection ΠC : J 1 (M) → T ∗ M. The generators of the algebra are then given
by the double points of ΠC (Λ), and the differential counts certain pseudoholomorphic disks in T ∗ M with boundary on ΠC (Λ). With a clever choice of
almost complex structure on T ∗ M, one gets that the homology of this complex
is a Legendrian invariant.
If M = R, then the count of pseudo-holomorphic disks reduces to combinatorics, as described by Chekanov in [Che02], but in higher dimensions the
Cauchy-Riemann equations give rise to non-linear partial differential equations, which are hard to solve. To simplify this, Ekholm in [Ekh07] introduced
Morse flow trees, which give a method to explicitly compute the differential
by considering certain gradient flows on the base manifold M. This is built on
similar ideas of Floer in [Flo89], and Fukaya and Oh in [FO97], where flow
trees are used to compute the differential in Lagrangian Floer Homology.
Since the differential in Legendrian contact homology is defined by counting pseudo-holomorphic disks, one can use the ellipticity of the ∂¯ -operator to
define coherent orientations of the moduli spaces of disks. This implies that
Legendrian contact homology can be defined with coefficients in Z. However,
when the dimension of the Legendrian is greater than 1, then it is difficult to
get an explicit understanding of these orientations.
1.1.4 A summary of the results
The main result of this thesis is that there exists a coherent orientation scheme
of moduli spaces of rigid Morse flow trees, and that this scheme can be chosen
9
so that it gives an explicit algorithm to compute the differential in Legendrian
contact homology with coefficients in Z. This is proved in Paper I.
This result is then used in Paper II to give an analytic derivation of the combinatorially defined DGA-maps in [Kál05], which were used to prove invariance of Legendrian contact homology over Z under Legendrian isotopies. Our
result makes use of the functorial properties of SFT, which briefly says that a
Lagrangian cobordism with Legendrian ends induces a morphism between the
associated DGAs of the ends.
In Paper III we prove a flexibility result for closed, immersed Lagrangians
in the symplectic space (R2 , dx ∧ dy).
1.2 Symplectic geometry and pseudo-holomorphic disks
A symplectic manifold (V, ω) is a 2n-dimensional manifold V equipped with
a closed, non-degenerate 2-form ω, called the symplectic form. The nondegeneracy condition means that ω ∧n gives a volume form on V .
A standard example of a symplectic manifold is R2n with the symplectic form ωstd = ∑ni=1 dxi ∧ dyi . Another example, which is a generalization
of (R2n , ωstd ) and comes from classical mechanics, is the cotangent bundle
T ∗ M of a manifold M. This space is equipped with the symplectic form
ω = ∑ni=1 dqi ∧ d pi , where q are local coordinates on M and p are the coordinates in the cotangent fibers.
By Darboux’s theorem, every symplectic manifold locally looks like the
space (R2n , ωstd ). This implies that the local behavior of a symplectic manifold
is known once we understand R2n , so the interesting questions are about global
properties. We refer to e.g. [MS95] for a wider introduction to the subject.
The non-degeneracy condition on ω implies that we get a natural one-toone correspondence between vector fields and 1-forms of V , given by
TV 3 X 7→ ω(X, ·) ∈ T ∗V.
A vector field X is called Hamiltonian if ω(X, ·) is exact, i.e. if there is a
function H : V → R so that ω(X, ·) = dH. In this case we write X = XH .
If Φt : V → V , t ∈ [0, 1], Φ0 = id, is an isotopy so that
Φt∗ ω = ω
for every t ∈ [0, 1],
then we say that Φt is a symplectic isotopy. If Φt is a Hamiltonian isotopy,
that is, if Φt solves
d
Φt = XHt (Φt ),
dt
Φ0 = id,
for some time dependent Hamiltonian Ht : V → R, t ∈ [0, 1], then Φt is symplectic, but the converse need not be true. However, if the first de Rham cohomology group of V is trivial, then every symplectic isotopy is Hamiltonian.
10
A submanifold L ⊂ (V, ω) is called Lagrangian if dim L = n and ω|T L vanishes. If moreover ω = dβ for some 1-form β , then V is an exact symplectic
manifold, and if β |L is exact then L is called an exact Lagrangian submanifold.
The definition of an immersed (exact) Lagrangian is similar.
The Lagrangian condition on the submanifolds give them a more rigid structure than ordinary submanifolds, and this makes them harder to study. For
example, Gromov showed in his seminal paper [Gro85] that there are no exact closed Lagrangian embeddings in standard symplectic R2n . This means,
for instance, that there are no Lagrangian embeddings of S3 into R6 . Another rigidity phenomena is that certain classes of Lagrangian submanifolds
are seen to intersect more times than they are obliged to by purely topological
reasons. The question about the minimal number of intersection points that
they are forced to have by the Lagrangian condition was one of the starting
points of the ongoing intense study of symplectic geometry, and is known as
Arnold’s conjecture.† The aim of solving this conjecture has lead to a great
amount of new mathematical theories, for example Floer homology, Fukaya
categories, and their relation to string theory via homological mirror symmetry. In contrast, Lagrangian immersions are more flexible since they satisfy an
h-principle. For a deeper discussion on this subject and related open problems,
see, e.g., the survey [Eli15].
The rigidity results of Gromov use pseudo-holomorphic curve techniques,
which has grown to become one of the main techniques in the study of Lagrangian submanifolds. To define the notion of a pseudo-holomorphic curve
of a symplectic manifold (V, ω), we first have to pick an almost complex structure of V . This is an endomorphism of TV satisfying J 2 = − id. We say that J
is compatible with ω if ω(·, J·) defines a Riemannian metric on X.
Definition 1.2.1. Let (Σ, j) be a Riemann surface, possibly with boundary and
punctures. We say that
u:Σ→X
is pseudo-holomorphic if it satisfies the Cauchy-Riemann equation
∂¯ u := du + J ◦ du ◦ j = 0.
(1.1)
If we want to emphasize which almost complex structure we are considering,
we say that u is J-holomorphic.
One of the most important insights of Gromov is that the moduli spaces of
pseudo-holomorphic curves with uniformly bounded energy can be compactified by broken pseudo-holomorphic curves, or curves with bubbles. This is
called Gromov compactness, and is a widely used tool in SFT.
† This conjecture is usually expressed as a statement about the minimal number of periodic orbits
of a 1-periodic Hamiltonian flow.
11
1.3 Contact geometry and exact Lagrangian cobordisms
A contact manifold (W, ξ ) is a 2n + 1-dimensional manifold W together with
a completely non-integrable hyperplane field ξ , called the contact structure.
The condition of ξ being completely non-integrable means that there are locally defined 1-forms α on W so that
ξ = Ker α,
α ∧ dα ∧n 6= 0.
In the case when α can be defined globally we say that (W, Ker α) is a cooriented contact manifold, and α is called a contact form. Henceforth, we
assume that all contact manifolds are co-oriented, and we will write (W, α) to
mean (W, Ker α).
For each contact manifold (W, α) there is an associated Reeb vector field
Rα , which satisfies
α(Rα ) = 1,
dα(Rα , ·) = 0.
Note that a fixed contact structure can be given as the kernel of several different
contact forms, and thus can have more than one associated Reeb vector field.
A natural question is to what extent the contact structure gives information
about the dynamics of the Reeb vector fields. The most famous question of
this form is known as the Weinstein conjecture, which states that the Reeb flow
of any contact form on a closed manifold must have at least one closed orbit.
The conjecture was proven to be true for 3-dimensional manifolds by Taubes
in 2007, see [Tau07], but is still open in the general case. The aim of proving
this conjecture has contributed to a versatile mathematical development.
A standard example of a (non-closed) contact manifold is given by the 1-jet
space
(J 1 (M), α) = (T ∗ M × R, dz − pdq)
of a manifold M, where z denotes the coordinate in the R-direction. In this
case we have that the Reeb vector field is given by Rα = ∂z . Similar to the case
of symplectic manifolds, there is a Darboux’s theorem for contact manifolds,
making them to locally behave as (J 1 (Rn ), Ker α) = (Cn × R, Ker(dz − ydx)).
Recall from Section 1.1 that a Legendrian submanifold Λ of a 2n + 1dimensional contact manifold is a submanifold of dimension n which is everywhere tangent to the contact structure. A Reeb chord of Λ is a flow segment of
the Reeb vector field, having its start and end point on Λ. In the case of Legendrian submanifolds of 1-jet spaces, one can use the Lagrangian projection
ΠC : J 1 (M) → T ∗ M
to study Legendrian submanifolds. The Legendrians are projected to exact,
immersed Lagrangians of T ∗ M under this projection, and the double points of
ΠC (Λ) correspond to Reeb chords of Λ. Moreover, after a small Legendrian
isotopy of Λ we may assume that it is chord generic, meaning that ΠC (Λ)
12
is an immersion with transverse double points as the only intersections. This
implies that if Λ is closed, then the number of Reeb chords of Λ is finite.
One way of studying Legendrian submanifolds of a contact manifold (W, α)
is to consider Lagrangian cobordisms in the symplectization (R × W, d(et α))
of (W, α). Here t denotes the coordinate in the R-direction.
Definition 1.3.1. We say that L is an exact Lagrangian cobordisms with cylindrical Legendrian ends Λ± if L is an exact Lagrangian submanifold of (R ×
W, d(et α)), asymptotic to R × Λ+ at t = +∞, and asymptotic to R × Λ− at
t = −∞.
Remark 1.3.2. Note that a Legendrian Λ of W gives rise to an exact Lagrangian
cylinder R × Λ in the symplectization.
We refer to e.g. [Gei08] for a more comprehensive introduction to contact
geometry.
13
2. Symplectic field theory
The concept of Symplectic Field Theory (SFT) was introduced by Eliashberg,
Givental and Hofer in the paper [EGH00]. Briefly, SFT is a package of homology theories which give invariants for objects in symplectic and contact
geometry.
The idea is to associate a differential algebra to the object under study,
where the differential counts certain pseudo-holomorphic curves. By using
Gromov compactness one then shows, in favorable situations, that the homology of the algebra gives a well-defined invariant. Examples of applications of
these invariants can be found in the study of the Weinstein and Arnold conjectures. Some of the invariants are also closely related to Gromov-Witten
invariants, Floer homology, etc.
There are still problems that remain to be solved to get the full general
setting of SFT. For example, there are questions concerning transversality for
pseudo-holomorphic curves. By recent results, some of these issues seem to
be solved, using Pardon’s techniques of virtual fundamental cycles from [Para]
and [Parb], or via the machinery of Polyfolds, invented by Hofer, Wysocki
and Zehnder (see e.g. [Hof] for an introduction, and the references therein).
Nevertheless, there are more work to be done to get the ideas of SFT to work
in full generality.
In some specific cases of the theories included in SFT, one can solve the
transversality issues by explicit methods. One instance of when this can be
done is in Legendrian contact homology, which is an invariant of Legendrian
submanifolds. One should mention that not even in this case is the theory
worked out in full generality. However, in [EES07] Ekholm, Etnyre and Sullivan established the analytical details in the case when the contact manifold
is of the form P × R, where P is an exact symplectic manifold satisfying some
technical, but not too restrictive constraints. In particular, this implies that
Legendrian contact homology is well-defined for Legendrian submanifolds of
1-jet spaces.
2.1 Legendrian contact homology in 1-jet spaces
Let Λ ⊂ J 1 (M) be a closed Legendrian submanifold, and assume that it is
chord generic. To define the Legendrian contact homology of Λ, we first associate a differential graded algebra A (Λ) to Λ.
14
2.1.1 Generators
Let R be a unital ring. The algebra A (Λ) is defined to be a unital, associative algebra over R, freely generated by the Reeb chords of Λ. If we want to
emphasize which ring we are considering we write A (Λ, R).
The generators are given the grading
|c| = CZ(c) − 1
where CZ(c) is the Conley-Zehnder index of c, see, e.g., [EES05a] or Paper I
for a definition.
2.1.2 The differential
The differential of A (Λ) is defined by counting pseudo-holomorphic disks of
the Legendrian Λ. This can be done in two different ways, either by counting
disks in the cotangent bundle T ∗ M, with the disks having boundary on the Lagrangian ΠC (Λ), or by counting disks in the symplectization of J 1 (M), with
the disks having boundary on the Lagrangian R × Λ. In [DR], DimitroglouRizell proves that for certain choices of almost complex structures these two
different set-ups give the same count of elements (without signs). In this section we give the definition of the count in the cotangent bundle, and refer to
Section 2.3 for the definition of the count in the symplectization.
Let J be an almost complex structure of T ∗ M, compatible with the standard symplectic structure. Let Dm+1 denote the punctured unit disk in C with
m + 1 punctures p0 , . . . , pm cyclically ordered along the boundary in the counterclockwise direction, starting at p0 = 1. Let a, b1 , . . . , bm be Lagrangian projections of Reeb chords.
Definition 2.1.1. We say that
u : (Dm+1 , ∂ Dm+1 ) → (T ∗ M, ΠC (Λ))
is a J-holomorphic disk of Λ with positive puncture a and negative punctures
b = b1 · · · bm if
• ∂¯J u := du + Jdu ◦ i = 0,
• u|∂ Dm+1 \{p0 ,...,pm } has a continuous lift ũ : ∂ Dm+1 \ {p0 , . . . , pm } → Λ,
• u(p0 ) = a, and ũ makes a jump from lower to higher z-coordinate when
passing through p0 in the counterclockwise direction,
• u(pi ) = bi , i = 1, . . . , m, and ũ makes a jump from higher to lower zcoordinate when passing through pi in the counterclockwise direction.
We let M (a, b) = MΠC (Λ) (a, b) denote the moduli space of J-holomorphic
disks of Λ with positive puncture a and negative punctures b. We consider two
disks in the moduli space to be equal if they only differ by a biholomorphic
reparametrization of the domain.
15
We define the differential of A (Λ, R) to be given by
∂ (a) =
∑
|M (a, b)|R b
(2.1)
dim M (a,b)=0
on generators a, and extend it to the whole of A (Λ) by the Leibniz rule. Here
|M (a, b)|R ∈ R is the algebraic count of elements in the moduli space, which
in the case of R = Z2 is given by the modulo 2 count.
2.1.3 Invariance over Z2
To get that the homology of (A (Λ), R) is a well-defined Legendrian invariant
it remains to prove the following:
(i.1) the sum in (2.1) is finite;
(i.2) ∂ 2 = 0;
(i.3) the homology is invariant under Legendrian isotopies.
In the case when R = Z2 this was proven by Ekholm, Etnyre and Sullivan in
[EES07] by using Fredholm theory for the ∂¯ -operator. That is, they proved that
there is a functional analytic setup so that the ∂¯J -operator is Fredholm. With
this established, they showed that for a generic choice of J, the linearized
∂¯ -operator Du ∂¯J is surjective for every J-holomorphic disk u (that is, 0 is a
regular value for ∂¯J ). From the infinite-dimensional implicit function theorem
it then follows that the moduli spaces M (a, b) are smooth manifolds. See e.g.
[MS12].
Definition 2.1.2. If ∂¯J is a Fredholm operator with 0 as a regular value, then
we say that the spaces M (a, b) are transversely cut out, and we call J a regular
almost complex structure. If moreover dim M (a, b) = 0 and u ∈ M (a, b),
then we say that u is a rigid disk.
The statements (i.1) and (i.2) now follows from a Gromov compactness
result proven in [EES07]. Indeed, this result implies that the 0-dimensional
moduli spaces consist of a finite number of elements, which makes the count in
the differential finite. It also implies that the boundaries of the 1-dimensional
moduli spaces are given by broken 0-dimensional curves, from which it follows that ∂ 2 = 0 if R = Z2 . In the same paper it is also proven that the
homology of (A (Λ), Z2 ) gives a Legendrian invariant, which in addition is
independent of the regular almost structure J.
2.1.4 Invariance over Z
To extend the definition of the differential ∂ in (2.1) to Z-coefficients, we have
to define coherent orientations of the 0-dimensional moduli spaces, that is,
orientations so that (i.2) and (i.3) hold.
16
Indeed, the count |M (a, b)|Z in (2.1) corresponds to a count of disks u ∈
M (a, b) where each disk is counted with a sign. Since the moduli spaces we
are considering are 0-dimensional manifolds, it follows that the sign of u can
be identified with the orientation of Tu M (a, b).
Since the 0-dimensional space M (a, b) is discrete it is clear that it can
be oriented, but the problem is to find a coherent orientation scheme. By
using Gromov compactness, we see that the rigid disks occur as boundaries
of 1-dimensional moduli spaces. Thus, if we orient the former objects as the
boundary of the latter we will get that their contribution to ∂ 2 cancel.
From the implicit function theorem it follows that if u ∈ M (a, b) is any
J-holomorphic disk then
Ker Du ∂¯ ' Tu M (a, b)
and hence an orientation of M (a, b) at u corresponds to an orientation of
Ker Du ∂¯ . If we now use the fact that Du ∂¯ is Fredholm and surjective, we see
that an orientation of Ker Du ∂¯ is nothing but an orientation of the determinant
line
max
max
^
^
det Du ∂¯ := (Ker Du ∂¯ ) ⊗ (Coker Du ∂¯ )∗
of Du ∂¯ . Thus the problem of giving coherent signs to 0-dimensional moduli
spaces can be seen as a problem of orienting the determinant line bundle of the
linearized ∂¯ -operator over the space of candidate maps. This space is given by
maps from the punctured disk with Lagrangian boundary conditions.
In the paper [EES05b], Ekholm, Etnyre and Sullivan defined something
called capping orientation of the determinant line of the linearized ∂¯ -operator,
and proved that this gives a coherent orientation scheme for the 0-dimensional
moduli spaces in the case when Λ is spin. This in turn uses the fact that the
determinant line bundle of the ∂¯ -operator over the space of trivialized Lagrangian boundary conditions on the closed unit disk in C is orientable, which
was proven by Fukaya, Oh, Ohta and Ono in [FOOO09].
The idea behind the definition of capping orientations is that if Λ is spin,
then we can choose well-defined trivializations for the Lagrangian boundary
conditions induced by the pseudo-holomorphic disks. Then we glue something called capping operators to the punctures of the disks, to close up the
boundary conditions to get ∂¯ -problems on the closed disk, with trivialized Lagrangian boundary conditions. With this done, we can use the result from
[FOOO09] to give a coherent orientation to the moduli spaces.
2.2 Morse flow trees
In general, it is hard to solve the ∂¯ -equation (1.1) for the pseudo-holomorphic
disks in the differential in Legendrian contact homology. One way to resolve
17
this problem is to use the technique of flow trees, which are trees built by
solution curves of certain gradient vector fields.
2.2.1 Background
That pseudo-holomorphic disks can be replaced by flow lines was first realized by Floer in his paper [Flo89], where he studied intersections of Lagrangian submanifolds in a cotangent bundle T ∗ M using pseudo-holomorphic
curve techniques. He proved that if f : M → R is a Morse function, then for
certain choices of almost complex structure J on T ∗ M and metric g on M, the
pseudo-holomorphic strips
u : (R × [0, 1], ∂ (R × [0, 1])) → (T ∗ M, d f ∪ M),
∂¯J (u) = 0,
lim u(τ,t) = x± ∈ d f ∩ M
τ→±∞
are in one-to-one correspondence with gradient flow lines
x : R → M,
d
x(τ) = −∇g ( f (x(τ)),
dτ
lim x(τ) = x± .
τ→±∞
Remark 2.2.1. Note that the section d f defines an exact Lagrangian in T ∗ M.
This result was generalized by Fukaya and Oh in [FO97] to hold for punctured pseudo-holomorphic disks with boundary on several Lagrangian sections
d f1 , . . . , d fk . To get the correspondence with flow lines, the authors introduced
flow trees. These are immersed trees in M whose edges are gradient flow lines
of pairwise differences of the functions f1 , . . . , fk , and where the vertices correspond to either intersection points of the Lagrangians, or to interior points
where one switches from one pair of functions to another.
In [Ekh07], Ekholm showed that these techniques can be applied in Legendrian contact homology, by establishing a one-to-one correspondence between
rigid flow trees of Λ and rigid pseudo-holomorphic disks of Λ. The main difficulty in this generalization is the presence of cusp edge singularities of the
Legendrians.
The advantage of using trees instead of pseudo-holomorphic disks is that
the former gives an explicit, flow-theoretical way to compute the differential,
instead of trying to solve the non-linear PDE associated to a rigid disk. In
the case when the dimension of the Legendrian equals 1, then one can use the
Riemann mapping theorem to reduce the problem of solving the ∂¯ -equation to
combinatorics, see e.g. [Che02], but in higher dimensions this is not possible.
In [EENS13], the methods of flow trees were used to compute the Legendrian
contact homology in the case of the co-normal lift of a knot, which gives a
Legendrian of dimension n = 2.
18
Below we give a sketch of the definition of Morse flow trees in Legendrian
contact homology, and also indicate how the correspondence with pseudoholomorphic disks can be seen. We refer to [Ekh07] and Paper I for a more
detailed description.
2.2.2 Morse flow trees in Legendrian contact homology
Let Λ ⊂ J 1 (M) be a chord generic Legendrian submanifold with simple front
singularities. We refer to [Ekh07] for a definition of the latter condition, but
briefly this means that the base projection
Π:Λ→M
is an immersion outside a co-dimension 1 singular set Σ ⊂ Λ, and that the
points in Σ give rise to standard cusp edge singularities.
Away from the singular set Σ, the pre-image of an open set U ⊂ M under Π
is given by the multi-1-jet lift of locally defined functions
f1 , . . . , fk : M → R,
that is,
Π−1 (U) =
k
[
{(x, d fi (x), fi (x)); x ∈ U}.
i=1
These locally defining functions of Λ are used to build the Morse flow trees.
More precisely, these trees are defined as follows.
Definition 2.2.2. A Morse flow tree is an immersed tree Γ in M satisfying the
following conditions.
• The tree is rooted and oriented away from the root. The root is 1- or
2-valent.
• Each edge γ of Γ is a solution curve of some local function difference:
γ̇ = γ̇i j (t) = −∇( fi − f j )(γi j (t)),
where fi > f j are locally defining functions of Λ.
• The edge γi j is given the orientation of −∇( fi − f j ).
• The cotangent lift of Γ gives an oriented closed curve in ΠC (Λ), in the
following way. Each edge γi j has two cotangent lifts
γ̂i j,k = {(x, d fk (x)); x ∈ γi j },
k = i, j.
If we give γ̂i j,i the orientation of γi j , and γ̂i j, j the negative orientation
of γi j , then the union of all the lifted edges of Γ are required to patch
together to give a closed curve in ΠC (Λ) ⊂ T ∗ M.
19
• The vertices of Γ have valence at most 3, and are of the following form.
– 1-valent punctures, which are critical points of the corresponding
local function difference,
– 2-valent punctures, which are critical points of the corresponding
local function difference,
– 3-valent Y0 -vertices, where flow lines γi j , γ jk , γik meet,
– 3-valent Y1 -vertices, similar to Y0 -vertices but contained in Π(Σ),
– 2-valent switch-vertices, contained in Π(Σ), with corresponding
flow lines which are tangent to Π(Σ) at the vertex,
– 1-valent end-vertices, contained in Π(Σ), with corresponding flow
lines which are transverse to Π(Σ) at the vertex.
• The root of the tree Γ is required to be a puncture, and is called the
positive puncture of the tree. All other punctures are called negative.
Since a puncture p is a critical point of a local function difference, we have
stable and unstable manifolds associated to p. These we denote by W s (p) and
W u (p), respectively.
The dimension of a Morse flow tree Γ with positive puncture a and negative
punctures b1 , . . . , bm can be computed using data from the tree, and is given by
m
dim(Γ) = 2 + dimW u (a) + ∑ (dimW s (b j ) − n + 1) + e(Γ) − s(Γ) −Y1 (Γ),
j=1
where e(Γ), s(Γ), Y1 (Γ) is the number of end-, switch- and Y1 -vertices of Γ.
Definition 2.2.3. A rigid flow tree is a Morse flow tree of dimension 0 which
is transversely cut out from the space of flow trees.
2.2.3 Correspondence between flow trees and disks
The correspondence between rigid flow trees and rigid pseudo-holomorphic
disks given in [Ekh07] is proven using the fiber scaling
sλ : J 1 (M) → J 1 (M),
(x, y, z) 7→ (x, λ y, λ z),
which pushes Λ towards the zero section. In fact, this correspondence is a
bijective correspondence between rigid flow trees of Λ and sequences of rigid
pseudo-holomorphic disks {uλ }λ →0 , where uλ is a pseudo-holomorphic disk
of the scaled Legendrian Λλ := sλ (Λ). The boundary of the disks converges
to the cotangent lift of Γ.
That there exists a family of pseudo-holomorphic disks converging to Γ is
proven using a sequence of pre-glued disks {wλ }λ →0 . These disks are built
out of local J-holomorphic disk models over neighborhoods of the vertices
20
of Γ, and they are constructed to have boundary on ΠC (Λλ ), arbitrarily close
to the cotangent lift of the corresponding part of Γ. Moreover, with a certain
choice of metric on M and after small perturbations of J and Λ, the Lagrangian
boundary conditions over these pieces can be described explicitly.
The local models are then glued together to give a disk wλ which is Jholomorphic except in the gluing regions. Then, by using elliptic estimates
together with a Floer-Picard Lemma, see e.g. [Flo95], it is proven that there
exists a true J-holomorphic disk uλ of Λλ , so that uλ → wλ as λ → 0.
The existence of the pre-glued disks wλ is repeatedly used in Paper I.
2.3 Exact Lagrangian cobordisms
Let L ⊂ R × J 1 (M) be an exact Lagrangian cobordism with cylindrical Legendrian ends Λ± . This cobordism setting fits into the package of SFT, meaning
that L induces a DGA-morphism
ΦL : A (Λ+ ) → A (Λ− )
by a count of pseudo-holomorphic disks.
More precisely, let ML (a, b) denote the moduli space of J-holomorphic
disks
u : (Dm+1 , ∂ Dm+1 ) → (R × J 1 (M), L)
mapping a neighborhood of the positive puncture p0 of Dm+1 asymptotically
to the Reeb chord strip R × a at t = +∞, and mapping a neighborhood of
the negative puncture p j asymptotically to the Reeb chord strip R × b j at t =
−∞, j = 1, . . . , m. Here J is some compatible almost complex structure of
R × J 1 (M), and we assume in addition that there is an N > 0 so that J is
cylindrical∗ for |t| > N. For generic J the moduli spaces are transversely cut
out manifolds of the expected dimension.
Let the coefficient ring be given by Z2 . We define ΦL on generators by
ΦL (a) =
∑
dim ML (a,b)=0
|ML (a, b)|Z2 b,
and extend it to the rest of the algebra by
ΦL (a + b) = ΦL (a) + ΦL (b)
ΦL (ab) = ΦL (a)ΦL (b).
In [EHK] it is proven, using the Gromov compactness results in [BEH+ 03]
and [Ekh08], that the definition of ΦL is functorial. That is, if Λ+ = Λ− = Λ
and L = R × Λ, then
ΦR×Λ = idA (Λ) ,
∗ An
almost complex structure of the symplectization of (W, α) is said to be cylindrical if it is
compatible with d(et α), invariant under t-translation, and J(Ker α) = Ker α, J(∂t ) = Rα .
21
and if L1 is an exact Lagrangian cobordism from Λ0 to Λ1 , and L2 is an exact Lagrangian cobordism form Λ1 to Λ2 , then these two cobordisms can be
concatenated to an exact Lagrangian cobordism L1 · L2 from Λ0 to Λ2 and
ΦL1 ·L2 = ΦL2 ◦ ΦL1 : A (Λ0 ) → A (Λ2 ).
(2.2)
Note that this concatenation is not canonical.
To prove that the map ΦL gives a DGA-morphism, i.e., that
ΦL ◦ ∂+ = ∂− ◦ ΦL
where ∂± is the differential for A (Λ± , Z2 ), one again uses Gromov compactness. But to be able to use these arguments, we first have to lift the definition of the differential given in Section 2.1 to be given by a count of pseudoholomorphic disks in the symplectization of J 1 (M). The latter count is defined in a similar way as the map ΦL . Namely, let J be a cylindrical almost
complex structure of the symplectization R × J 1 (M), and consider the trivial
Lagrangian cobordism R × Λ. Then we define the differential
∂ : (A (Λ), Z2 ) → (A (Λ), Z2 )
to be given by
∂ (a) =
∑
dim MR×Λ (a,b)=1
|MR×Λ (a, b)/R|Z2 b
on generators, and extend it by the Leibniz rule to the rest of the algebra.
Here the R-action comes from the fact that the almost complex structure is
translation-invariant in the t-direction.
Remark 2.3.1. If we want to emphasize which differential of A (Λ) we are
using, we write ∂s for the one defined by the count of disks in the symplectization, with corresponding moduli spaces MR×Λ (a, b), and ∂l for the one
defined by a count of disks in the Lagrangian projection, with corresponding
moduli spaces MΠC (Λ) (a, b). In [DR] it is proven that for certain choices of
almost complex structures, these two different settings give the same count of
disks. That is, it is proven that the projection
πP : R × (T ∗ M × R) → T ∗ M
induces a diffeomorphism
πP : MR×Λ (a, b)/R → MΠC (Λ) (a, b),
u 7→ πP (u).
The DGA-morphisms induced by exact Lagrangian cobordisms can be used
to derive results about the geometry and topology of Legendrian and Lagrangian submanifolds. For example, in [EHK] Ekholm, Honda and Kálmán
22
used these techniques to prove that there exist embedded exact Lagrangians in
R × R3 , which are smoothly isotopic and bound the same Legendrian knot, but
which are not isotopic through exact Lagrangian embeddings. To prove this,
the authors used Morse flow tree techniques to give an explicit description of
the DGA-morphisms induced by the following elementary cobordisms:
(L1) the trace of a triple point move;
(L2) the trace of pairwise cancellation of Reeb chords;
(L3) the trace of pairwise creation of Reeb chords;
(L4) a disk bounding the standard Legendrian unknot;
(L5) a Lagrangian that corresponds to a zero resolution of a contractible Reeb
chord.
In Paper II we extend the definition of ΦL to Z-coefficients, and calculate the
signs for the maps that correspond to (L1) – (L3).
23
3. On the appended papers
3.1 Paper I
In this paper we give a coherent orientation scheme for the moduli space of
rigid flow trees associated to a spin Legendrian Λ ⊂ J 1 (M). The orientation scheme uses orientations of stable and unstable manifolds associated to
the punctures of the trees, and by transversality and rigidity it follows that
the orientation of a tree can be given by oriented intersections of these flowmanifolds.
To make this rigorous, we modify the techniques of capping orientations for
pseudo-holomorphic disks from [EES05b] to fit with the correspondence between trees and disks given in [Ekh07]. Recall from Section 2.2.3 that this correspondence is given by a correspondence between rigid trees and sequences
of rigid disks uλ of the scaled Legendrian sλ (Λ).
The main result of Paper I is the following.
Theorem 3.1.1 (Paper I, Theorem 1.1). Let Λ ⊂ J 1 (M) be a spin Legendrian
submanifold, and fix a spin structure s of Λ. Assume that Γ is a rigid flow
tree of Λ, and let uλ , λ → 0, be a sequence of rigid pseudo-holomorphic disks
converging to Γ in the sense of [Ekh07]. Let Os (uλ ) be the capping orientation
of uλ with respect to s. Then there is a combinatorial way to compute this
capping orientation for each λ sufficiently small, and in particular, there is a
λ0 > 0 so that Os (uλ ) is independent of λ for λ ≤ λ0 .
To prove this, we consider the capping orientation of the pre-glued disks
wλ used in [Ekh07] to establish the existence of the sequence uλ , as briefly
explained in Section 2.2.3. By considering the capping orientations of the local
models that wλ is built out of, we get the combinatorial orientation scheme.
Then, by using the rigidity of the disks together with Fredholm properties of
the associated linearized ∂¯ -operators, we prove that the capping orientation of
wλ coincides with the capping orientation of uλ for small λ .
To derive the combinatorial algorithm, we use the explicit description of the
boundary conditions associated to the local models of wλ over the vertices.
That is, for each vertex v of Γ let wv denote the corresponding piece of wλ . By
considering the linearized ∂¯ -problem at this piece we get a ∂¯ -problem
Dwv ∂¯ : H2 [wv ] → H1 [wv ],
where Hk [wv ], k = 1, 2, denote Sobolev spaces associated to the linearization.
These Sobolev spaces can be weighted in a way so that the operators Dwv ∂¯
24
become surjective, with kernels that can be identified with subspaces of T M
along Γ via evaluation as follows.
Ker Dwv ∂¯ :
Type of vertex v:
1-valent, positive puncture
TvW u (v)
1-valent, negative puncture
TvW s (v)
2-valent puncture
0
Y0 -vertex
Tv M
Y1 -vertex
Tv Π(Σ)
switch-vertex
Tv Π(Σ)
end-vertex
Tv M
It follows that the capping orientation of the vertices can be understood as
orientations of these subspaces.
To get the capping orientation of the tree Γ out of this, we glue the pieces
back together again, piece by piece. This will give us intermediate disks wC ,
which correspond to connected parts of Γ with vertices C = v1 , . . . , vk . To calculate the capping orientations of the disks wC we use linear gluing sequences.
These occur when we glue two disks wA , wB together at a common special∗
puncture p, to create a glued disk wC = wA # p wB . If the Sobolev spaces associated to the disks wA and wB both have a small negative exponential weight
at the puncture p, then this gluing gives rise to an exact sequence
0 → Ker DwA # p wB ∂¯ → Ker DwA ∂¯ ⊕ Ker DwB ∂¯
→ Coker Dw ∂¯ ⊕ Tp M ⊕ Coker Dw ∂¯ → Coker Dw
A # p wB
B
A
∂¯ → 0. (3.1)
In particular, if the operators DwA ∂¯ , DwB ∂¯ , DwA # p wB ∂¯ are all surjective and
have kernels that can be identified with subspaces of Tp M via evaluation, then
Ker DwA # p wB ∂¯ can be interpreted as the fibered product Ker DwA ∂¯ × p Ker DwB ∂¯ .
This means that
Ker Dw # w ∂¯ ' ev p (Ker Dw # w ∂¯ ) = ev p (Ker Dw ∂¯ ) ∩ ev p (Ker Dw ∂¯ ),
A p B
A p B
A
B
and the orientation of Ker DwA # p wB ∂¯ can be computed from the orientations of
Tp M and Ker Dwi ∂¯ , i = A, B, via the sequence (3.1) which now reads
0 → Ker DwA # p wB ∂¯ → Ker DwA ∂¯ ⊕ Ker DwB ∂¯ → Tp M → 0.
∗ When
we cut wλ into pieces, we create punctured disks which inherit the punctures from
wλ , but which also will have additional punctures representing the cutting points. The latter
punctures are called special punctures.
25
Unfortunately, after having glued together pieces of wλ so that we get a
disk with at least 4 punctures, the corresponding linearized ∂¯ -operator will
not be surjective anymore. The reason is that we have to consider the space
of conformal structures of the disk when we linearize the ∂¯ -operator. For a
disk with m ≥ 3 punctures, this space is of dimension m − 3, and hence the
linearized ∂¯ -operator at an m times punctured rigid disk will be injective with
an m − 3-dimensional cokernel.
To get a geometric interpretation of the cokernel, we stabilize the linearized
¯∂ -operators so that they become surjective. We do this by adding a space Vcon
to the domain of the operator, and then we extend the operator over this space.
The space Vcon represents the space of conformal variations, and the vectors in
this space can be interpreted as vector fields in M along Γ, corresponding to
moving the 2-valent punctures and the 3-valent vertices along the tree.
In this way we get a stabilized operator
Dwλ ,s ∂¯ : H2 [wλ ] ⊕Vcon → H1 [wλ ]
(3.2)
which is an isomorphism. Moreover, if wC represents a part of wλ obtained by
cutting Γ into 2 pieces at a point p, then the corresponding stabilized operator
DwC ,s ∂¯ will be surjective with kernel that can be identified with its evaluated
image in Tp M.
Remark 3.1.2. For pieces of wλ corresponding to vertices of Γ, the stabilized
∂¯ -operator coincides with the un-stabilized operator.
In summary, this means that when we glue the pieces of wλ back together,
if we at each gluing consider the stabilized problem (3.2) then we can replace
the sequence (3.1) by the sequence
0 → Ker DwA # p wB ,s ∂¯ → Ker DwA ,s ∂¯ ⊕ Ker DwB ,s ∂¯ → Tp M → 0
(3.3)
and get that
Ker DwA # p wB ,s ∂¯ ' Ker DwA ,s ∂¯ × p Ker DwB ,s ∂¯ .
This allows us to inductively compute the capping orientation of wλ by
considering oriented intersections of subspaces in M along Γ. And by rigidity
we get that the final gluing, which is the gluing where we recover wλ , gives a
0-dimensional intersection and hence only a sign. This will be the orientation
of Γ.
The essential difficulty in the proof of Theorem 3.1.1 is that the domains of
the disks wλ and uλ vary with λ . This implies, for example, that we have to
interpret the sequences (3.1) and (3.3) as sequences of vector bundles over the
parameter space (0, λ0 ].
Moreover, the stabilizations of the linearized ∂¯ -operators give rise to additional signs that must be added to Ker DwA # p wB ,s ∂¯ after the gluing. That is, the
26
stabilized orientation of Ker DwA # p wB ,s ∂¯ will not be given from the orientation
induced by the sequence (3.3) solely, we also have to multiply by signs coming from the fact that the stabilized glued orientation not necessarily equals
the glued stabilized orientation. This gives rise to quite long sign formulas,
but despite this, the formulas are computable and expressed in terms of data
coming from the tree. These formulas are given in Section 3 of Paper I.
3.2 Paper II
In this paper we study DGA-morphisms
ΦL : A (Λ+ , Z) → A (Λ− , Z)
induced by the trace L of a Legendrian isotopy Λ+ ' Λ− of Legendrian knots
in R3 . By using the Morse flow tree techniques from [EK08] and [EHK]
together with the orientation results from Paper I, we prove that the combinatorially defined DGA-maps in [Kál05] can be derived analytically.†
More precisely, in [EHK] it is proven that the definition of ΦL as a count
of pseudo-holomorphic disks with boundary on L can be replaced by a count
of certain Morse flow trees. These are flow trees associated to a Morse cobordism LMO which is exact Lagrangian isotopic to L. Using arguments similar
to those in [Ekh07], it is proven that the rigid Morse flow trees of LMO are
in bijective correspondence with the rigid disks of LMO . Hence, if we let
MT,L (a, b) denote the moduli space of flow trees of LMO with positive puncture a and negative punctures b, then the DGA-morphism ΦL can be given
by
ΦL (a) =
∑
|MT,L (a, b)|b.
(3.4)
dim MT,L (a,b)=0
With the orientation scheme of Morse flow trees from Paper I we can define
|MT,L (a, b)| to be an algebraic count of elements.
By using abstract perturbations of the flow trees, as described in [EK08]
and [EHK], we prove the following.
Theorem 3.2.1 (Paper II, Theorem 1.4). Let L be an exact Lagrangian cobordism induced by an elementary Legendrian isotopy, given by one of the moves
(L1) – (L3) in Section 2.3. Then there is a choice of orientation conventions
so that
ΦL : A (Λ+ , Z) → A (Λ− , Z)
defined by (3.4) is chain homotopic to the corresponding DGA-map given in
[[Kál05], Theorem 3.2].
† We
give an explicit description of these maps in Paper II, Section 4.
27
In [Kál05], these maps were used to prove invariance of the homology of
(A (Λ), Z) under Legendrian isotopies, with the differential defined combinatorially using the shading rule given in [ENS02]. We give a reminder of how
this rule works in Section 4 of Paper II.
In [EES05b] it is proven that this shading rule can be derived analytically,
by using capping orientations of the disks in the ∂l -differential. We prove,
after a slight modification of the orientation conventions in [EES05b], that
this orientation scheme can be lifted to the moduli spaces associated to the
differential ∂s . Moreover, this modification will not affect the shading rule.
It follows that there is an orientation scheme for moduli spaces of pseudoholomorphic disks so that
ΦL ◦ ∂s,+ = ∂s,− ◦ ΦL .
All this is summarized by the following result.
Theorem 3.2.2 (Paper II, Theorem 1.2). Let Λ ⊂ J 1 (M) be a spin Legendrian
submanifold. Pick almost complex structures on T ∗ M and R × J 1 (M), respectively, satisfying the assumptions in [[DR], Theorem 2.1]. Then the moduli
spaces Ml,Λ (a, b) = MΠC (Λ) (a, b) and Ms,Λ (a, b) = MR×Λ (a, b)/R can be
given orientations so that
πP : Ms,Λ (a, b) → Ml,Λ (a, b)
u 7→ πP (u)
is orientation preserving, and so that
∂i (a) =
|Mi,Λ (a, b)|b,
∑
i = l, s,
(3.5)
dim Mi,Λ (a,b)=0
satisfies ∂i2 = 0. Here |Mi,Λ (a, b)| is the algebraic number of disks in the
moduli space, where the signs come from the coherent orientation scheme.
If M = R and Λ is a knot, then this orientation scheme gives the shading
rule of Λ as defined in [ENS02].
Remark 3.2.3. This is not completely straightforward, compare [[DR], Remark 2.4]. The complication comes from the fact that the orientation conventions in [EES05b] do not give a differential of the form (3.5), but instead
∂l (a) =
∑
(−1)(n−1)(|a|+1) |Ml,Λ (a, b)|b.
dim Ml,Λ (a,b)=0
3.3 Paper III
This paper concerns immersed, closed Lagrangians in the plane (R2 , dx ∧ dy).
We prove that the only obstruction for there to be a symplectic isotopy
φt : R2 → R2 ,
28
t ∈ [0, 1],
φ0 = id
taking a Lagrangian L1 to a Lagrangian L2 is the area of the disks that L1 and
L2 bounds.
More precisely, a closed, connected Lagrangian in R2 is an immersed circle,
and hence it divides the plane into a number of bounded components (disks),
and one unbounded component. Since a symplectic isotopy is area-preserving,
a necessary condition for there to be a symplectic isotopy taking L1 to L2 is
that there is a smooth isotopy
ψt : R2 → R2 ,
t ∈ [0, 1],
ψ0 = id,
ψ1 (L1 ) = L2 ,
(3.6)
so that
area(D) = area(ψ1 (D))
(3.7)
for every disk D that L1 bounds. In Paper III we prove that this is also a
sufficient condition.
Since every symplectic isotopy in the plane is a Hamiltonian isotopy, the
result can be restated as follows.
Theorem 3.3.1 (Paper III, Theorem 1.1). Let L1 and L2 be two self-transverse,
closed, connected, immersed Lagrangian submanifolds of (R2 , dx ∧ dy). Assume that there is an isotopy ψt : R2 → R2 satisfying (3.6) and (3.7). Then
there is a Hamiltonian isotopy φt : R2 → R2 , t ∈ [0, 1], taking L1 to L2 .
29
4. Att orientera modulirum av flödesträd för
symplektisk fältteori, Summary in Swedish
4.1 Bakgrund
Ända sedan Lord Kelvins idé om att atomer är ”knutar i etern” så har matematiker försökt klassificera geometriska objekt. Två objekt som kan deformeras
till varandra kallas för isotopa och den vanligaste metoden för att studera isotopiproblem är att koppla invarianter till objekten. En invariant är en kvantitet
som bevaras under de tillåtna deformationerna och kan exempelvis ges av ett
tal, ett polynom, en algebra eller ett topologiskt rum. Objekt som har invarianter som skiljer sig åt kan inte vara isotopa, men att två objekt har invarianter
som överensstämmer behöver inte nödvändigtvis innebära att de är deformationer av varandra.
Detta studium tillhör idag den del av matematiken som kallas topologi och
har under åren som gått generaliserats i många olika riktningar, bland annat till
högre dimensioner och till att innefatta olika typer av geometriska bivillkor.
Ett exempel på ett sådant bivillkor ges av kontaktstrukturen hos en kontaktmångfald. En kontaktmångfald är ett par (W, ξ ) där W är en uddadimensionell
glatt mångfald och ξ är ett totalt icke-integrerbart hyperplanfält definierat över
W . Hyperplansdistributionen ξ kallas för mångfaldens kontaktstruktur.
Denna avhandling handlar i huvudsak om invarianter av Legendriska delmångfalder. Dessa är delmångfalder till W som uppfyller villkoret att de
tangerar ξ och som dessutom är av maximal dimension för att kunna göra
detta. Vi ger en mer utförlig beskrivning av vad allt detta innebär i nästa avsnitt.
4.2 Kontaktgeometri
Studiet av Legendriska delmångfalder tillhör ämnet kontaktgeometri, ett ämne
som har sitt ursprung i Sophus Lies arbete om system av differentialekvationer.
Kontaktgeometri har även kopplingar till optik och termodynamik där bland
annat vågfronter kan modelleras av Legendriska delmångfalder.
Den geometriska innebörden av en kontaktstruktur kan beskrivas på följande sätt. Låt dimW = 2n + 1. Att ξ är ett hyperplanfält betyder att för varje
punkt p i W så ger ξ oss ett 2n-dimensionellt delrum av tangentrummet Tp M,
så att dessa delrum varierar glatt över W . Att ξ är totalt icke-integrerbart betyder i sin tur att det (lokalt) kan ges som kärnan till en (lokalt definierad) 1-form
30
α som uppfyller α ∧ (dα)∧n 6= 0. Detta villkor gör att det ej går att hitta någon 2n-dimensionell delmångfald av W som är tangent till dessa hyperplan,
inte ens lokalt. Dessutom följer det att n är den maximala dimensionen för en
delmångfald som kan vara tangent till ξ och en sådan maximal delmångfald
kallas alltså för en Legendrisk delmångfald.
I de fall då ξ kan ges som kärnan till en globalt definierad 1-form α säger
vi att kontaktmångfalden är ko-orienterad. Formen α kallas för en kontaktform och vi skriver (W, α) för att mena (W, Ker α). Observera att olika val av
kontaktform kan ge samma kontaktstruktur.
4.3 Symplektisk geometri och pseudoholomorfa kurvor
Kontaktgeometri är nära besläktat med symplektisk geometri, vilket är ett ämne
som har sitt ursprung i klassisk mekanik och som bland annat används för att
förstå geometrin hos fasrum.
Mer bestämt så ges en symplektisk mångfald (V, ω) av en jämndimensionell
mångfald V tillsammans med en sluten, icke-degenererad 2-form ω, den symplektiska formen. Ett standardexempel ges av kotangentknippet T ∗ M till en
mångfald M, tillsammans med den symplektiska formen ω = ∑i dqi ∧ d pi där
q är lokala koordinater på M och p är koordinater i kotangentfibrerna.
Symplektisk geometri kan användas för att förstå geometrin hos kontaktmångfalder, och vice versa. Bland annat så går det att symplektisera en given
kontaktmångfald (W, α) för att få en symplektisk mångfald (R ×W, d(et α)).
Motsvarigheten till Legendriska delmångfalder i symplektisk geometri ges
av Lagrange-delmångfalder och dessa är delmångfalder L ⊂ V som uppfyller
1
dim L = dimV, ω|T L = 0.
2
Immerserade Lagrangianer definieras på ett liknande sätt.
Lagrange-delmångfalder är rigida objekt, vilket gör dem svåra att få grepp
om. Detta insågs bland annat av Arnold som formulerade en förmodan om
det minsta antal skärningspunkter som två Lagrange-delmångfalder måste ha,
och av Gromov, som bland annat visade att det ej går att inbädda S3 som
en Lagrange-delmångfald av R6 . Immerserade Lagrangianer är däremot mer
flexibla, ty de uppfyller en så kallad h-princip.
Beviset av Gromovs rigiditetsresultat använder pseudoholomorfa kurvor.
Se [Gro85]. Detta har senare visat sig vara ett mycket användbart verktyg
för att studera Lagrange-delmångfalder och vidareutvecklades av Floer, bland
andra, för att bevisa delar av Arnolds förmodan.
För att kunna beskriva vad en pseudoholomorf kurva är för slags objekt, så
måste vi först välja en nästan komplex struktur på V . Detta är en endomorfi
J : TV → TV som uppfyller att J 2 = − id. Om nu (Σ, j) är en Riemannyta så
säger vi att
u : (Σ, j) → (V, J)
31
är pseudoholomorf om Cauchy-Riemann-ekvationen
∂¯J := du + J ◦ du ◦ j = 0
är uppfylld.
4.4 Det Legendriska isotopiproblemet
Två Legendriska delmångfalder Λ0 , Λ1 ⊂ W är Legendriskt isotopa om det går
att finna en glatt familj av Legendriska delmångfalder Λ̃t ⊂ W , t ∈ [0, 1], som
uppfyller att Λ̃i = Λi , i = 0, 1. Familjen {Λ̃t }t∈[0,1] kallas för en Legendrisk isotopi. Att klassificera alla Legendriska delmångfalder av en kontaktmångfald,
upp till Legendrisk isotopi, är ett av huvudproblemen inom kontaktgeometrin.
För att finna en lösning på detta problem söker man efter Legendriska invarianter, det vill säga invarianter som är oförändrade under Legendriska isotopier.
Två Legendriskt isotopa delmångfalder är även isotopa i den glatta kategorin. Genom att undersöka vilka krav som kontaktstrukturen ställer på
de Legendriska isotopierna har två klassiska Legendriska invarianter kunnat
definieras. Dessa kallas för Thurston-Bennequin invarianten samt rotationsnumret. Vi hänvisar till [Gei08] (i fallet dim(Λ) = 1) och [EES05a] (för högre
dimensioner) för deras definitioner. Med dessa invarianter är det enkelt att ge
exempel på Legendriska delmångfalder som är isotopa i den glatta kategorin
men som ej är Legendriskt isotopa.
4.5 Legendrisk kontakthomologi
Genom att använda Gromovs resultat om pseudoholomorfa kurvor har mer
sofistikerade Legendriska invarianter kunnat definieras. Dessa invarianter ingår under begreppet Symplektisk fältteori (SFT), vilket först introducerades av
Eliashberg, Givental och Hofer i artikeln [EGH00]. Kort sagt så är SFT en
teori som använder pseudoholomorfa kurvor för att ge algebraiska invarianter
för objekt i symplektisk- och kontaktgeometri.
Det finns delar av de grundläggande koncepten för SFT som fortfarande
behöver utredas. Bland annat saknas det verktyg för att lösa transversalitetsproblem hos pseudoholomorfa kurvor i det generella fallet, även om teorin
om polyfalder, se exempelvis [Hof], samt Pardons arbeten [Para, Parb] om
virtuella fundamentalcykler, tycks kunna lösa delar av dessa frågor. I vissa
situationer går det dessutom att komma runt dessa problem med explicita
metoder.
Ett exempel på när detta är möjligt är i definitionen av Legendrisk kontakthomologi för Legendriska delmångfalder Λ i 1-jetrummet
J 1 (M) = T ∗ M × R
32
av en mångfald M. Detta är en teori som associerar en graderad differentialalgebra A (Λ) till Λ, där A (Λ) är fritt genererad av dubbelpunkterna till Λ
under Lagrange-projektionen
ΠC : J 1 (M) → T ∗ M
och där differentialen räknar punkterade, pseudoholomorfa diskar i T ∗ M med
rand på ΠC (Λ).
För generiska val av kompatibel∗ nästan komplex struktur J på V så fås
att modulirummen av dessa pseudoholomorfa diskar är ändligtdimensionella
mångfalder med rand som består av brutna diskar. Detta medför att om vi
definierar differentialen som en modulo 2-räkning av diskar i 0-dimensionella
modulirum så ger homologin av A (Λ) en Legendrisk invariant med koefficienter i Z2 .
Eftersom differentialen för A (Λ) definieras genom att studera pseudoholomorfa diskar så går det att använda ellipticiteten hos ∂¯ -operatorn för att härleda
koherenta orienteringsscheman för diskarnas modulirum. Detta innebär att
Legendrisk kontakthomologi kan definieras med koefficienter i Z, vilket görs
rigoröst av Ekholm, Etnyre och Sullivan i [EES05b]. Detta bygger i sin tur
på Fukaya, Oh, Ohta och Onos resultat från [FOOO09], som säger att determinantlinjeknippet för ∂¯ -operatorn över den kompakta enhetsdisken i C med
trivialiserade Lagrange-randvillkor är orienterbart.
4.6 Flödesträd
I fallet då M = R kan räkningen av pseudoholomorfa diskar i differentialen
i Legendrisk kontakthomologi reduceras till kombinatorik via Riemanns avbildningssats. I högre dimensioner ger dock Cauchy-Riemann-ekvationerna
upphov till icke-linjära partiella differentialekvationer som i allmänhet är svåra
att lösa. För att undgå detta problem utvecklade Ekholm i [Ekh07] en teknik
för att beräkna differentialen genom att använda flödesträd. Detta reducerar
∂¯ -problemet till ett Morse-teoretiskt problem som kan lösas genom att förstå
vissa speciella gradientflöden i basmångfalden M. Detta ger ett kraftfullt verktyg för att beräkna Legendrisk kontakthomologi med koefficienter i Z2 även i
högre dimensioner.
4.7 Sammanfattning av avhandlingens resultat
I Artikel I definierar vi ett koherent orienteringsschema för modulirum av
rigida flödesträd för Legendriska delmångfalder Λ ⊂ J 1 (M). Orienteringsschemat använder orienteringar av de stabila och instabila flödesmångfalderna
∗J
är kompatibel med ω om ω(·, J·) definierar en riemannmetrik på V .
33
associerade till träden. Från transversalitets- och rigiditetsegenskaper hos träden följer det att orienteringen kan ges av orienterade skärningar av dessa
flödesmångfalder och på detta sätt får vi ett kombinatoriskt verktyg för att
beräkna differentialen för A (Λ) med Z-koefficienter i högre dimensioner.
För att göra allt detta rigoröst så använder vi orienteringsteknikerna från
[EES05b] tillsammans med en bijektion från [Ekh07] som förknippar rigida
träd med en viss typ av följder av rigida diskar. För att konstruera ett kombinatoriskt teckenschema krävs dock vissa modifikationer av konstruktionerna i
[EES05b]. Dessutom behöver vi använda att den Legendriska delmångfalden
trycks ned mot nollsektionen i J 1 (M) under bijektionen i [Ekh07], något som
även komplicerar konstruktionen. Orsaken till det senare är att följden av
diskar som svarar mot träden degenererar då vi går i gräns.
I Artikel II ger vi en analytisk härledning av de kombinatoriskt definierade
DGA-morfismerna† i [[Kál05], Sats 3.2]. Dessa DGA-morfismer användes av
Kálmán samt av författarna i [ENS02] för att visa invarians för Legendrisk
kontakthomologi över Z i fallet då dim(Λ) = 1 och M = R.
Vi utnyttjar att spåret av en Legendrisk isotopi Λ+ ' Λ− ger upphov till en
exakt Lagrange-kobordism i symplektiseringen av J 1 (R). Från de funktoriella
egenskaperna hos SFT följer det att en sådan kobordism L inducerar en DGAmorfism
ΦL : A (Λ+ ) → A (Λ− ),
vilken definieras genom att räkna pseudoholomorfa diskar med rand på L.
Genom att använda flödesträdskonstruktioner från [EK08] och [EHK] tillsammans med orienteringsschemat från Artikel I visar vi att ΦL är kedjehomotop
med Kálmáns motsvarande avbildning.
Artikel III rör symplektisk geometri och är fristående från de andra två
artiklarna. Här visar vi ett flexibilitetsresultat för slutna, immerserade Lagrangianer i (R2 , dx ∧ dy).
† Vi
34
använder förkortningen DGA för graderade differential-algebror.
5. Acknowledgements
First of all, I would like to thank my supervisor Tobias Ekholm for all support,
and for always taking time to discuss problems and come with new ideas. I
am also grateful for the financial support from the Knut and Alice Wallenberg
Foundation. Furthermore, I would like to express my appreciation to my assistant supervisor Ryszard Rubinsztein, especially for the excellent guidance
in the existing literature.
I will miss my colleagues at the department; administrators, researchers
and teachers. A special thanks goes to Håkan for all excursions related to
analysis and fitness training, to Ove, Emilia, Sebastian and Andreas for all
entertaining coffee breaks with discussions about topology and other, more
or less, important things, to Martin and Björn for all beer and soccer, and
to Seidon and Djalal for an algebraic but also nice atmosphere at the office. I
would also like to thank Lars-Åke who made me realize the magic of math, and
Johan B for tricking me to write a bachelor thesis in topology. I am also very
thankful to Bolle, Bertil and Leif for supporting my interest in math during
my school time in Finspång.
Without the support from my family, I would never have written this thesis.
I am very grateful to my mom, who still tries to teach me to think critically, and
to my dad, who encouraged me in my studies and who never stopped hoping
that I would return home as an engineer. A big thanks also goes to my second
family SJRK (med omkrets), for keeping my mind open and for all the joy of
socializing. In addition, I would like to thank Valentina (and Cauchy) for all
the years we spent together as flat-mates. And a special thanks to Marcus for
being such a good friend, entertaining neighbor and wise colleague.
Last, but definitely not least, I would like to thank Tomas for always standing by my side, and for reminding me that there is more in life than math.
35
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