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FILTERED FLOER AND SYMPLECTIC HOMOLOGY VIA GROMOV–WITTEN THEORY A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Lu´ ıs Miguel Pereira de Matos Geraldes Diogo August 2012

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Page 1: FILTERED FLOER AND SYMPLECTIC HOMOLOGY VIA …ldiogo/research/ThesisLuisDiogo.pdf · 5 Relation with Gromov{Witten numbers48 5.1 Pseudo-holomorphic curves in R Y and N, meromorphic

FILTERED FLOER AND SYMPLECTIC HOMOLOGY

VIA GROMOV–WITTEN THEORY

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MATHEMATICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Luıs Miguel Pereira de Matos Geraldes Diogo

August 2012

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Abstract

We describe a procedure for computing Floer and symplectic homology groups, with

action filtration and algebraic operations (coming from a version of Floer’s equation

on Riemann surfaces), in an important class of examples. Namely, we consider closed

monotone symplectic manifolds X with smooth symplectic divisors Σ, Poincare dual

to a positive multiple of the symplectic form (satisfying a few more technical assump-

tions). We express the Floer homology of X and the symplectic homology of XzΣ, for

a special class of Hamiltonians, in terms of absolute and relative Gromov–Witten in-

variants of the pair pX,Σq, and some additional Morse-theoretic information. The key

point of the argument is a relation between solutions of Floer’s equation and pseudo-

holomorphic curves, both defined on the symplectization of a pre-quantization bundle

over Σ. As an application, we compute the symplectic homology rings of cotangent

bundles of spheres, and compare our results with an earlier computation in string

topology.

iv

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Acknowledgements

It is a great pleasure for me to thank the many people who have had a decisive impact

in my life and work during these years as a graduate student.

I have to begin with my advisor, Yasha Eliashberg, who has taught me so much

for so long. I look up to his generosity with his time and ideas, to his creativity,

energy, humor, kindness, and to his example of persistence attacking problems until

they are solved. Having the opportunity with learn so much from Yasha has been an

enormous privilege, my appreciation of which has not ceased to increase with time.

A special mention is also due to Sam Lisi for his friendship, his patience in teaching

me so many things, and for long hours of discussions about this project. The work

presented in this thesis is at least as much his as mine (except for the mistakes, for

which I claim sole authorship).

My work has benefited enormously from interactions with many mathematicians.

In particular, Strom Borman, Frederic Bourgeois, Kai Cieliebak, Tobias Ekholm,

Oliver Fabert, Joel Fish, Janko Latschev, Mark McLean, Alex Oancea, Leonid Polte-

rovich, Paolo Rossi, Nick Sheridan and Dimitri Zvonkine. I want to thank especially

to Eleny Ionel, for her excellent classes and for always making time to answer my

questions.

I had the privilege of starting my graduate studies at the University of Chicago,

where I had great teachers and great friends. I want to thank them all, and in

particular to Shmuel Weinberger. I also want to thank all my teachers and fellow

students at Stanford University, for a fantastic learning environment. I was quite

lucky to have had the opportunity to spend the year 2009/2010 at the Mathematical

Sciences Research Institute. This was an extraordinary experience, for which I am

v

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very grateful. I also want to thank Leonid Polterovich and Strom Borman, at the

University of Chicago, Frederic Bourgeois and Samuel Lisi, at the Universite Libre

de Bruxelles, and Miguel Abreu, at Instituto Superior Tecnico, for their wonderful

hospitality.

I am extremely lucky for all the friends I have made, in Portugal, Chicago, Stan-

ford, Berkeley and in many conferences in many places. I have to apologize for not

naming them here, but I would certainly leave out many important people if I tried.

I want to thank them in any case, and I am sure that they know that I am referring

to them. I want to make one exception, though, and thank my dear friends Frank

and Sylvia Soler, for letting me be part of their family.

I want to write a warm hug to my family, that I missed and miss so much, namely

Ze, Alice, Augusto, Nuno, Carla, Bruna, Duarte, Maria, Joao, Rosa and Quim. Um

grande abraco e obrigado por tudo!

The final words have to go to Ana, who makes me so happy, for her constant

presence and for allowing things to make sense. This is for her.

vi

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This thesis was partly supported by the fellowship SFRH / BD / 28035 / 2006 of

Fundacao para a Ciencia e a Tecnologia, Portugal.

vii

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Contents

Abstract iv

Acknowledgements v

1 Introduction 1

2 Floer homology and symplectic homology 6

2.1 Filtered Floer homology groups . . . . . . . . . . . . . . . . . . . . . 6

2.2 Symplectic homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Operations on Floer theory; relation with string topology . . . . . . . 11

3 Split Floer homology 14

3.1 Symplectic divisors on monotone manifolds . . . . . . . . . . . . . . . 14

3.2 Degenerating the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Splitting the manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 An ansatz for split Floer trajectories 32

4.1 Several types of punctures and marked points . . . . . . . . . . . . . 32

4.2 Pseudo-holomorphic curves in R Y . . . . . . . . . . . . . . . . . . 34

4.3 The cylinder equation . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Floer trajectories in R Y . . . . . . . . . . . . . . . . . . . . . . . . 39

4.5 Excluding unwanted solutions . . . . . . . . . . . . . . . . . . . . . . 45

viii

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5 Relation with Gromov–Witten numbers 48

5.1 Pseudo-holomorphic curves in R Y and NΣ, meromorphic sections

of holomorphic line bundles and Gromov–Witten numbers of Σ . . . . 48

5.1.1 Gromov–Witten numbers and quantum cohomology . . . . . . 49

5.1.2 Meromorphic sections of holomorphic line bundles and pseudo-

holomorphic curves in R Y . . . . . . . . . . . . . . . . . . 52

5.1.3 Pseudo-holomorphic curves in NΣ . . . . . . . . . . . . . . . . 58

5.2 Pseudo-holomorphic curves in W and relative Gromov–Witten num-

bers of pX,Σq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2.1 Relative Gromov–Witten numbers . . . . . . . . . . . . . . . . 59

5.2.2 Pseudo-holomorphic curves in W . . . . . . . . . . . . . . . . 61

5.3 Floer and symplectic homology via Gromov–Witten theory . . . . . . 63

5.3.1 Symplectic homology . . . . . . . . . . . . . . . . . . . . . . . 63

5.3.2 Floer homology . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6 The example of cotangent bundles of spheres 71

6.1 T S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1.1 Relevant Gromov–Witten numbers . . . . . . . . . . . . . . . 72

6.1.2 The group SHpTS2q . . . . . . . . . . . . . . . . . . . . . . 74

6.1.3 The ring SHpTS2q . . . . . . . . . . . . . . . . . . . . . . . 77

6.2 T Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2.1 The topology of QN . . . . . . . . . . . . . . . . . . . . . . . 81

6.2.2 Gromov–Witten numbers of QN . . . . . . . . . . . . . . . . . 90

6.2.3 The group SHpTSnq . . . . . . . . . . . . . . . . . . . . . . 96

6.2.4 The ring SHpTSnq . . . . . . . . . . . . . . . . . . . . . . . 101

Bibliography 110

ix

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List of Figures

3.1 S- and J-shaped Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Various pieces in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Two split Floer differentials . . . . . . . . . . . . . . . . . . . . . . . 31

4.1 Types of punctures on a broken augmented pair-of-pants . . . . . . . 33

4.2 Unwanted configuration and steep Hamiltonian . . . . . . . . . . . . 46

5.1 Configurations given by coefficients cpk, k1 , . . . , kl

; k, k1 , . . . , klq . 54

5.2 Configurations given by coefficients cpk 1; k, 1q . . . . . . . . . . . . 55

5.3 Configurations given by coefficients cpk |d|; kq . . . . . . . . . . . . 57

5.4 Pseudo-holomorphic curves in X and W . . . . . . . . . . . . . . . . 62

5.5 The differential d_M . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.6 The differential d_^ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.7 Augmented and non-augmented pairs-of-pants contributing to the prod-

uct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.8 Broken pair-of-pants . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.9 Constant orbit contributing to the product of two non-constant orbits 69

6.1 Broken pair-of-pants on T S2 . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Broken pair-of-pants on T Sn . . . . . . . . . . . . . . . . . . . . . . 102

x

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Chapter 1

Introduction

Floer and symplectic homology groups are very important tools in the study of sym-

plectic manifolds pM,ωq, respectively closed and open. These groups are the Morse

homologies of the symplectic action functional associated with a Hamiltonian func-

tion H : S1 M Ñ R, which is defined on (a cover of) the free loop space LM .

These invariants have a very rich structure. On one hand, fixing H, there is a fil-

tration of the Floer chain complex of H by the symplectic action. This can be used

to define spectral invariants, which have many applications in symplectic topology,

via, for instance, symplectic quasi-morphisms and quasi-states. On the other hand,

these homology groups have a rich algebraic structure, defined in terms of spaces of

solutions of elliptic equations over punctured Riemann surfaces. In particular, one

can use the pair-of-pants to define a product on Floer and symplectic homology.

Despite the usefulness and richness of these invariants, they are frequently very

hard to compute. One important reason is that, to define them rigorously, one often

needs to study equations with perturbation terms that make them very hard to solve

explicitly. The goal of this thesis is to prove a version of the following.

Theorem 1.1. There is an explicit description of the Floer and symplectic homology

groups, with their action filtration and algebraic structures, in a certain important

class of examples. This description involves Gromov–Witten numbers, both absolute

and relative, and some Morse-theoretic data.

We will apply these techniques to compute symplectic homology rings of cotangent

1

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CHAPTER 1. INTRODUCTION 2

bundles of spheres, and recover a result of Cohen–Jones–Yan, in [CJY04]. One could

object that the definition of Gromov–Witten invariants also often involves the study

of solutions of perturbed equations. But there are many cases in which they can

effectively be computed, using for example tools from algebraic geometry or complex

analysis (see [Bea95] and [Zin11], for instance).

Before we give a more concrete description of our work, we should stress that all

of it is joint with Samuel Lisi, and that a large portion of this material will appear in

[DL12]. This is part of a larger project, joint with Strom Borman, Yakov Eliashberg,

Samuel Lisi and Leonid Polterovich. Borman’s upcoming thesis [Bor] contains a

different approach to some of the topics discussed in this text.

We should also point out that this approach to Floer theory is very much inspired

by the work of F. Bourgeois and A. Oancea, in [BO09b] and [BO09a], of F. Bourgeois,

T. Ekholm and Y. Eliashberg, in [BEE09], and of Y. Eliashberg and L. Polterovich,

in [EP10]. It is also related with what P. Seidel explains in Section 1 of [Sei02].

Let us now sketch the main steps in this work. We will consider the follow-

ing setting: pX,ωq is a monotone closed symplectic manifold, with integral rωs P

ImageH2pX;Zq Ñ H2pM ;Rq

. Σ X is a monotone smooth symplectic submani-

fold of codimension 2, which is Poincare-dual to Krωs, for some integer K ¡ 0. We

will also assume that Σ admits a perfect Morse function and that H1pΣ;Rq 0.

We will compute the Floer homology of X and the symplectic homology of W ,

the completion of W : XzΣ, for a certain class of Hamiltonians (which will be called

S- and J-shaped). These Hamiltonians are degenerate, which is usually not the case

in Floer theory. So, the first thing we need to do is describe what we mean by Floer

theory for such Hamiltonians. This will involve a Morse–Bott version of the Floer

chain complex, following work of [Bou02], [BO09b] and [BEE09].

The next step is to split X and W along certain contact-type hypersurfaces. Under

our assumptions, we can identify a neighborhood of Σ X with a normal disk bundle

of Σ. The boundary of this bundle is a contact-type hypersurface Y X, and choices

can be made so that Y is an S1-bundle over Σ (Y is a pre-quantization bundle). In

fact, there is an isosymplectic embedding of a piece of the symplectization of Y ,

pa, bq Y ãÑ X, for some interval pa, bq R. We will split X along two parallel

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CHAPTER 1. INTRODUCTION 3

copies of Y , and W along one copy of Y , in a sense similar to that of symplectic field

theory (see [BEH03]). Therefore, if we start with X, we will have three pieces: W ,

the symplectization R Y and the normal bundle NΣ. If we start with W instead,

we will have two pieces: W and R Y . An argument similar to that in [BEH03]

allows us to describe what happens to the solutions of Floer’s equation in X and W ,

as we split the manifolds. We get split Floer trajectories, with components in R Y ,

and possibly also in W and NΣ. This gives an alternative description of the Floer

and symplectic homology, which we refer to as split Floer and symplectic homology

(see Figure 3.3 below).

A key point will be that, in both Floer and symplectic homologies, for the classes

of Hamiltonians H under consideration, the supports of the Hamiltonian vector fields

H will be contained in R Y . Therefore, the components of split Floer trajectories

that are contained in W and in NΣ satisfy a (perturbed) pseudo-holomorphic curve

equation, whereas the components in R Y satisfy Floer’s equation.

Next, we show that, under a certain symmetry assumption on the almost complex

structure J in RY , the components v of split Floer trajectories contained in RYare in bijective correspondence with (equivalence classes of) pairs pu, fq, where u is

a punctured pseudo-holomorphic curve in R Y and f is a function with values in

R S1 that solves an auxiliary equation (we will say that f is a cylinder solution).

Furthermore, when we restrict our attention to components of rigid Floer solutions,

which are the ones used in the definition of the differential and operations in Floer

theory, the corresponding cylinder solutions f can be understood rather explicitly.

This reduces the problem of computing (rigid) Floer solutions to that of computing

punctured pseudo-holomorphic curves in R Y .

The final step in our description of Floer and symplectic homology is to relate

pseudo-holomorphic curves in W , R Y and NΣ with Gromov–Witten numbers.

Pseudo-holomorphic curves in W , asymptotic at punctures to Reeb orbits of Y , can be

equivalently described by maps from closed Riemann surfaces into X, intersecting Σ

with certain tangency conditions (see Figure 5.4 below). These are precisely described

by relative Gromov–Witten numbers of the pair pX,Σq.

As for pseudo-holomorphic curves u in RY , they project to pseudo-holomorphic

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CHAPTER 1. INTRODUCTION 4

maps w : CP 1 Ñ Σ. The reason why these are defined on all of CP 1 is that punctures

of u asymptote to Reeb orbits of Y , which are fibers of the bundle S1 Ñ Y Ñ Σ.

Now, RY can be thought of as the complement of the zero section on a complex line

bundle E Ñ Σ. The pseudo-holomorphic curve u then corresponds to a meromorphic

section of the bundle wE Ñ CP 1. Therefore, we reduce the problem of finding

pseudo-holomorphic curves in R Y to that of finding maps w : CP 1 Ñ Σ and

meromorphic sections of wE Ñ CP 1. On one hand, the counts of maps w are

precisely those that contribute to Gromov–Witten numbers of Σ. On the other hand,

meromorphic sections of a holomorphic line bundle over CP 1 are well understood:

they form a C-family, once we fix the positions and multiplicities of the zeros and

poles. We can thus reduce the problem of finding pseudo-holomorphic maps u to that

of finding Gromov–Witten numbers of Σ.

Finally, we need to study pseudo-holomorphic maps into NΣ. In many cases,

one can argue that the components in NΣ of rigid configurations can only be simple

covers of fibers of NΣ Ñ Σ. For more general configurations, an argument similar of

that of the previous paragraph reduces the problem again to finding Gromov–Witten

numbers of Σ. This completes the description of how to relate Floer trajectories in

X and W with Gromov–Witten invariants of the pair pX,Σq.

We use the procedure outlined above to compute symplectic homology rings, in

the case of pX,Σq pQn, Qn1q, where Qn is the n-dimensional complex projective

quadric. We will review the topology of these manifolds, and collect the relevant

Gromov–Witten numbers from [Bea95]. In this case, W is symplectomorphic to

T Sn, and a theorem of A. Abbondandolo and M. Schwarz implies that the symplectic

homology of W is isomorphic to the homology of the free loop space of Sn, as rings

(see [AS10]). R. Cohen, J. Jones and J. Yan computed these rings (see [CJY04]),

and our results match theirs. One interesting point is that our computation of the

pair-of-pants product needs to include some broken configurations (as represented in

Figure 5.8). One might at first hope that counts of pseudo-holomorphic pairs-of-

pants in R Y might be enough to describe the Floer product, but that is not the

case even in these simple examples. We will not compute Floer homology groups of

closed manifolds in this thesis, but refer the interested reader to [EP10] for the case

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CHAPTER 1. INTRODUCTION 5

of Q2 CP 1 CP 1, which includes applications to quasi-states. We also refer to

Borman’s upcoming thesis [Bor] for computations in other examples.

Summing up our discussion, here is a schematic description of the argument:$''&''%non-degenerate

Floer

trajectories

,//.//- AÐÑ

$''&''%degenerate

Floer

trajectories

,//.//- BÐÑ

$''&''%split

Floer

trajectories

,//.//- CÐÑ

CÐÑ

$''&''%holomorphic curves

&

cylinder solutions

,//.//- DÐÑ

$''&''%GW numbers

&

relative GW numbers

,//.//-In Chapter 2, we will quickly review Floer and symplectic homologies, their alge-

braic structures and the action filtration. We will describe our setup in more detail

and (briefly) explain correspondences A and B in Chapter 3. Correspondence C will

be explained in Chapter 4. In Chapter 5, we will quickly review (absolute and rela-

tive) Gromov–Witten numbers and explain correspondence D. We will also sum up

the argument with a description of the symplectic homology differential and pair-of-

pants product in terms of Gromov–Witten theory. In Chapter 6, we illustrate our

results with the computation of the (previously know) symplectic homology rings of

cotangent bundles of spheres.

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Chapter 2

Floer homology and symplectic

homology

In this chapter, we will review the construction of Floer and symplectic homology,

with their ring structures.

2.1 Filtered Floer homology groups

We begin with a review of Hamiltonian Floer homology. For details, we refer to

[Sal99]. Let pM2n, ωq be a closed symplectic manifold, so that ω P Ω2pMq satisfies

dω 0 and ωn is a volume form on M . We will assume our manifolds to be monotone,

which means that there is a real constant λ ¡ 0 such that

xω,Ay λ xc1pTXq, Ay

for all A P H2pX;Qq.Let J be an almost complex structure in M , compatible with ω. This means

that J P EndpTMq, J2 Id and ωp., J.q is a Riemannian metric. Call a function

H : S1 M Ñ R a Hamiltonian. One can use H to define an S1-dependent vector

field XHt in M , by the relation ωp., XHtq dHt. Abbreviate XHt to XH . The goal of

Floer theory is to study 1-periodic XH-orbits in M .

6

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CHAPTER 2. FLOER HOMOLOGY AND SYMPLECTIC HOMOLOGY 7

The Floer complex is morally a Morse complex for the action functional :

AH : yLM Ñ R

pγ, uq ÞÑ

»D2

»S1

Hpt, γptqqdt.

where yLM is a cover of the space L0M of contractible loops in M , given by pairs

pγ, uq, where γ : S1 Ñ M and u : D2 Ñ M is such that u|BD2 γ P L0M (under a

certain equivalence relation). The critical points of this functional are precisely the

(capped) 1-periodic orbits of XH . We fix, for each 1-periodic orbit γ, a capping plane

uγ. Denote by PH the set of 1-periodic orbits orbits of XH .

Remark 2.1. There are also versions of Floer theory for non-contractible orbits.

One could, for instance, consider pairs pγ, uq such that u : S Ñ M , where S is a

compact surface with one boundary component and u|BS γ. One might also be

interested in studying periodic orbits that define non-trivial elements of π1pMq or

H1pM ;Zq. One could decompose the space of 1-periodic orbits into homotopy classes

of free loops, or, put differently, conjugacy classes in π1pMq, as is done in [BO09b].

These equivalence classes are preserved by the Floer homology differential. Another

option would be to decompose the space of periodic orbits into homology classes, as

done in [EGH00] in the context of symplectic field theory. If some orbits define torsion

elements in H1pM ;Zq, then one could use a fractional grading for the elements of the

Floer complex, as pointed out in Section 2.9.1 of [EGH00]. This is the approach that

we will take in our setting.

We need to specify the coefficient ring for the Floer chain complex. We will take

the Novikov ring

Λ : Zrt, t1s

of Laurent polynomials in t. We are now ready to define the Floer chain complex as

CFpHq ΛxPHy

by which we mean the free Λ-module generated by the 1-periodic orbits of XH .

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CHAPTER 2. FLOER HOMOLOGY AND SYMPLECTIC HOMOLOGY 8

This complex has a grading, prescribed by

degpγ tmq µRSpγq 2mN, (2.1)

Here µRS is the Conley–Zehnder index (under the conventions specified by Robbin and

Salamon in [RS93], hence our notation) of γ associated with the trivialization of TM |γ

that is given by the capping uγ (see [Sal99]). The number N is the minimal Chern

number of M , given by the minimum of the set xc1pTMq, Ay : A P H2pM ;Zq

(XZ¡0.

We should think of a monomial γ tm as the periodic orbit γ with the capping given

by the connect sum of uγ with a surface of Chern class mN . We also take

AHpγ tmq AHpγ, uγq λmN. (2.2)

A justification for (2.1) and (2.2) will be given at the end of this section.

To define the differential in CFpHq, we count solutions of Floer’s equation

V : R S1 ÑM

BsV JpV qBtV XH

0

(2.3)

for the variables ps, tq P R S1. This can be thought of as the positive gradient flow

equation for the action AH . Given 1-periodic orbits γ and γ, let

Mpγ, γq V : R S1 ÑM |V solves p2.3q and lim

sÑ8V ps, tq γptq

(R

where we take a quotient by domain translations of the variable s P R. Since M is

a monotone manifold, the Mpγ, γq are manifolds for generic choices of H and J .

The spaceMpγ, γq might have multiple components of different dimensions, which

depend on the homology class in H2pM ;Zq obtained by gluing the Floer cylinders to

the capping disks uγ and uγ (with the opposite orientation on the latter). Denote by

Mpγ, uq, pγ, uq

the space of V PMpγ, γq such that u is homologous to V Y

u. Denote by M0pγ, γq the union of the components of Mpγ, γq of dimension

zero. These zero-dimensional spaces turn out to be compact, and can be given an

appropriate orientation, so that one can define their signed counts #M0pγ, γq.

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CHAPTER 2. FLOER HOMOLOGY AND SYMPLECTIC HOMOLOGY 9

These numbers are used to define the Floer differential:

d : CFkpHq Ñ CFk1pHq

x P PH ÞѸyPPH

#M0px, yq . y tjpx,yq (2.4)

where 2jpx, yqN µRSpxq µRSpyq 1

. Given V PM0px, yq, we can also write

jpx, yqN xc1pTMq, puxq Y V Y uyy.

Theorem. (Floer) d 2 0. The Floer homology of M is HpCFpHq, dq, and it does

not depend on the generic choices of H, J . In fact, it is isomorphic to HnpM ; Λq

(singular cohomology with Novikov coefficients).

Denote the Floer homology of M with respect to the Hamiltonian H by HFpHq.

We have mentioned that (2.3) is the equation for the positive gradient flow of AH .

In particular, AH increases along solutions V , as s increases, and d decreases AH .

This implies that one can filter the Floer complex CFpHq by values of the action,

and define, for a P R, subcomplexes

CF ak pHq

#¸i,j

ci,j xi tj P CFkpHq

@i,j AHpxi tjq ¤ a

+

Denote the homology of this subcomplex by HF a pHq. It is then the case that

HFkpHq limaÑ8

HF ak pHq.

Even though HFpHq is isomorphic to HnpM ; Λq, one can extract very useful

symplectic (and not just topological) information from the Floer complex, when tak-

ing into account the action filtration. It is particularly useful to consider the spectral

invariants of H, with which one can define, for instance, symplectic quasi-morphisms

and quasi-states in M . For more details, see for example [Oh97] and [EP03].

We finish this section with a statement of the facts that motivate definitions (2.1)

and (2.2), and the expressions for j in (2.4). For more details, see [MS04] and [Sal99].

Proposition. Let V : R S1 Ñ M be a solution of (2.3), connecting the 1-periodic

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CHAPTER 2. FLOER HOMOLOGY AND SYMPLECTIC HOMOLOGY 10

orbits γ and γ, let u : D2 ÑM be cappings for γ and let V Y u be the capping

for γ that is induced by V and u. Then,

• dimMpγ, V Y uq, pγ, uq

µRSpγ, V Y uq µRSpγ, uq;

• EpV q : 12

³RS1 |BsV |

2 |BtV XH |2dsdt AHpγ, V Y uq AHpγ, uq.

Now, let A P H2pM ;Zq and denote by u#A the connect sum. Then,

• µRSpγ, u#Aq µRSpγ, uq 2 xc1pTMq, Ay;

• AHpγ, u#Aq AHpγ, uq xω,Ay.

2.2 Symplectic homology

Symplectic homology is a version of Hamiltonian Floer homology for (completions

of) a certain class of symplectic manifolds with boundary, called Liouville domains.

We will review the construction and some properties of this invariant, referring to

[Oan04] [Sei08] for more details.

A Liouville domain is a symplectic manifold with boundary pW,ωq with a vector

field V pointing outward along Y BW , such that LV ω ω. The 1-form α :

pιV ωq|Y is a contact form on Y . Write ξ kerα for the contact structure and R for

the Reeb vector field of α, defined uniquely by the conditions ιRdα 0 and αpRq 1.

One can use V to form the completion W : W YY r0,8q Y . The Liouville form

η : ιV ω, a primitive for ω, can be extended to p0,8q Y W as erα, where r is

the coordinate on p0,8q. Denote this extension also by η and the symplectic form

dη P Ω2pW q by ω. For technical reasons, we will further assume that c1pTW q is a

(possibly vanishing) torsion element in H2pW ;Zq.Let J be an almost complex structure in W , compatible with ω. We require that

J preserve ξ and that JBr R, on p0,8qY (one can sometimes relax this condition

and require J to be only asymptotically cylindrical). To define Floer homology, one

needs a Hamiltonian in W . We consider H : S1 W Ñ R such that:

• H|W is an S1-independent C2-small Morse function;

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CHAPTER 2. FLOER HOMOLOGY AND SYMPLECTIC HOMOLOGY 11

• H|p0,8qY is a small perturbation, near the 1-periodic orbits, of hperq, for some

function h : RÑ R such that limτÑ8 h1pτq 8.

The symplectic homology of W is the Floer homology of such Hamiltonians. Since

pW,ωq is exact and c1pTW q is torsion, we can use integer coefficients in the definition,

instead of a Novikov ring. We can split the Floer complex into summands indexed by

free homotopy classes of orbits, each of which admitting a Z-grading, as in [BO09b].

In the case when all periodic orbits define torsion elements in H1pW ;Zq, one can define

instead a Q-grading, using the ideas in Section 2.9.1 of [EGH00]. The advantage of

this is to reduce the number of choices necessary to grade the symplectic homology

complexes, which is practical for computations. Symplectic homology can be shown

to be independent of H and J . We denote it by SHpW q.

If pW1, η1q and pW2, η2q are two completions of Liouville domains (with Liouville

forms ηi and symplectic forms dηi) for which there is a diffeomorphism φ : W1 Ñ W2

such that φη2 η1, then SHpW1q SHpW2q (see Section 7 in [Sei08]). Therefore,

for completions W such that H1pW ;Rq 0, as in the examples that we will consider

in Chapter 6, symplectic homology is a symplectomorphism invariant.

2.3 Operations on Floer theory; relation with string

topology

We recall now how to use spaces of solutions of elliptic equations defined over punc-

tured Riemann surfaces to define operations on Floer and symplectic homology. A

reference in the case of Floer homology is the thesis of Schwarz [Sch95]. For symplec-

tic homology, a reference is Abbondandolo and Schwarz’s [AS10]. We will use Seidel’s

approach to operations on Floer theory (see [Sei08] and [Rit11]).

Fix a Hamiltonian H : S1 M Ñ R and an almost complex structure J in

M . Let ΓF tz1 , . . . , zku and ΓF tz1 , . . . , z

ku be two disjoint finite subsets

of a fixed Riemann surface σ (which for our purposes will always be CP 1). Write

ΓF : ΓF YΓF and S : σzΓF . Fix conformal parametrizations ϕi : RS1 Ñ S of

neighborhoods of the zi P ΓF . Choose a 1-form β P Ω1pSq, such that pϕi qβ ci dt,

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CHAPTER 2. FLOER HOMOLOGY AND SYMPLECTIC HOMOLOGY 12

for some constants ci ¡ 0, and dβ ¤ 0 (with respect to the conformal structure on

S). Seidel’s generalization of Floer’s equation (2.3) is

v : S ÑM

pdv XH b βq0,1 0(2.5)

Note that, in the case when S R S1 and β dt, this equation becomes (2.3).

By counting rigid solutions of (2.5), one can define operations

HFpc1 Hq b . . .bHFpc

kHq Ñ HFpc

1 Hq b . . .bHFpc

kHq

when M is closed. When M is a completion of a Lioville domain, the same proce-

dure defines operations on symplectic homology. Composition of these operations

corresponds to gluing of domains (see [Rit11]).

As a particular case, one can let σ CP 1, ΓF t0, 1u, ΓF t8u, and β ψdt,

for a branched cover ψ : CP 1zΓF Ñ R S1, and define an operation

HFpHq bHFpHq Ñ HFp2Hq.

Using a continuation map (from an interpolation of Hamiltonians), we can construct

an isomorphism HFpHq Ñ HFp2Hq. Inverting this map, we get a product

HFpHq bHFpHq Ñ HFpHq.

A similar structure can be defined on symplectic homology. The continuation map

argument is a bit more subtle in this context (see the Appendix 3 in [Rit11]).

These ring structures on Floer and symplectic homology are often related with

other structures. Indeed, we have the following enhancement of Floer’s theorem.

Theorem. 1. [PSS96] If pM,ωq is a closed semi-positive (a generalization of mono-

tone) symplectic manifold, then, over Q,

HFpMq QHnpMq

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CHAPTER 2. FLOER HOMOLOGY AND SYMPLECTIC HOMOLOGY 13

as rings (we will recall later how to use Gromov–Witten invariants to define the

quantum cohomology ring QHpMq, which is HpM ; Λq as an abelian group).

2. [AS10] If N is a closed spin manifold, then

SHpTNq HpLNq

as rings (where HpLNq is the homology of the free loop space of N , with the

Chas–Sullivan product; this is part of the string topology of N).

The string topology rings of some manifolds have been computed. As an example,

the following was proven in [CJY04].

Theorem (Cohen-Jones-Yan). If n ¡ 1, the ring HpLSnq is isomorphic to

• pΛrbs b Zra, vsq pa2, ab, 2avq, for some a P H0pLSnq, b P Hn1pLS

nq and v P

H3n2pLSnq, if n is even,

• Λras b Zrus, for a P H0pLSnq and u P H2n1pLS

nq, if n is odd.

If we shift the grading by n, then the product preserves the grading. In Chapter

6, we will see how to use our techniques to give an alternative proof of this result.

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Chapter 3

Split Floer homology

This chapter describes the assumptions that we make on our manifolds, and gives

an indication of how to degenerate both the Hamiltonians and the manifolds. Our

description of the degenerations will not contain most details, but is included to

motivate what will be done in later chapters.

3.1 Symplectic divisors on monotone manifolds

We now describe the particular class of Liouville domains that will be of interest

to us. Let pX,ωq be a closed connected symplectic manifold, with integral rωs P

ImageH2pX;Zq Ñ H2pM ;Rq

and with a connected closed symplectic submanifold

Σ of codimension 2. Assume that Σ is Poincare-dual to Krωs, for some integer K ¡ 0.

Many interesting examples can be obtained by taking as X2n a complete intersection

in CP nr, and as Σ the intersection of X with a projective hypersurface.

Note. Donaldson showed that every symplectic manifold with an integral symplectic

form admits a symplectic submanifold Poincare-dual to Krωs, for K ¡ 0 sufficiently

large (see [Don96]). The examples we will consider in the last chapter are all polar-

ized Kahler manifolds, in the sense of Biran (see [Bir01]). These consist of quadruples

pX,ω, J,Σq, such that pX,ω, Jq is a Kahler manifold with integral symplectic form,

and Σ is a smooth and reduced complex hypersurface Poincare-dual to a positive in-

teger multiple of ω. We should point out that we will work with almost complex

14

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CHAPTER 3. SPLIT FLOER HOMOLOGY 15

structures J that are not necessarily integrable.

Let us assume that pX,ωq is monotone, and let λX ¡ 0 be such that xω,Ay

λXxc1pTXq, Ay for all A P H2pX;Qq. We will denote the normal bundle to Σ X

by NΣ and the boundary of a disk tubular neighborhood of Σ by Y . This is an

S1-bundle over Σ.

Lemma 3.1. For every A P H2pΣ;Qq,

xω,Ay λX

1K λXxc1pTXq, Ay.

Proof. Since the first Chern class is additive, c1pTΣq c1pTXq|Σ c1pNΣq. Given

A P H2pΣ;Qq,

xc1pTΣq, Ay xc1pTXq, Ay xc1pNΣq, Ay 1λXxω,Ay Kxω,Ay

p1λX Kqxω,Ay

which is what we wanted to show. We have used the fact that xc1pNΣq, Ay #pΣX

Aq xKω,Ay, based on the assumption that Σ PDpKωq.

This result implies that pΣ, ωq is also monotone (with λΣ λX

1K λX), if K λX 1.

Important assumptions. Throughout the rest of this text, pX,ωq will be a closed

connected monotone symplectic manifold, with integral rωs P H2pX;Zq, and with a

monotone smooth connected symplectic divisor Σ, Poincare-dual to Krωs for some

integer K ¡ 0. We will further assume that Σ admits a perfect Morse function and

that H1pΣ;Rq 0.

The following result will also be useful.

Lemma 3.2. Let W : XzΣ. Write λX pq for some p, q P Z.

• rωs|W 0 P H2pW ;Rq and c1pTW q is torsion in H2pW ;Zq;

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CHAPTER 3. SPLIT FLOER HOMOLOGY 16

• if H1pX;Zq has no torsion, then pKpq c1pTW q 0;

• W is the interior of a Liouville domain.

Proof. Let A P H2pW ;Qq. Then,

xω,Ay 1

K#pΣX Aq 0

because A does not intersect Σ. Also, using the monotonicity of X,

xc1pTW q, Ay xc1pTXq, Ay 1

λXxω,Ay 0.

This implies the first part of the Lemma.

For the second part, we write part of the long exact sequence for the pair pX,Σq:

H2pX,W ;Zq ϕÑ H2pX;Zq ψ

Ñ H2pW ;Zq.

Denoting by Φ P H2pNΣ,Σ;Zq H2pX,W ;Zq the Thom class of the bundle NΣ, we

have ϕpΦq PDpΣq Krωs Kpqc1pTXq (the first identity follows from Proposition

6.24 in [BT82]; the third identity is a consequence of the assumption that H1pX;Zqhas no torsion, which implies that H2pX;Zq has no torsion). Thus, by exactness,

0 pψ ϕqpqΦq pKpq c1pTW q, as wanted.

We now show that W is the interior of a Liouville domain. One can think of Was the interior of a compact manifold W , such that BW is diffeomorphic to Y . Since

Σ PDpKωq, for some K ¡ 0, BW is a convex boundary, which means that W has

a Liouville vector field defined on a collar neighborhood p0, 1sY Ñ W of BW . This

means that, on p0, 1s Y , we have a vector field V1, such that LV1ω ω. Therefore,

in that neighborhood of BW , η1 : ιV1ω is a primitive for ω. On the other hand, since

we just saw that rωs 0, we know that ω dη2, for some global 1-form η2 P Ω1pW q.

Now, choose a function β : p0, 1s Ñ r0, 1s which is identically 0 near 0 and identically

1 near 1. Then,

dβη1 p1 βqη2

ω dβ ^ pη1 η2q.

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CHAPTER 3. SPLIT FLOER HOMOLOGY 17

Observe that dpη1η2q 0 in p0, 1sY . Suppose that η1η2 df , for some function

f : p0, 1sY Ñ R (whose existence will be shown later). Take now g : p0, 1sY Ñ Rsuch that g f in psupp dβq Y , g 0 very near t1u Y BW and g 1 very

near t0u Y . Define η : βη1 p1 βqη2 βdg P Ω1pW q. Then,

dη ω dβ ^ pη1 η2q dβ ^ dg ω.

Since η η1 near BW , η is a Liouville form on W . To conclude the proof, we

just need to show the existence of the function f above. This follows from the fact

that H1pY ;Rq 0, which we now prove. The Gysin sequence for the fibration

S1 Ñ Y Ñ Σ yields

0 Ñ H1pΣ;Rq Ñ H1pY ;Rq Ñ H0pΣ;Rq Y c1pYÑΣqÝÑ H2pΣ;Rq.

By monotonicity of Σ, the map on the right is non-zero. Since Σ admits a perfect

Morse function, HpΣ;Zq has no torsion, so the map on the right is an injection.

Therefore, H1pΣ;Rq Ñ H1pY ;Rq is an isomorphism. Since we assume in this text

that H1pΣ;Rq 0, we conclude that H1pY ;Rq 0, which finishes the proof of the

Lemma.

This result implies that one can define symplectic homology of the completion W

of W , with integer coefficients and with a rational grading, without needing Novikov

coefficients. If H1pX;Zq has no torsion and K p 1, as will be the case in Chapter

6, then c1pTW q 0 and there is an integer grading on symplectic homology.

Remark 3.1. It should be possible to extend the results that will be explained in

this text to the general case of a complex projective manifold X with an ample (or

positive) smooth divisor Σ (see [Huy05] for the definitions). In this case, XzΣ also

has convex boundary (see Section 2.7 in [CE]).

The assumptions that Σ has a perfect Morse function and that H1pΣ;Rq 0 are

both used in the proof above, but in a rather weak way, and one should be able to do

away with them, at least in some important cases.

Given X and Σ as above, we can choose a Hermitian metric on NΣ, for which

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CHAPTER 3. SPLIT FLOER HOMOLOGY 18

a connection 1-form defines a contact form α in Y . We can further assume that

the Reeb flow corresponds to flowing along the fibers of S1 Ñ Y Ñ Σ (say that

Y is a pre-quantization bundle). By the symplectic tubular neighborhood theorem

(see Section 9.3.2 in [EM02] and Section 2.1 in [Bir01]), there is an isosymplectic

embeddingpa, bq Y, dperαq

ãÑ pX,ωq, for some interval pa, bq R. Biran has

shown that, in the setting of a polarization, there is such an embedding with full

volume in X (see [Bir01]). One should be able to extend this result to a symplectic

(not necessarily Kahler) setting, using methods from [Gir02]. For any x P pa, bq, Br

is a local Liouville vector field near txu Y , so we say that this is a contact-type

hypersurface in X. Therefore, txuY separates X into two pieces: a convex filling of

Y , corresponding to the side where r x, and whose completion is symplectomorphic

to W ; and a concave filling, on the side where r ¡ x, which is symplectomorphic to

a disk normal bundle of Σ in X.

Remark 3.2. Since we want the symplectization coordinate r inR Y, dperαq

to

grow as one approaches Σ, we will think of the bundle S1 Ñ Y Ñ Σ as having first

Chern class equal to c1pNΣq. Note that RY can be thought of as the complement

of the zero section on a complex line bundle E Ñ Σ that is dual to NΣ.

At this point, we would like to say some words about the monotonicity assumptions

on X and Σ. On one hand, these are useful to ensure transversality for the spaces

of pseudo-holomorphic curves and Floer trajectories that we will consider. On the

other hand, monotonicity will also lie behind the fact that the Novikov parameter in

the Floer homology differential counts intersections of Floer trajectories in X with

the divisor Σ, as will be explained at the end of the next section.

3.2 Degenerating the Hamiltonian

Recall that we mentioned, when defining Floer homology and symplectic homology,

that our Hamiltonians might need to be S1-dependent, so that they can satisfy a

certain non-degeneracy condition. Nonetheless, for our purposes, we will need to

consider certain degenerate Hamiltonians, which will require an adjustment in our

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CHAPTER 3. SPLIT FLOER HOMOLOGY 19

description of Floer theory. We now define the classes of Hamiltonians that we will

use (see also Figure 3.1).

Definition 3.1. A function H : X Ñ R is called S-shaped, if:

1. there are values r1, r2 P pa, bq, where pa, bq Y ãÑ XzΣ W , such that the

support of dH is contained in pr1, r2q Y ;

2. on pr1, r2q Y , Hpr, Y q hperq, for some monotone increasing function h :

RÑ R;

3. h1 has one absolute maximum M , which is not an integer;

4. for all 0 c M , there are exactly two values of r P pr1, r2q such that h1perq

c.

A function H : W Ñ R is called J-shaped if

1. there is a value r1 P pa, bq, where pa, bqY ãÑ XzΣ W , such that the support

of dH is contained in pr1,8q Y W ;

2. on pr1,8q Y , Hpr, Y q hperq, for some monotone increasing function h :

RÑ R;

3. h2 is positive on pr1,8q and limrÑ8 h1prq 8;

4. for all 0 c, there is exactly one value of r P pr1,8q such that h1perq c (this

actually follows from the previous conditions).

We will want to define Floer and symplectic homology for S- and J-shaped Hamil-

tonians, respectively. Notice that, on the support of the derivative of such H, the

Hamiltonian vector field is XHpr, yq h1perqRpyq, where Rpyq is the Reeb vector

field at y P Y . Therefore, on the support of dH, the 1-periodic orbits of XH come

in S1-families (because H is time-independent) and correspond to Reeb orbits in Y .

The S1-families of XH-orbits on the level tru Y correspond precisely to the Reeb

orbits in Y of period T h1perq. Since the Reeb flow on the pre-quantization bundle

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CHAPTER 3. SPLIT FLOER HOMOLOGY 20

er1 er2

er

Hprq

Σ

E

0

slope M

er1 er

Hprq

0

Figure 3.1: S- and J-shaped Hamiltonians

Y goes around the orbits of the fibration S1 Ñ Y Ñ Σ, for each positive integer k

there is a Y -family of (parametrized) Reeb orbits of period k.

To define the symplectic action and the Floer grading, it will be useful to specify

cappings for our orbits. Recall that, given A P H2pΣ;Zq, we have #pA X Σq

Kxω,Ay. Therefore, if we fix a point p P Σ, and A P H2pΣ;Zq such that xω,Ay 0,

then we can lift a representative of A that intersects Σ only at p (to order Kxω,Ay)

to a surface in Y whose only boundary component is a pKxω,Ayq-cover of the fiber

over p. As a consequence, the pKxω,Ayq-multiple of every Reeb orbit in Y can be

capped by a surface in W , and it vanishes on H1pW ;Zq. For this reason, we will

say that every Reeb orbit has a fractional capping inside W . Notice also that every

non-constant periodic orbit in X admits a capping via (a multiple of) a fiber of (a

disk bundle of) NΣ. This capping disk is oriented in such a way that its intersection

number with Σ is negative. It will be useful to think of (symplectic homology) orbits

in W as having fractional cappings in W , and of non-constant (Floer homology) orbits

in X as having both a fractional capping in W and a capping in X that intersects Σ

negatively. Constant orbits have constant cappings.

Lemma 3.3. The 1-periodic orbits of an S-shaped Hamiltonian H in X are of two

types:

• constant: corresponding to the points p P Xzpsupp dHq. When given a trivial

capping, these orbits have action AH Hppq;

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CHAPTER 3. SPLIT FLOER HOMOLOGY 21

• non-constant: for each integer 0 k M maxh1, there are two values

r P pr1, r2q such that k h1perq. There is a Y -family of 1-periodic XH-orbits

contained in truY and another Y -family contained in truY . When given

a rational capping inside W , these orbits have action AH erh1per

qhper

q,

respectively. When capped by a disk whose intersection number with the divisor

Σ is k, their action is AH erh1per

q hper

q kK.

The 1-periodic orbits of a J-shaped Hamiltonian H in W are of two types:

• constant: corresponding to the points p P W zpsupp dHq, with action AH

Hppq;

• non-constant: for each integer k ¡ 0, there is one value r P pr1,8q such that

k h1perq. There is a Y -family of 1-periodic XH-orbits contained in tru Y .

These orbits have action AH erh1perq hperq, with respect to a fractional

capping in W .

Proof. We have already described the periodic XH-orbits, and are only left with

justifying the values of their actions. Start with the case of W , where ω is exact

(recall Lemma 3.2), so we can write ω dη for some η P Ω1pW q. Fix A P H2pΣ;Zqsuch that xω,Ay 0. Given a non-constant 1-periodic orbit γ of the Hamiltonian

H, we saw above that the pKxω,Ayq-cover of γ, denoted by γKxω,Ay, is trivial on

H1pW ;Zq. Let uKxω,Ay : S Ñ W be a capping for γKxω,Ay, where S is a surface with

one boundary component and u|BS γKxω,Ay. Then»S

puKxω,Ayqω

»S1

Hdt

»S1

pγKxω,Ayqη HpγKxω,Ayqdt

Kxω,Ay »

S1

γη Hpγqdt

and thus AHpγq ³S1 γ

ηHdt, with respect to the fractional capping 1Kxω,Ay

uKxω,Ay

(note that this is independent of capping). The non-constant orbits are contained in

a half-infinite piece of RY , where η erα (where α is the contact form on Y ). For

a 1-periodic XH-orbit γptq r, γpTtq

R Y , where γ : RTZ Ñ Y is a closed

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CHAPTER 3. SPLIT FLOER HOMOLOGY 22

Reeb orbit of period T h1perq, we have

AHpγq »S1

erαp 9γq hperqdt erh1perq hperq.

The fact that constant orbits with constant cappings have action given by the value

of H is immediate from the definition of action.

Consider now the case of X. If we again choose fractional cappings contained

inside W (where ω dη) for the non-constant 1-periodic XH-orbits, then the compu-

tation of the action of these orbits is the same as the one done above for symplectic

homology. The exactness of ω|W again implies that the action is independent of the

choice of fractional capping in W . Now, let γk be an XH-orbit corresponding to a

Reeb orbit of multiplicity k and denote by u1k the capping of γ by a plane that inter-

sects k times Σ. There is a corresponding capping u1kKxω,Ay for a pKxω,Ayq-cover

of γk, which also admits a capping ukKxω,Ay : S Ñ X inside W . The difference in

actions computed with respect to these cappings is given by the Proposition at the

end of Section 2.1:

1

Kxω,Ay

»D2

pu1kKxω,Ayqω

»S

pukKxω,Ayqω

1

Kxω,Ay

Aω,

u1kKxω,Ay Y pukKxω,Ayq

E

1

Kxω,Ay

1

K#

ΣXu1kKxω,Ay Y pukKxω,Ayq

kKxω,Ay

K2xω,Ay

k

K.

Therefore, AHpγ, u1q AHpγ, ulqkK, as wanted. The computation of the actions

of constant orbits (with constant cappings) is analogous to the one for W .

Since the 1-periodic orbits of XH come in manifold families, there are two ap-

proaches one can take to define Floer and symplectic homology chain complexes:

either perturb H to a non-degenerate time-dependent Hamiltonian, or use a Morse–

Bott version of Floer and symplectic homology. Both approaches should give isomor-

phic homology groups. We will use the latter. In [BO09b], Bourgeois and Oancea

describe a Morse–Bott chain complex that computes symplectic homology for time-

independent Hamiltonians, and show that it is isomorphic, as an abelian group, to

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CHAPTER 3. SPLIT FLOER HOMOLOGY 23

symplectic homology, as defined for non-degenerate Hamiltonians. On one hand, we

need a stronger form of their result, allowing for Hamiltonians that are non-degenerate

not only because they are time-independent, but also because they are constant on

large subsets of X and W , and because the Reeb flow itself is degenerate on the

pre-quantization bundle Y . On the other hand, the manifolds that we consider are

not as general as those considered in [BO09b], since they do not restrict their atten-

tion to Liouville domains whose boundaries are pre-quantization bundles. The fact

that the Reeb flow is Morse–Bott non-degenerate (as in [Bou02]) is useful to achieve

transversality for the relevant spaces of Floer trajectories and pseudo-holomorphic

curves.

We will now describe the version of Floer theory that is appropriate for our setting,

but without a justification of the construction or of the independence of the choices

involved.

Recall that, by definition, if H is an S-shaped Hamiltonian, then there are two

separating contact-type hypersurfaces Y1 tr1u Y and Y2 tr2u Y , such that

dH is supported between Y1 and Y2 (see Figure 3.1). Denote by A and B the two

connected components of XzpY1 Y Y2q where H is constant, as in Figure 3.2. A is

a tubular neighborhood of Σ and B is diffeomorphic to W XzΣ. Similarly, a J-

shaped Hamiltonian is constant on a subset of W that is diffeomorphic to W , and

that we also denote as B.

The closures A and B are manifolds with boundary, on which we choose auxiliary

functions fA : AÑ R and fB : B Ñ R, that are constant on the boundaries. Suppose

that these functions are Morse–Smale on the interiors A and B, respectively, and also

that fA attains its minimum on BA Y2 and that fB has its maximum on BB Y1.1

Lemma 3.3 and Morse–Bott homology (see [BH11] and [BO09b]) suggest that we

define the chain complex CFpHq for Floer homology of the S-shaped Hamiltonian

1Alternatively, we could choose a Morse–Smale function f : X Ñ R such that one of its level setsis a copy of Y , and whose maximum is attained at the connected component of XzY that containsΣ.

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CHAPTER 3. SPLIT FLOER HOMOLOGY 24

Σ

Y2

Y1

B

A

X

Figure 3.2: Various pieces in X

H (as an abelian group) as follows:

CFpHq CMpfBqrns `CCpY q

M ` CCpY q M r1s

` (3.1)

`CCpY q

M r1s ` CCpY q M

` CMpfAqrns

where CMpgq is the Morse complex of a function g (we say more about the gradings

on the Morse complexes in (3.1) below, in Lemma 3.4). CCpY q M is the truncation of

the chain complex for contact homology of Y , which is generated by Reeb orbits of pe-

riod less than M .2 Since Y Ñ Σ is a pre-quantization bundle, and its (parametrized)

periodic Reeb orbits come in Y -families, we take an auxiliary Morse-Smale function

fΣ : Σ Ñ R on Σ. Then CCpY q has a generator pk for every critical point p P Σ and

every multiplicity k ¡ 0. For each multiplicity 0 k M , there are two Y -families

of 1-periodic XH-orbits corresponding to Reeb orbits of period k, one on the concave

part and one on the convex part of H. The function fΣ lifts to a Morse–Bott function

on Y , whose critical manifolds are circles. We take auxiliary Morse functions on these

circles, with two critical points. This justifies our need for four copies of CCpY q M .

The degree shifts account for the degrees of the critical points of the auxiliary Morse

functions on fibers of S1 Ñ Y Ñ Σ, and for the fact that some orbits are located on

2We will see below that Y has no bad orbits, in the sense of symplectic field theory.

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CHAPTER 3. SPLIT FLOER HOMOLOGY 25

the convex part (those that generate CCpY q M ` CCpY q

M r1s), and some on the

concave part of H (those generating CCpY q M r1s ` CCpY q

M). Summing up,

every critical point p P Σ and multiplicity 0 k M gives rise to four generators of

CFpHq, denoted by qpcvxk , ppcvxk , qpccvk and ppccvk . As in Lemma 3.3, we think of constant

orbits as having constant cappings, and non-constant orbits as having fractional cap-

pings inside W , or disk cappings intersecting Σ negatively. We take all the pieces in

the above direct sum to be generated over the Novikov ring Λ Zrt, t1s.

Similarly, we define the chain complex CSpHq for Floer homology of a J-shaped

Hamiltonian H (as an abelian group) as

CSpHq CMpfBqrns ` CCpY q ` CCpY qr1s. (3.2)

We now have two copies of CCpY q, with unbounded periods, because for each k ¡ 0,

there is a Y -family of 1-periodic XH-orbits corresponding to Reeb orbits of period k.

This time, we can take coefficient over Z, instead of Λ.

The following result tells us what the gradings are on these chain complexes.

Lemma 3.4. The grading of the generators of CFpHq is as follows:

• if x P CritpfIq, where I A or B, then degpxq indfI pxq n;

• given qi P CritpfΣq such that indfΣpqiq i, let qi,k denote the corresponding

Reeb orbits of multiplicity k. Then, for XH-orbits on the convex part of H

degpqqcvxi,k q 2

1

KλX 1

k n 1 i

and

degppqcvxi,k q 2

1

KλX 1

k n 2 i

with respect to a fractional capping inside W . With respect to a capping by a

disk intersecting Σ (negatively), we have instead

degpqqcvxi,k q 2k n 1 i

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CHAPTER 3. SPLIT FLOER HOMOLOGY 26

and

degppqcvxi,k q 2k n 2 i

As for orbits on the concave part of H, we have degpqqccvi,k q degpqqcvxi,k q 1 and

degppqccvi,k q degppqcvxi,k q 1.

The grading of the generators of CSpHq is as follows:

• if x P CritpfBq, then degpxq indfI pxq n;

• given qi P CritpfΣq such that indfΣpqiq i, let qi,k denote the corresponding

Reeb orbits of multiplicity k. Then, for XH-orbits of H

degpqqi,kq 2

1

KλX 1

k n 1 i

and

degppqi,kq 2

1

KλX 1

k n 2 i

for a fractional capping inside W .

Proof. We will use a combination of results from other authors. As with the proof

of Lemma 3.3, we begin with the case of symplectic homology of W . Recall that we

assume constant orbits to have constant cappings and non-constant orbits to have

fractional cappings inside W . The formula for constant orbits is given in Lemma 7.2

in [SZ92]. The formula for non-constant orbits can be explained as follows:

degpqqi,kq µRSpqqi,kq µRSpqi,kq.

The term on the right is the Robbin–Salamon index for a Reeb orbit (whereas the term

in the middle is the Robbin–Salamon index for a Hamiltonian orbit). The formula is

justified in Lemma 3.4 of [BO09b] (although some of our conventions are different).

Similarly,

degppqi,kq µRSppqi,kq µRSpqi,kq 1.

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CHAPTER 3. SPLIT FLOER HOMOLOGY 27

Now, since qi,k is associated with a Morse–Bott family of (unparametrized) k-periodic

Reeb orbits that is parametrized by Σ (which we denote as Σk), Lemma 2.4 in [Bou02]

yields

µRSpqi,kq µRSpΣkq pn 1q i

where µRSpΣkq is the Robbin–Salamon index of a Reeb orbit in Σk. Therefore, the

formulas for degpqqi,kq and degppqi,kq follow from the fact that µRSpΣkq 2

1KλX

1k,

under a fractional capping contained in Y (and thus in W ), which we now justify.

Let γk be a Reeb orbit in Σk. To assign an index to γk, we will argue as in

Section 2.9.1 of [EGH00] and Section 9.1 in [Bou02]. Take A P H2pΣ;Zq such that

xω,Ay 0, and use it to construct a capping ukKxω,Ay for the multiple γkKxω,Ay, as

explained before. Such capping lies inside Y , and we think of it as inside W . With

respect to the induced trivialization of TW |γkKxω,Ay, µRSpΣkKxω,Ayq 2 xc1pTΣq, Ay k

(the index vanishes with respect to the product framing, and it changes by c1 under

change of framing). We then take µRSpΣkq 2 xc1pTΣq,AyKxω,Ay

k, which can sometimes be

fractional. Now,xc1pTΣq, Ay

Kxω,Ayxω,AyλΣ

Kxω,Ay

1

KλΣ

and the fact that λΣ λX

1KλX(see Lemma 3.1) implies that 1

KλΣ 1

KλX1. Therefore,

µRSpΣkq 2

1KλX

1k, as wanted.

The proof in the case of Floer homology of X is completely analogous. We can take

a capping for the XH-periodic orbit corresponding to qqi,k by a disk u1k that intersects

k times the divisor Σ, and an analogous capping u1kKxω,Ay for a pKxω,Ayq-cover of

this orbit. Using the Proposition at the end of Section 2.1, we see that the difference

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CHAPTER 3. SPLIT FLOER HOMOLOGY 28

in indices given by the two choices of (fractional) cappings for qqi,k is

1

Kxω,Ay2@c1pTXq, u

1kKxω,Ay Y pukKxω,Ayq

D

1

Kxω,Ay2

1

λX

@ω, u1kKxω,Ay Y pukKxω,Ayq

D

1

Kxω,Ay2

1

KλX#

ΣXu1kKxω,Ay Y pukKxω,Ayq

1

Kxω,Ay2

1

KλXpkKxω,Ayq

2k

KλX.

Adding this to the index formula with respect to the fractional capping inside W ,

we get the (integer) index with respect to a capping by a disk intersecting k times

Σ. The only point that still needs justification is the relation between the degrees of

convex and concave generators. This is once again due to our Morse–Bott setting,

and the fact that the second derivative of a concave function is negative.

Remark 3.3. Since the Floer differential connects elements with index difference

one, the symplectic homology chain complex splits as a sum over the fractional parts

of the indices of the generators. This is analogous to writing the symplectic homology

complex as a sum over free homotopy classes of orbits (in which case each of the

summands can be given an integer grading).

Since the Hamiltonian H is Morse–Bott, the Floer differential should count cas-

cades, with components solving Seidel’s equation, connected at removable singularities

to gradient flow lines of the auxiliary Morse functions that were chosen. One can also

describe the pair-of-pants product for the Hamiltonian H in terms of cascades. We

will not give a more explicit description of these configurations at this point, but will

provide more details in Section 5.3.1 below.

Proposition 3.1. The chain complex CFpHq for an S-shaped Hamiltonian H com-

putes the Floer homology HFpXq; the complex CSpHq for a J-shaped Hamiltonian

H computes the symplectic homology SHpW q.

We will not present a proof of Proposition 3.1 here. A possible approach to prov-

ing this result would be via a continuation map constructed from the interpolation

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CHAPTER 3. SPLIT FLOER HOMOLOGY 29

between a degenerate (S- or J-shaped) Hamiltonian and a non-degenerate small per-

turbation.

In the case of Floer homology of the closed manifold X, the choice of fractional

cappings contained in W , for the non-constant 1-periodic orbits in X, has an im-

portant consequence. It implies that the Novikov variable t (of degree 2N) in the

Floer homology differential keeps track of how many times a Floer cylinder intersects

Σ PDpKrωsq. More explicitly, one can decompose the Floer differential as

d ¸i¥0

di t iK λXN (3.3)

where di counts Floer cylinders intersecting i times the divisor Σ (possibly after a

small perturbation to ensure a transverse intersection). Recall from (2.4) that the

exponent of t in the contribution of an orbit y to the Floer differential of an orbit x is

xc1pTXq, puxqYV YuyyN , where ux and uy are cappings, V is a Floer trajectory

connecting y and x, and N is the minimal Chern number of X. Formula (3.3) now

follows from monotonicity of X and the fact that rωs PDpΣKq. Compare this with

Section 3 in [EP10]. If we chose instead cappings by disks intersecting Σ (negatively),

then the exponents of t would also involve the multiplicities of the periodic orbits. If

x corresponds to a Reeb orbit of period k and y corresponds to one of period l, then

the contribution of y to dpxq can be written as

¸i¥0

pdipxq, yq t iklK λXN (3.4)

where pdipxq, yq is a signed count of rigid Floer trajectories connecting y and x that

intersect i times the divisor Σ.

3.3 Splitting the manifold

We have just described how to define Floer and symplectic homology for a class of

degenerate Hamiltonians (omitting many details about the differential). Before we

can relate these chain complexes with Gromov–Witten numbers, we need to also

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CHAPTER 3. SPLIT FLOER HOMOLOGY 30

degenerate the manifold. This construction is inspired by the work of Bourgeois and

Oancea (namely Section 5 of [BO09a]).

Let H be an S-shaped Hamiltonian on X. Recall that there are two values r1, r2 P

pa, bq such that supp dH pr1, r2qY , and that we denote Yi triuY , for i 1, 2

(see Figure 3.2). We wish to split X along the Yi, in a way similar to symplectic field

theory splitting, explained in Section 3.4 of [BEH03].

Recall that [BEH03] describes limits of pseudo-holomorphic curves on a symplec-

tic manifold M (possibly with cylindrical ends), as one stretches the neck of M along a

compact contact-type hypersurface V , or, more generally, a stable Hamiltonian struc-

ture. In the limit, one gets pseudo-holomorphic buildings, possibly including pieces

in the symplectization R V . In our case, we want to describe the limits of Floer

trajectories as one splits X along Y1 and Y2. To this end, recall that Floer trajecto-

ries can themselves be thought of as pseudo-holomorphic curves (an idea inspired by

Gromov’s paper [Gro85]). For instance, a Floer cylinder R S1 Ñ X is the same

as a holomorphic section of pR S1q X, for a certain almost complex structure on

the product (see [EKP06], Section 4.12.1). Then, one can try to apply the results of

[BEH03] to the splitting of pR S1q X along pR S1q Y1 and pR S1q Y2.

The reason that this argument needs further justification, which we will not provide

here, is that the pR S1q Y1 are not compact, and so the compactness results in

[BEH03] do not apply as stated. However, note that, since H is constant near the

Yi, the Floer equation actually coincides with the pseudo-holomorphic curve equa-

tion in those regions. This justifies that one should be able to argue as if one were

splitting pseudo-holomorphic curves along compact contact-type hypersurfaces, as in

[BEH03]. In his upcoming thesis [Bor], Borman describes a compactness result that

applies to a setting similar to ours.

In order to identify spaces of Floer trajectories in X with spaces of split Floer

trajectories, one needs, in addition to the compactness statement aluded to above,

also a gluing argument. We will not provide the details of these arguments.

The process of splitting X with an S-shaped Hamiltonian H does not affect the

non-constant 1-periodic XH-orbits. We can thus say that a chain complex coinciding

with (3.1) as an abelian group computes the Floer homology of X. The differential

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CHAPTER 3. SPLIT FLOER HOMOLOGY 31

x

y

dx y . . .

w

z

dw z . . .

Figure 3.3: Two split Floer differentials

and the algebraic operations can now be defined in terms of symplectic-field-theory-

type buildings, possibly with components in RY , in NΣ and in W , connected with

gradient flow lines of auxiliary Morse functions. Call these split Floer trajectories and

refer to this description as split Floer homology. Figure 3.3 contains some split Floer

trajectories contributing to the split Floer differential. In the picture on the left, the

top disk represents a plane in NΣ, the intermediate piece is in R Y and the two

bottom pieces are planes in W . On the right, the cylinder is in R Y and the rest is

in W . The letters x, y and w represent non-constant periodic Floer orbits. The letter

z represents a critical point of an auxiliary Morse function in the region where H is

constant; the segment connecting z to a plane is a gradient flow line for the same

function. The other periodic trajectories depicted are asymptotic Reeb orbits in Y .

In Section 5.3.1, we will depict more split trajectories, but first we will need to relate

them with pseudo-holomorphic curves.

We should point out that, when splitting, the function fB is replaced by a function

on fW : W Ñ R that grows at infinity. Similarly, fA is replaced by a function

fNΣ : NΣ Ñ R, that decreases at infinity. In R Y , gradient flow lines should be

thought of as vertical lines.

There is an analogue of the above discussion in which one splits W along a single

copy of Y . This leads to the split symplectic homology of W .

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Chapter 4

An ansatz for split Floer

trajectories

We will now relate split Floer trajectories for S- and J-shaped Hamiltonians (men-

tioned above) with pseudo-holomorphic curves. Recall that we split our manifolds in

such a way that the only component with a non-constant Hamiltonian is the sym-

plectization RY . In this chapter, we will show how Floer trajectories in RY can

be related with (perturbed) pseudo-holomorphic curves and solutions of an auxiliary

equation. We will describe the relevant moduli spaces of pseudo-holomorphic curves

and solutions of the auxiliary equation, and then relate them with Floer trajectories.

We begin with a description of the various types of punctures and marked points on

the domains of our maps to R Y . These results will appear in [DL12], joint with

Samuel Lisi.

4.1 Several types of punctures and marked points

Even though we will not be careful with these distinctions in subsequent sections, it

is important to point out that the configurations we will consider have punctures of

different types, which we now describe.

To define the differential and operations on Floer and symplectic homology, one

considers solutions of Seidel’s equation (2.5) on a Riemann surface S CP 1zΓF ,

32

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 33

ΓF

ΓF,r

ΓF

ΓF,r

ΓcR

ΓaR

Figure 4.1: Types of punctures on a broken augmented pair-of-pants

where ΓF CP 1 is a finite set. We decompose ΓF ΓF Y ΓF into positive and

negative punctures.

When we consider S- and J-shaped Hamiltonians, as in Section 3.2, the periodic

XH-orbits are not isolated, forming manifolds, possibly with boundary. For this

reason, we need to replace counts of solutions of Seidel’s equation with counts of

cascades, in the sense of Morse–Bott homology (see [BO09a]). These have some

components solving Seidel’s equation, connected to gradient flow lines of auxiliary

Morse functions on the manifolds of orbits. We will think of the points that connect

to gradient flow lines as removable singularities, and denote the set of those by ΓF,r.

When we split the manifold, some solutions of Seidel’s equation split into different

pieces, with new punctures asymptotic to Reeb orbits of Y . These new punctures are

of two types. Some are capped by holomorphic planes in W or in NΣ, and we denote

the set of such Reeb punctures by ΓaR (the superscript stands for ‘augmentation’,

although one usually reserves this term for planes in the convex filling W , not on

the concave filling NΣ). The other case is when the puncture affects the conformal

structure of the domain. The set of those punctures is called ΓcR. Figure 4.1 sketches

an example of a broken pair-of-pants where all types of punctures occur.

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 34

4.2 Pseudo-holomorphic curves in R Y

To define (perturbed) pseudo-holomorphic curves in R Y , we will choose some

additional structure on the closed symplectic manifold pΣ, ωq, and lift it to R Y .

We will need a generic ω-compatible almost complex structure J in Σ. For m ¥

3, let Mm be the moduli space of stable Riemann surfaces S of genus 0 (nodal

surfaces of genus 0, with at least three nodes and marked points on each irreducible

component), and let Um be the corresponding universal curve bundle. We will consider

perturbation forms ν P ΓUm,Hom0,1pTS, TΣq

, or, put differently, Mm-dependent

forms ν P Ω0,1pS, TΣq. We further assume that these perturbations are supported

away from the nodes and marked points in S. We will see in Section 5.1.1 that the

Gromov–Witten numbers of Σ can be defined using moduli spaces of maps w : S Ñ Σ

that solve the perturbed equation

dw J dw j ν.

The form ν is chosen so that homologically trivial curves in Σ are not constant.

This will allow us to choose only one Morse function fΣ on Σ and, for generic J and ν,

to have the transversality required for constructing fiber products of moduli spaces of

holomorphic curves and stable/unstable manifolds of fΣ, with respect to evaluation

maps at various marked points. Since there are no constant holomorphic curves, there

will be no trees of three or more gradient flow lines meeting at one point.

If m 3, then we do not have a space of stable curves, so we cannot talk about

the universal curve bundle. This is not a problem, because in this case it will be

enough to consider solutions of dw J dw j 0, with no perturbation term ν.

The almost complex structure J in Σ can be lifted uniquely to a cylindrical almost

complex structure J on R Y . Note that the projection P : Y Ñ Σ is such that the

pullback bundle P TΣ coincides with the contact distribution ξ on Y . Therefore, J

is determined by the conditions J |ξ P J and JBr R. We can also lift ν to ν in

R Y , by making ν trivial in the Br and R directions. Let now Z tz1, . . . , zku be

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 35

a finite set of points in CP 1. We will be interested in studying maps

u pa, uq : CP 1zZ Ñ R Y

du J du i ν(4.1)

Transversality for ν-perturbed J-holomorphic curves in Σ will imply transversality

for ν-perturbed J-holomorphic curves in R Y .

4.3 The cylinder equation

Let u pa, uq : Sztother puncturesu Ñ R Y be a solution of (4.1), where S

CP 1zΓF . Fix conformal parametrizations ϕi : R S1 Ñ S of neighborhoods of the

zi P ΓF and β P Ω1pSq, as in the discussion of Seidel’s equation (2.5).

We will be interested in solutions of the following auxiliary equation:

f pf1, f2q : S Ñ R S1

df1 df2 i h1peaf1qβ i 0.(4.2)

Since f takes values in RS1, we call this the cylinder equation. Denote by Cpuq the

space of cylinder solutions associated with u.

Proposition 4.1. Let u be a perturbed pseudo-holomorphic curve in R Y . Then,

1. equation (4.2) is a Fredholm problem, of index

indpCpuqq 2 kcvx kccv,

where kcvx is the number punctures in ΓF converging to non-constant orbits in

the region where H is convex, and kccv is the number punctures in ΓF converging

to non-constant orbits in the region where H is concave;

2. if indpCpuqq ¤ 0, then the kernel of the linearized operator is 1-dimensional, and

is spanned by the generator of the S1 action;

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 36

3. if indpCpuqq ¡ 0, then the linearized operator is surjective, and its kernel in-

cludes the generator of the S1 action.

A proof of this statement will be given in [DL12]. The index formula follows from

the study of the relevant asymptotic operators, obtained from the linearization of

equation (4.2). The proof of second and third statements involves the study of the

eigenvalues of the asymptotic operators and uses automatic transversality, namely

Proposition 2.2 in [Wen10].

To study rigid Floer trajectories, it will also be very useful to understand solutions

of equation (4.2) in families of low index.

Proposition 4.2. 1. If indpCpuqq 0, then Cpuq is cobordant to a single point.

2. If indpCpuqq 1, then Cpuq is isomorphic to R.

Proof. We will leave the details of the proof of the first part to [DL12]. Let us

just sketch the argument. We need to recall some information and introduce some

notation. For each puncture z P ΓF , we have the data pTz, bzq P R2 such that

h1pebzq Tz. Then, we have in cylindrical coordinates near the puncture z,

aps, tq P Tzs cz W 1,p,δpR S1q,

for a constant cz and for δ ¡ 0 small. Recall that in cylindrical coordinates, β κzdt,

at least sufficiently close to the puncture. Cylinder solutions pf1, f2q are in the function

space

f1 P Ts c r W 1,p,δc pSq, f2 P W

1,p,δc pSq.

Let µ : S Ñ R be a smooth function supported near the punctures such that,

in cylindrical coordinates, µ Tzs cz bz close enough to the puncture z. Let

g1 f1 µ and g2 f2. Then, our problem may be reformulated as:

dg1 dg2 j dµ h1pepaµqg1qβ j 0.

The proof of the first part of this Proposition consists of studying the family of

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 37

equations

dg1 dg2 j τdµ h1pepaµqg1qβ j

0

for τ P r0, 1s. One can show that the space of solutions is invariant under change in

τ . When τ 1, we have the problem we are interested in. When τ 0, we get the

standard Cauchy–Riemann equation. The asymptotic conditions imply that there is

a unique solution when τ 0, and thus also when τ 1, as wanted.

Now we prove the second part of the Proposition. If pa, uq : R S1 Ñ R Y is a

trivial cylinder, then it has the form

pa, uqps, tq Ts C, γpTsq

for some Reeb orbit γ of period T and some constant C P R. Equation (4.2) can be

written as $&%Bsf1 Btf2 h1peaf1q 0

Btf1 Bsf2 0

for pf1, f2q : R S1 Ñ R S1 satisfying the appropriate asymptotic conditions. It

will be more useful to consider instead the functions pb, cq pf1 a, f2q, which solve

the system of equations #Bsb Btc h1pebq T 0

Btb Bsc 0(4.3)

with asymptotic conditions limsÑ8pb, cqps, tq pr, kq, for some constants r P Rand k P S

1.

We begin by showing that there can be no solution pb, cq such that k k.

Suppose that pb, cq solves (4.3), with the required asymptotics. Since c : RS1 Ñ S1

is null-homotopic, we can find a lift c : R S1 Ñ R. Define I : R Ñ R such that

Ipsq ³S1 cps, tqdt. Then,

dI

ds

»S1

Bscps, tqdt

»S1

Btbps, tqdt 0.

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 38

and we conclude that I is a constant function. But

limsÑ8

Ipsq kpmod 1q and limsÑ8

Ipsq kpmod 1q

hence k k, as wanted.

We continue with the study of (4.3). If we assumed c k to be constant, and

bps, tq bpsq to be independent of t, then we would get the ordinary differential

equationdb

ds h1pebq T 0. (4.4)

The asymptotic conditions at 8 would imply h1perq T . Since H is S-shaped,

the equation h1perq T has only two solutions. Since Floer trajectories increase the

symplectic action, it is necessarily the case that r ¡ r. This implies that the Floer

trajectories pb, vq given by these pa, uq and pf1, f2q will converge to an orbit on the

concave part of H as s Ñ 8 and to an orbit on the convex part of H as s Ñ 8.

Since (4.4) is autonomous, we get a family of solutions parametrized by s P R (which

is compatible with the fact that we want index 1-families of solutions pf1, f2q). We

will show that these are indeed all the solutions to this problem.

Observe that the system of equations (4.3) is a Floer equation on R S1 for

the Hamiltonian Hpx, yq ³h1pexqdx Tx and the standard symplectic form and

complex structure. Therefore, the solutions can be thought of as holomorphic curves

R S1 Ñ M : pR S1q pR S1q, for some twisted almost complex structure on

the target (by the trick of Gromov that was alluded to in Section 3.3). The projection

of these curves to the first cylinder factor is the identity. Since the solutions of (4.4)

come in R-families and k is not restricted, we get a foliation of the open subset

pR S1q pr, rq S1

M by holomorphic curves. We want to show that

these are all the solution of the system of differential equations. Call them gradient

solutions (because they are solutions of an ordinary differential equation).

Now, suppose we fix a constant k and find a solution G pb1, c1q of the systems

of equations that is not a gradient solution, with limsÑ8 c1ps, tq k. If c1 k, then

b would again solve (4.4) and G would be a gradient solution. Therefore, c1 k, and

the graph of G intersects the graph of a gradient solution F pb2, c2q with c2 k.

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 39

Take now another gradient solution F pb3, c3q such that c3 k. We can homotope

G to F , since they are both null-homotopic maps R S1 Ñ R S1. Since G and

F have the same asymptotics, we can make sure that the homotopy is C0-small on

neighborhoods of 8 S1 R S1. This implies that the graph of G will be

homotoped to the graph of F in M , and that the intersections with the graph of F

will remain in a compact region of M . So, we have an equality of signed counts

#GraphpGq XGraphpF q

#

GraphpF q XGraphpF q

0,

since F and F have different asymptotics. But then positivity of intersection for holo-

morphic curves in 4-dimensions implies that the graphs of G and F do not intersect,

which gives a contradition. Therefore, there is no such G, as we wanted to show.

4.4 Floer trajectories in R Y

We now study solutions of a perturbed version of Seidel’s Floer equation, in R Y :

v pa, vq : Sztpuncturesu Ñ R Y

pdv XH βq0,1 ν.

(4.5)

Consider the moduli space

H !pa, uq, pf1, f2q

|pa, uq satisfies (4.1) and pf1, f2q satisfies (4.2)

)C

where we take the quotient by the C-action given by

eρiθ.pa, uq, pf1, f2q

pa ρ, φθ uq, pf1 ρ, f2 θq

,

φθ being the Reeb flow on Y for time θ. Let also

F pb, vq satisfying (4.5q

(.

Our goal is to prove the following result.

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 40

Theorem 4.1. The map

Φ : HÑ F

pa, uq, pf1, f2q ÞÑ pa f1, φf2 uq (4.6)

is well-defined and a diffeomorphism.

We will split the proof of this result into the following parts:

(i) Φ is well-defined;

(ii) Φ is a bijection;

(iii) Φ is differentiable and an immersion.

Proof of (i). It is clear that Φ is C-equivariant. To see that Φ is well-defined, let

pu, fq pa, uq, pf1, f2q

P H. Write v pb, vq : Φpu, fq. We need to show that

v P F . First, notice that, if we denote by π1 and π2 the projections associated with

the splitting

T pR Y q RxBry ` RxRy

` ξ

then

π1pdu J du iq Br b pda uα iq R b pda uα iq i 0 (4.7)

and

π2pdu J du iq π2du J π2du i ν (4.8)

Since H is S- or J-shaped, we have XHpr, yq h1perqRpyq, so

pdv XH βq0,1 dv J dv i h1pebqR b β h1pebqBr b β i.

Therefore,

π1

pdv XH βq

0,1 Br b pdb vα i h1pebqβ iq

R b pdb vα i h1pebqβ iq i.

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 41

Formula (4.6) implies that

vα α dpφf2 uq df2 uα

so

π1

pdv XH βq

0,1 Br b pda df1 df2 i uα i h1peaf1qβ iq

R b pda df1 df2 i uα i h1peaf1qβ iq i

Br bpda uα iq pdf1 df2 i h1peaf1qβ iq

R bpda uα iq pdf1 df2 i h1peaf1qβ iq

i 0

because of (4.7) and (4.2). On the other hand,

π2

pdv XH βq

0,1 π2dv J π2dv i π2dφf2 du J π2dφf2 du i

dφf2 pπ2du J π2du iq dφf2 ν ν

using the fact that dφf2 commutes with π2 and J , (4.8) and the fact that ν is R-

invariant. This concludes the proof that v solves (4.5).

Now that we have seen that Φ is well-defined, we show that it is bijective.

Proof of (ii). Fix v pb, vq P F .

Claim 1. For every f pf1, f2q that solves

df1 df2 i h1pebqβ i 0, (4.9)

there is a unique solution u pa, uq of (4.1) such that Φpu, fq v.

This implies that, to understand Φ1pvq, we should study the solutions of (4.9).

Proof of Claim 1. Such u is uniquely determined by (4.6) to be

u pa, uq b f1, φf2 v

.

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 42

The computations in the proof of (i) above also show that u solves (4.1).

The fact that Φ is a bijection will now follow from the fact that (4.9) always has

a C-family of solutions.

Claim 2. Given v P F , there is a solution of (4.9), unique up to the C-action

eρiθ.pf1, f2q pf1 ρ, f2 θq.

Proof of Claim 2. We want a solution pf1, f2q of (4.9) such that, near zi P Γ, with a

parametrization ϕi : R S1 Ñ CP 1,

pf1, f2qps, tq pTis ci, diq

where Ti is the period of the Reeb orbit associated with the puncture zi and the

Seidel solution pb, vq and ci P R, di P S1 are constants. To make sense of this, we

fix functions ψi P C8pS,Rq such that ψips, tq 1 near zi and ψi 0 away from a

neighborhood of the zi that is contained in ϕipR S1q. Then, we say that

pg1, g2q :f1ps, tq

¸i

ψips, tqpTis ci,1q, f2ps, tq ci,2

P W 1,p,δpS,R S1q.

Denote ηps, tq °i ψips, tqpTis ci,1q. Finding pf1, f2q is equivalent to finding

pg1, g2q, for some constants ci,1, ci,2. Now, equation (4.9) is equivalent to

dg1 dg2 i h1pebqβ i dη.

The left side of this equation defines a Fredholm operator of index 2, whose kernel is

precisely given by constants in R2. Therefore, the cokernel is trivial, and the operator

is surjective. This implies that there is a required unique solution of this equation.

The fact that Φ is a bijection follows immediately from the two claims, and from

the definition of H as a quotient by C.

We already know that Φ is a bijection. We now prove that it is also a diffeomor-

phism.

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 43

Proof of (iii). We will show that Φ gives a bijection of tangent spaces. We first

describe the tangent spaces to F and H. These are both spaces of solutions of elliptic

differential equations, so their tangent spaces are kernels of linearized operators.

J and ω define a Hermitian metric on RY . Denote the corresponding connection

by ∇. With respect to this connection, the linearized operator associated with the

pseudo-holomorphic curve equation (4.1), at a solution u pa, uq, is (see [Dra04])

Dholou : W 1,p,δ

S, uT pR Y q

Ñ Lp,δ

S,Λ0,1pT Sq bJ u

T pR Y q

ζ ÞÑ ∇ζ J∇j.ζ p∇ζJq du j ∇ζ ν

Writing ζ pλ, µq in terms of the splitting uT pRY q

RxBry`RxRy

`uξ,

we get

ζ ÞÑ p∇λ i∇j.λq ∇µ J∇j.µ p∇µJqπ du j ∇µν

Denote Dholo

u Dholo1 Dholo

2 , under this splitting. Write λ λ1 iλ2. Then

Dholo1 λ ∇λ i∇j.λ pdλ1 dλ2 jq ipdλ2 dλ1 jq

pdλ1 dλ2 jq ipdλ1 dλ2 jq j

and

Dholo2 µ ∇µ J∇j.µ p∇µJqπ du j ∇µν

The linearized operator associated with the cylinder equation (4.2) at a solution

f pf1, f2q is

Dcylpu,fq : W

1,p,δpS,Cq Ñ Lp,δS,HompT S,Rq

pF1, F2q ÞÑ dF1 dF2 j h2peaf1qeaf1F1β j.

Therefore, the tangent space to H at ru, f s is given by the quotient of kerDholou `

kerDcylpu,fq by the R2-action induced by the C-action on pairs pu, fq.

Finally, we consider the linearization of the operator corresponding to Seidel’s

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 44

equation (4.5), at a solution v pb, vq:

DFloerv : W 1,p,δ

S, vT pR Y q

Ñ Lp,δ

S,Λ0,1pT Sq bJ v

T pR Y q

ζ ÞÑ ∇ζ J∇j.ζ p∇ζJq dv j ∇ζXH b β ∇ζpJXHq b β j ∇ζ ν

Writing ζ pρ, σq in terms of the splitting vT pRY q

pRxBry`RxRy

`vξ,

we get

ζ ÞÑ∇ρ i∇j.ρ p∇ρh

1perqRq b β p∇ρh1perqBrq b β j

∇σ J∇j.σ p∇σJqπ dv j ∇σν

Denote DFloer

v DFloer1 DFloer

2 , under this splitting. Write ρ ρ1 iρ2. Then

DFloer1 ρ ∇ρ i∇j.ρ p∇ρh

1perqBrq b β p∇ρh1perqRq b β j

pdρ1 dρ2 j ρ1h2perqerβ jq ipdρ2 dρ1 j ρ1h

2perqerβq

pdρ1 dρ2 j ρ1h2perqerβ jq ipdρ1 dρ2 j ρ1h

2perqerβ jq j

and

DFloer2 σ ∇σ J∇j.σ p∇σJqπ dv j ∇σν

We now write the differential of the map Φ : H Ñ F at a point ru, f s

rpa, uq, pf1, f2qs, such that Φru, f s v pb, vq. Under the splittings uT pR Y q

RxBry ` RxRy` uξ and v

T pR Y q

RxBry ` RxRy

` vξ, as above, and

writing λ λ1 iλ2, we have

DΦpλ, µq, pF1, F2q

pλ1 F1q ipλ2 F2q, Dpφf2qµ

.

Since H and F are manifolds of the same dimension, it is enough to show that DΦ

is surjective. Consider then pρ, σq P TvF kerDFloerv , where v Φrpa, uq, pf1, f2qs.

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 45

Take pλ, µq 0, Dpφf2qσ

and F1 iF2 ρ. Note that

Dholou pλ, µq Dholo

2 µ ∇µ J∇j.µ p∇µJqπ dv j ∇µν

∇Dpφf2qσ J∇j.Dpφf2qσ ∇Dpφf2 qσ

Jπ dv j ∇Dpφf2 qσ

ν

p;q Dpφf2q

∇σ J∇j.σ p∇σJqπ dv j ∇σν

Dpφf2qDFloer2 σ 0

In p;q, we have used the fact that ∇ is the Levi-Civita connection for the metric

ωp., J.q on R Y , that the flow φ of the Reeb vector field on Y is by isometries of

R Y , and that J and ν are invariant under φ.

We also have

Dcylpu,fqpF1, F2q dF1 dF2 j h2peaf1qeaf1F1β j π1pD

Floer1 ρq 0

so pλ, µq P kerDholou TuM and pF1, F2q P kerDcyl

pu,fq. Furthermore,

DΦpλ, µq, pF1, F2q

pρ, σq

which completes the proof that DΦ is surjective and that Φ is a diffeomorphism.

4.5 Excluding unwanted solutions

According to Proposition 4.1 there are certain configurations in the split Floer homol-

ogy differential of X whose associated cylinder equation is not transverse. Namely,

if kcvx 1 kccv, which implies that the index of the linearized cylinder equation

is zero, and both kernel and cokernel are one-dimensional. We now explain how to

exclude those configurations, with an argument involving the action functional.

The only configurations in the split Floer homology differential whose correspond-

ing cylinder equation is not transverse are given by punctured cylinders

V pb, vq : R S1ztpu Ñ R Y

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 46

aa δ

er

Hprq

Σ

b

0

slope M

b η

θ

a ε

x

x

|x

xxFigure 4.2: Unwanted configuration and steep Hamiltonian

where the puncture converges to a Reeb orbit at the 8 end of the symplectization,

which is capped by a plane on a fiber of the normal bundle NΣ. In Figure 4.2, we

sketch one such configuration, and an S-shaped Hamiltonian H with steep slope M ,

for which we will be able to show that such configurations cannot exist.

Suppose that limsÑ8 vps, tq xptq, for some Reeb orbits x in Y . One of

the hypothetic configurations that we are trying to rule out would contribute with

x t1

K λXN to the differential of x. The energy of such a configuration would be

E AHpxq 1K AHpxq.

Suppose that x is an l-cover of a fiber of S1 Ñ Y Ñ Σ. Then, x is an pl 1q-cover

of a fiber, since the component of the split Floer trajectory that is contained in RYprojects to a contractible map to Σ, by index reasons. Put differently, this component

is a perturbation of a cover of a trivial cylinder.

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CHAPTER 4. AN ANSATZ FOR SPLIT FLOER TRAJECTORIES 47

Therefore, according to Lemma 3.3, and using the notation in Figure 4.2,

E pa εql b η 1K pa δqpl 1q θ

a εl b η 1K δpl 1q θ.

Since the energy of a non-constant Floer trajectory is positive, we get that

b a 1K η θ δ pε δql.

Therefore, we conclude that, if we fix small η, θ, δ and ε, and large L and b, we can

conclude that such configurations cannot exist for 0 l L. Heuristically, this

means that if H is close to a step function and b is large, then we can exclude the

existence of configurations as in Figure 4.2, as long as l is not too large.

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Chapter 5

Relation with Gromov–Witten

numbers

We saw above that Floer trajectories in RY can be understood in terms of punctured

pseudo-holomorphic curves in RY and solutions of the auxiliary cylinder equation.

In this chapter, we relate pseudo-holomorphic curves in R Y , in NΣ and in W

with Gromov–Witten numbers of Σ and relative Gromov–Witten numbers of pX,Σq.

Then, we explain how to use this information to express the differentials in split Floer

and symplectic homology. We also give an indication of the analogous results for the

pair-of-pants products.

5.1 Pseudo-holomorphic curves in R Y and NΣ,

meromorphic sections of holomorphic line bun-

dles and Gromov–Witten numbers of Σ

We will now see how to relate pseudo-holomorphic curves in RY with meromorphic

sections of line bundles over CP 1 and with Gromov–Witten numbers of Σ.

48

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 49

5.1.1 Gromov–Witten numbers and quantum cohomology

Let us quickly review some basics about Gromov–Witten theory. Although we use

a slightly different point of view (namely with respect to the perturbations that we

take, and to the fact that we will use a Morse chain version), we refer the reader to

[MS04] for more details.

Recall that pΣ2n2, ωq is a closed symplectic manifold on which we chose a generic

ω-compatible almost complex structure J and a family ν of perturbation 1-forms,

parametrized by the universal curve and supported away from nodes and marked

points (see Section 4.2).

We are interested in maps w : CP 1 Ñ Σ that solve the perturbed equation

dw J dw j ν.

The spaces of such maps realizing homology classes A P H2pΣq, modulo domain

automorphisms preserving m marked points, define moduli spaces MA,mpΣq. These

come equipped with m evaluation maps at the marked points, and (for generic J and

ν) define pseudo-cycles in the product of m copies of Σ:

ev :MA,mpΣq Ñ Σm.

These pseudo-cycles have dimension 2n 2 2 xc1pTΣq, Ay 2m 6. To define

Gromov–Witten invariants of Σ, one intersects such pseudo-cycles with homology

classes in Σ. To that end, one can take pseudo-cycle representatives for generators of

the homology of Σ (see for instance [MS04] and [Sch99]), and work with intersections

of pseudo-cycles. We are interested in a slightly more general chain-level definition

(which is why we use the term ‘numbers’ and not ‘invariants’). Let fΣ : Σ Ñ R be a

Morse–Smale function with respect to a Riemannian metric g on Σ. The critical points

of fΣ generate a chain complex that computes the singular homology of Σ. Schwarz

showed this fact in [Sch99], by proving that the unstable manifolds associated with a

Morse cycle form a pseudo-cycle in Σ, which in turn defines a unique homology class.

He also showed that, if we fix a pseudo-cycle P in Σ, then for generic pairs pfΣ, gq,

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 50

P intersects the Morse pseudo-cycles of pfΣ, gq transversely, and in the interior of

the top dimensional strata of the Morse pseudo-cyles (recall that a Morse pseudo-

cycle is given by the union of stable or unstable manifolds of finitely many critical

points, of possibly different indices). For details on this point, see Theorem 4.9 in

[Sch99]. We can then argue that, if we fix a countable collection of pseudo-cycles

Pi inside powers of Σ, then for a Baire set of pairs pfΣ, gq, products of stable and

unstable manifolds of critical points of f intersect the Pi transversaly. To define

Gromov–Witten numbers of Σ, we let the Pi be the pseudo-cycles associated with the

relevant moduli spaces of perturbed pseudo-holomorphic curves, and intersect those

with the stable and unstable manifolds of a generic pair pfΣ, gq. Given a homology

class A P H2pΣq and Morse chains C1, . . . , Cm, we denote by GWΣA,mpC1, . . . , Cmq the

corresponding Gromov–Witten number, obtained by intersecting the pseudocycles

given by MA,mpΣq and by C1 . . . Cm, in Σm.

Remark 5.1. In this text, we made the simplifying assumption that Σ admits a

perfect Morse function, which holds in the examples that we will consider. The main

consequence is that there is no need to distinguish between Gromov–Witten numbers

and invariants, which can be read from the quantum cohomology rings. See also

Remarks 3.1 and 5.4 for other implications of this assumption.

At this point, we should remark that, since we assume our symplectic manifolds

to be monotone, in principle we would not need the perturbation term ν to achieve

transversality for the spaces of perturbed pseudo-holomorphic curves, and to define

Gromov–Witten numbers. A key feature of Gromov–Witten numbers is that, if we

choose the Ci to be homology classes, instead of chains, then GWΣA,mpC1, . . . , Cmq

does not depend on generic J and ν (and is called an ‘invariant’). In fact in Chapter

6, we will compute some Gromov–Witten invariants in a setting where ν 0. The

reason why we introduce the perturbation terms ν in our definition is to have enough

transversality of the evaluation maps at marked points to the stable and unstable

manifolds of a single Morse function in Σ. For this to be the case, we need homolog-

ically trivial curves in Σ not to be constant, which can be achieved by the term ν in

the equation.

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 51

If m 3, then we do not have a space of stable curves, so we cannot talk about

the universal curve bundle. But in this case, it turns out to be enough to consider

unperturbed pseudo-holomorphic curves, which solve dw J dw j 0. Just as

before, we can use moduli spaces of solutions of this equation to define Gromov–

Witten numbers with one and two marked points.

One important property of Gromov–Witten numbers, which we will use, is the

divisor equation. This states that, given A P H2pΣq, homology classes C1, . . . , Cm P

HpΣq and H P H2n2pΣq,

GWΣA,mpC1, . . . , Cmq

1

#pAXHqGWΣ

A,m1pC1, . . . , Cm, Hq.

The meaning of this is that each curve in class A intersecting C1, . . . , Cm will intersect

#pAXHq times the class H, and will therefore contribute with the factor #pAXHq

to the count of curves in homology class A intersecting C1, . . . , Cm, H.

Another important point about Gromov–Witten invariants (on homology, not

on the chain level) is that they have also been defined using methods of algebraic

geometry. The advantage of this approach is that it is often much more computable

than its symplectic counterpart. It has been shown that in the case of complex

projective Kahler varieties, where both the symplectic and the algebraic definitions

make sense, the invariants are the same (see for instance [LT99]). The point of our

work is precisely to describe, in certain examples, the Floer differential and pair-of-

pants product in terms of Gromov–Witten numbers, that can sometimes be computed

explicitly using tools from algebraic geometry (as in [Bea95] or [Zin11]).

Let us recall also how to use Gromov–Witten invariants to construct a deformation

of the cup product on cohomology. Assume that the symplectic manifold pM,ωq is

monotone, with minimal Chern number N . Consider a Novikov ring ΛQ : Qrt, t1s,

where t is a variable of degree 2N , as in Section 2.1. Define a product on QHpMq :

HpM ;Qq b ΛQ as follows: given a P HkpM ;Qq and b P H lpM ;Qq,

a b :¸

APH2pM ;Zq

pa bqA txc1pTMq,AyN

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 52

where pa bqA P HpM ;Qq is specified by saying that, for any c P HpM ;Qq,»M

pa bqA Y c GWMA,3pPDpaq,PDpbq,PDpcqq.

QHpM ;Qq with this ring structure is called the quantum cohomology algebra. As we

have recalled in Section 2.3, quantum cohomology of monotone symplectic manifolds

is isomorphic to Floer homology. In Chapter 6, we will use results about quantum

cohomology algebras (from [Bea95]) to extract the Gromov–Witten invariants that

will be necessary for our symplectic homology computations.

5.1.2 Meromorphic sections of holomorphic line bundles and

pseudo-holomorphic curves in R Y

As was pointed out in Remark 3.2, the symplectization R Y can be thought of as

the complement of the zero section of a complex line bundle E Ñ Σ (associated with

the S1-bundle Y Ñ Σ).

Let now u pa, uq : S Ñ R Y be a pseudo-holomorphic curve, where S is the

complement of a finite subset of CP 1. One can project u to the divisor Σ, and obtain

a pseudo-holomorphic map w : S Ñ Σ. Since punctures of finite energy pseudo-

holomorphic curves asymptote to Reeb orbits in Y , and these are multiple covers of

the fibers of Y Ñ Σ, w extends to a map from CP 1. Now, pa, uq can be identified with

a section s of the bundle wE Ñ CP 1, with prescribed zeros and poles at the points

in CP 1zS. This section is complex linear, by (4.7). Since every complex line bundle

over CP 1 admits a unique holomorphic strucutre (see Exercise 3.3.7 in [Huy05]), we

conclude that s is actually a meromorphic section. This proves the following result.

Lemma 5.1. Every pseudo-holomorphic curve u pa, uq : S Ñ R Y defines a

pseudo-holomorphic map w : CP 1 Ñ Σ and a meromorphic section of wE Ñ CP 1

with zeros and poles on the finite set CP 1zS.

We can now reduce the question of counting punctured pseudo-holomorphic maps

u to that of counting pseudo-holomorphic curves w : CP 1 Ñ Σ, together with mero-

morphic sections s of wE. The count of maps w is related with the computation

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 53

of Gromov–Witten numbers of Σ. We can also give a complete description of the

relevant meromorphic sections s. Recall that, given a divisor of points D and a

holomorphic line bundle L over CP 1, where both D and L have degree d, there is a

C-family of meromorphic sections of L such that the divisor associated with each

section is D. One can justify this fact by reducing it to the simplest case of trivial

L: use a trivialization of L over C CP 1 to identify meromorphic sections of L with

meromorphic functions on CP 1 (see pages 342–345 of [Mir95]).

Remark 5.2. The fact that meromorphic sections of line bundles over CP 1 come in

C-families implies that the contact homology differential (without point constraints)

should vanish for pre-quantization bundles (see [EGH00]), because the moduli spaces

of non-constant holomorphic curves with cylindrical ends have an S1-action without

fixed points. This is not the case in our setting, though, because we impose marker

conditions on our asymptotic limits (we should think of non-equivariant contact ho-

mology, as in Section 3.2 of [BO09a]).

From this point on, let us restrict our attention to cylinders in RY contributing

to the split Floer or symplectic homology differential, possibly with punctures capped

by planes in W or in NΣ. For an example, see Figure 5.1. In this picture and in all

the ones that will follow, the periodic orbits represent asymptotic Reeb orbits. The

configuration depicted here contains components in NΣ (on top), in R Y (in the

middle) and in W (on the bottom).

Given a non-vanishing Gromov–Witten number GWΣA,lpC1, . . . , Clq, where the Ci

are stable or unstable manifolds of critical points of a Morse function fΣ : Σ Ñ R,

there are rigid pseudo-holomorphic maps w : pCP 1; z1, . . . , zlq Ñ pΣ;C1, . . . , Clq,

modulo automorphisms. Since we are describing the differential, we have l ¥ 2. Let

us try to describe which rigid maps

u : R S1ztl 2 puncturesu Ñ R Y,

with fixed markers at 0 and 8, this gives rise to. Let l be the number of punctures

that are capped by a plane in NΣ and let l l 2 l be the number of punctures

capped in W . Fix a map w as above and assume without loss of generality that

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 54

pqk qk1 qkl

qqk qk1 qkl

Figure 5.1: Configurations given by coefficients cpk, k1 , . . . , kl

; k, k1 , . . . , klq

z1 0 and z2 8. Given b P S1, we can define another pseudo-holomorphic map w,

such that wpzq wpbzq. This gives us one degree of freedom to fix one marker, either

at 0 or at 8. Each of these maps w defines a line bundle wE Ñ CP 1, of degree

dpAq : xc1pE Ñ Σq, Ay. Since prescribing zeros and poles gives a C-family of

meromorphic sections, we have another S1-parameter that can be used to fix another

marker. This motivates the following definition (see Figure 5.1).

Definition 5.1. Let cpk, k1 , . . . , kl

; kk1 , . . . , klq be the number of pseudo-holo-

morphic curves u : RS1ztl2 puncturesu Ñ RY , with one positive Floer puncture

asymptotic to a Reeb orbit of multiplicity k, l positive punctures (asymptotic to

Reeb orbits of multiplicity k1 , . . . , kl

, respectively) to be capped by planes in NΣ,

one negative Floer puncture asymptotic to a Reeb orbit of multiplicity k, and l

negative punctures (asymptotic to Reeb orbits of multiplicity k1 , . . . , kl

, respectively)

to be capped by planes in W , associated with a non-zero Gromov–Witten number

GWΣA,lpC1, . . . , Clq.

Note that we should have

dpAq xc1pE Ñ Σq, Ay k

l

i1

ki k

l

j1

kj ,

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 55

to obtain the correct difference of the number of zeros and poles. The following result

will be useful in our computations of symplectic homology differentials.

Lemma 5.2. cpk 1; k, 1q 1 and cpk |d|; kq |d|, where d dpAq 0 for some

A P H2pΣ;Zq.

Proof. We begin with the proof that cpk1; k, 1q 1. This corresponds to saying that,

given a rigid augmentation plane in W capping the simple Reeb orbit corresponding

to the critical point q P Σ, we have a contribution pqk to the differential of qqk1, for

any multiplicity k ¥ 1, coming from the configuration in Figure 5.2.

pqk q1

qqk1

Figure 5.2: Configurations given by coefficients cpk 1; k, 1q

The component u : CP 1zt0,8, λu Ñ R Y (where λ is the augmented puncture)

projects to a null-homologous map w : CP 1 Ñ Σ, so the line bundle wE Ñ CP 1 is

trivial. The relevant meromorphic sections are in this case meromorphic functions,

which can be written explicitly as:

s : Czt0, λu Ñ C

z ÞÑ azk1pz λq

where a reiθ P C. This extends to a map s : CP 1 Ñ CP 1. The zeros correspond

to the points 0 and λ on the domain, and represent pqk and q1, respectively; the pole

is attained at 8 and corresponds to qqk1. Over CP 1z0, with holomorphic coordinate

y 1z, the section s becomes

y ÞÑ a1 λy

yk

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 56

Write λ ρeiϕ and observe that we have four real degrees of freedom: r, θ, ρ and

ϕ. We need to quotient out our space of configurations by R-translations on the

domain and on the target, so we can assume r ρ 1. Fixing two markers will

rigidify these configurations. We will show that we indeed get a unique configuration

(corresponding to unique values of θ and ϕ) when we fix markers.

To fix the markers at qqk1 and pqk, we can, for instance, force markers aligned with

positive real directions on the domain to map to markers aligned with positive real

directions on the target. For pqk, let z t, for t ¡ 0 small. We get

spzq sptq eiθtk1pt eiϕq tk1eipθϕq.

For the argument of the image to coincide with the prescribed marker on the target,

we get θ ϕ π. For qqk1, let y t, for t ¡ 0 small. We get

spyq sptq eiθ1 eiϕt

tk eiθtk1

and we can again force the argument to match with the corresponding prescribed

marker on the target. This implies θ 0 and ϕ π. Therefore, after prescribing the

markers, we get unique values for θ and ϕ, which justifies cpk 1; k, 1q 1.

We now wish to show that cpk |d|; kq |d|, where d xc1pE Ñ Σq, Ay 0 for

some A P H2pΣ,Zq. This means that, if p, q P Σ are critical points of fΣ, then any rigid

holomorphic sphere in Σ contributing to GWΣA,2pW

uppq,W spqqq, gives a contribution

of p|d| . pqkq to the differential of qpk|d|. These terms correspond to pseudo-holomorphic

cylinders u : CP 1zt0,8u Ñ R Y projecting to maps w : CP 1 Ñ Σ such that

rws A, and are represented in Figure 5.3. The line bundle wE has degree d 0.

The maps u are given by composing meromorphic sections of wE Ñ CP 1 with

reparametrizations

CP 1 Ñ CP 1

z ÞÑ bz

for |b| 1 (since in non-equivariant contact homology our cylinders have asymptotic

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 57

pqk

qpk|d|

Figure 5.3: Configurations given by coefficients cpk |d|; kq

markers, these domain rotations are not automorphisms we can mod out by).

Let us write down coordinates for the domain, CP 1zt0,8u, and target, the degree

d 0 line bundle wE Ñ CP 1. On the domain, take a coordinate z on C

CP 1z0 and y 1z on C CP 1z8. On the target, take coordinates pζ, uq on

pCP 1z0qpCP 1z0q and pξ, vq p1ζ, uξ|d|q on pCP 1z8qpCP 1z8q. We will consider

meromorphic sections of wE with a prescribed zero of order k (corresponding to a

k-multiple of the fiber of Y Ñ Σ over q P Σ), say at 0, and a prescribed pole of order

k |d| (corresponding to a pk |d|q-multiple of the fiber of Y Ñ Σ over p P Σ), say

at 8. These sections can be written as

upζq aζk and vpξq upζqξ|d| aξk|d|.

Therefore, we can write the map CP 1zt0,8u Ñ wE as

z ÞÑbz, apbzqk

and y ÞÑ

yb, abk|d|yk|d|

.

Since we quotient by R-translation on the symplectization direction, which in our

case corresponds to the radial direction on the fiber, we can assume |a| 1. Fixing

the two markers, say by requiring that the positive real lines map to the positive real

lines, we get

abk 1 and abk|d| 1

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 58

which implies that b|d| 1 and a bk. Therefore, writing η e2πi|d|,

pa, bq P p1, 1q, pηk, ηq, . . . , pηkp|d|1q, η|d|1q

(.

These are the |d| solutions we were after, and this is why cpk |d|; kq |d|.

5.1.3 Pseudo-holomorphic curves in NΣ

Split Floer trajectories may also contain components in NΣ, where the Hamiltonian

is constant. Therefore, components in NΣ are pseudo-holomorphic curves, possibly

connected with gradient flow lines of a Morse function fNΣ in NΣ. The simplest such

components are cappings of punctures of pseudo-holomorphic curves in R Y , as in

Figure 5.1. These are pseudo-holomorphic curves in NΣ, converging to Reeb orbits

in Y . For such cappings to be rigid, they should correspond to fibers of NΣ Ñ Σ.

This implies that such planes should asymptote to simple Reeb orbits, since otherwise

one would have non-rigid families of capping planes in NΣ, given by covers of fibers

of this line bundle. As a consequence, we can conclude that ‘positive augmentation’

punctures of rigid holomorphic curves in RY must converge to simple Reeb orbits.

There could also be configurations with more complicated components in NΣ. An

argument similar to that of the previous section implies that the pseudo-holomorphic

components of such configurations correspond to pseudo-holomorphic maps w : CP 1 Ñ

Σ, together with meromorphic sections of wNΣ.

5.2 Pseudo-holomorphic curves in W and relative

Gromov–Witten numbers of pX,Σq

When we split X and W , we get a piece that is symplectomorphic to W , with a con-

stant Hamiltonian. So, as in NΣ, the components of split Floer trajectories contained

in W are pseudo-holomorphic curves, possibly connected to gradient flow lines of a

Morse function fW in W . Pseudo-holomorphic curves in W for a J that is cylindri-

cal at infinity can be identified with closed holomorphic curves in X intersecting Σ.

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 59

These are the curves described by relative Gromov-Witten numbers.

5.2.1 Relative Gromov–Witten numbers

Let us quickly review relative Gromov–Witten theory. For details, we refer the reader

to [IP03]. Similarly to the absolute case, discussed above, we will use a Morse–Bott

chain level version of relative Gromov–Witten numbers.

Let fΣ : Σ Ñ R be a Morse function and fX : X Ñ R be a Morse–Bott func-

tion. Suppose that Σ is a critical manifold of fX , with no other critical manifolds of

dimension greater than zero, and that the global maximum of fΣ is attained at Σ.

We will sometimes not be careful in distinguishing fX from fW . We also choose a

Riemannian metric g in X, with respect to which we define the Morse flows of fX and

fΣ. Denote these flows by φsfX and φsfΣ, respectively, where time is measured by the

variable s P R. The critical points of fX and fΣ should be thought of as the critical

points of a Morse function on X, obtained by using fΣ to perturb fX near Σ (as in

[BH11]). Given x P CritpfΣq, let W sfXpxq : W s

fΣpxq and

W ufXpxq : W u

fX

W ufΣpxq

!a P X| lim

sÑ8φsfX paq P W

ufΣpxq

).

As in Morse theory, one can use Morse–Bott chains formed by stable and unstable

manifolds to define pseudo-cycles in X, which can be used to compute its singular

homology (see [Fra04] and [BH11]).

We will also need an almost complex structure in X, such that Σ is an almost

complex submanifold. Take one that extends the previously chosen J in Σ, and denote

this extension also by J . Given k P Z¥0, ~s ps1, . . . , slq P Zl¡0 and A P H2pXq, we

can define moduli spacesMA,k,~s pX; Σq of pseudo-holomorphic map CP 1 Ñ X (whose

image is not entirely contained in Σ), with k l disjoint marked points, m of which

mapping into Σ, with orders of tangency to Σ prescribed by the entries of the vector

~s. These spaces have evaluation maps defining pseudo-cycles

ev :MA,k,~s pX; Σq Ñ Xk Σl

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 60

of dimension 2n 2 xc1pTXq, Ay 2k l

°li1 si

6. Given Morse–Bott chains

C1, . . . , Ck in X and Morse chains B1, . . . , Bl in Σ, we define the relative Gromov–

Witten number GWX,ΣA,k,~s

C1, . . . , Ck;B1, . . . , Bl

, to be the intersection number of

the pseudo-cycles defined by MA,k,~s pX; Σq and by C1 . . . Ck B1 . . . Bl, in

Xk Σl.

A useful property of relative Gromov–Witten numbers is that, when the intersec-

tions with Σ are all transverse, they can be expressed in terms of absolute Gromov–

Witten numbers of X: if ~s p1, . . . , 1q and #pAX Σq |~s|,1 then

GWX,ΣA,k,~s pC1, . . . , Ck;B1, . . . , B|~s|q GWX

A,k|~s| pC1, . . . , Ck, B1, . . . , B|~s|q. (5.1)

This requires a slight extension of the above description of Morse chain level absolute

Gromov–Witten numbers, to the case when X has a Morse–Bott function and its

critical manifolds have auxiliary Morse functions.

One important point in the proof of (5.1) is to justify that there are generically

no relevant holomorphic curves contained inside Σ. These might contribute to the

absolute Gromov–Witten number, but not to the relative number, by definition. The

following result rules out this possibility.

Lemma 5.3. Fix a vector ~s p1, . . . , 1q and A P ImagepH2pΣ;Zq Ñ H2pX;Zqq such

that2 #pA X Σq ¥ |~s|. Suppose that there are chains Ci in X and Bj in Σ, so that

GWXA,k|~s| pC1, . . . , Ck, B1, . . . , B|~s|q 0. Generically, there are no maps u : CP 1 Ñ X

contributing to this count, such that Imagepuq Σ.

Proof. If the absolute Gromov–Witten number in non-zero, then,

dimMA,k|~s|pXq k

i1

p2n dimCiq

|~s|

j1

p2n dimBjq

1The assumption #pA X Σq |~s| cannot be removed: consider the example of pX,Σq pCP 3,CP 2q. Denoting the complex-oriented generator of H2pCP 3;Zq by L, we have

GWCP 3

L,2 ppt, ptq 1, but GWCP 3,CP 2

L,0,p1,1q ppt, ptq 0, because a holomorphic curve in homology class L

going through two points in CP 2 is contained in CP 2 and therefore does not contribute to a relativeGromov–Witten number.

2The condition #pAX Σq ¥ |~s| does not hold in the example of the previous footnote.

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 61

and so

2n 2 xc1pTXq, Ay 2pk |~s|q 6 2npk |~s|q k

i1

dimCi

|~s|

j1

dimBj. (5.2)

We wish to rule out the existence of holomorphic curves contained entirely in Σ that

might contribute to this non-zero count. If such configurations existed, then we would

have

dimMA,k|~s|pΣq ¥k

i1

p2n 2q pdimCi 2q

|~s|

j1

p2n 2q dimBj

.

Therefore,

2n 2 2 xc1pTΣq, Ay 2pk |~s|q 6 ¥ 2npk |~s|q k

i1

dimCi

|~s|

j1

dimBj 2|~s|.

This fact, together with xc1pTΣq, Ay xc1pTXq, Ay xc1pNΣq, Ay, xc1pNΣq, Ay

#pAX Σq ¥ |~s| and (5.2), implies a contradiction and proves the lemma.

Remark 5.3. As observed by Maulik and Pandharipande in [MP06], the relative

Gromov–Witten numbers of a pair pX,Σq can often be obtained from the (absolute)

Gromov–Witten numbers of X and of Σ, and from the map HpXq Ñ HpΣq.

5.2.2 Pseudo-holomorphic curves in W

Let J be a generic almost complex structure in X, for which Σ is an almost complex

submanifold and J is cylindrical near Σ. Such J also defines a cylindrical almost

complex structure on W .

The following result tells us that pseudo-holomorphic curves inW can be described

in terms of relative Gromov–Witten numbers of pX,Σq. Fix points p1, . . . , pm P Σ,

corresponding to simple Reeb orbits γ1, . . . , γm in Y .

Proposition 5.1. Given positive integers k1, . . . , km, there is a bijective correspon-

dence between J-holomorphic spheres in X that intersect Σ precisely at the points pi

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 62

Σ

X W

p1

k1

p2

k2

γk11 γk2

2

Figure 5.4: Pseudo-holomorphic curves in X and W

with order of tangency ki, and J-holomorphic curves in W of genus 0 with punctures

asymptoting to ki-covers of the γi.

Relative Gromov–Witten numbers are important for our purposes for three rea-

sons. The first is that, as a consequence of the previous Proposition, they count aug-

mentation planes in W capping negative punctures of Floer and pseudo-holomorphic

curves in R Y . The second reason is that they contain information about the split

Floer homology differential connecting non-constant orbits in R Y with constant

orbits in W (see Figure 5.5). Finally, they also describe some broken configurations

contributing to the pair-of-pants product, as we will see later. For the moment, let

us focus on the second application of relative numbers, to the differential connecting

non-constant and constant orbits.

Suppose that the relative number GWX,ΣA,1,pkq

W ufXpxjq;W

ufΣpqiq

is non-vanishing,

for certain critical points qi P Σ and xj P XzΣ. Let the pseudo-holomorphic map

w : CP 1 Ñ Σ, such that wp0q P W ufΣpqiq and wp8q P W u

fXpxjq, contribute to this

count. The next result is analogous to Lemma 5.2.

Lemma 5.4. The map w gives a contribution pk . xjq to the split symplectic homology

differential of qqi,k.

Proof. For each b P S1, we have a new pseudo-holomorphic map w, such that wpzq

wpbzq. Since the symplectic homology differential fixes markers, as in non-equivariant

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 63

qqi,k

xj

qqi,k

Figure 5.5: The differential d_M

contact homology, we need to determine for how many values of b P S1 we can fix the

marker of qqi,k. Since w intersects Σ with order of tangency k, it can be written, up

to lower order terms, as

z ÞÑ pzk, 0 . . . , 0q.

near z 0 P CP 1 (on the domain) and near the intersection point (on the target).

We now see that we can impose a marker condition on qqi,k for k values of b P S1,

which implies the result.

5.3 Floer and symplectic homology via Gromov–

Witten theory

We are now ready to write formulas for the differential and product in split symplectic

and Floer homologies, in terms of holomorphic curves and gradient flow lines.

5.3.1 Symplectic homology

Assume fW : W Ñ R is a Morse function and fΣ : Σ Ñ R is a perfect Morse function

on Σ with critical points

CritpfΣq tq1, . . . , qmu.

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 64

Recall that this means that the Morse differential vanishes. Since Y is a pre-quantization

bundle over Σ, then (if M is not an integer),

CCpY q M

mài1

tMuàk1

Λ@qi,k

Dis the truncation of the contact homology chain complex by orbits of period less than

M . According to (3.2), the chain complex for split symplectic homology of W is

CSpW q ~CHpY q `zCHpY q ` CMpfW qrns

where CHpY q is a copy of the chain complex CHpY q in which we denote the

generators as qqi,k and yCHpY q is a copy of the chain complex CHpY qr1s (degree

shift of 1) in which we denote the generators as pqi,k. CMpfW qrns is the Morse

complex of fW with a degree shift of n.

The differential in CSpW q is the 3 3 matrix

d

0 0 0

d_^ 0 0

d_M 0 dM

The vanishing of so many terms is justified by index considerations and by the fact

that Y has no bad orbits, in the sense of [EGH00]. If such orbits existed, then there

would be a non-zero term d^_, on the first row and second column of the differential

matrix (see Lemma 4.28 in [BO09b]). There are no bad orbits in our setting, as a

consequence of the index computation in Lemma 3.4. Recall that, according to this

result, if we choose trivializations along periodic Reeb orbits by using capping planes

that intersect Σ, then the parity of the index of a Reeb orbit does not change if one

changes the multiplicity of the orbit.

The term dM is the Morse differential for the function fW on W . The term

d_^ : CHpY q Ñ yCHpY q counts cascades of pseudo-holomorphic buildings. It has

two types of contributions: from pseudo-holomorphic cylinders in RY (possibly with

punctures capped in W ) and from Morse flow lines in the spaces of orbits (see Figure

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 65

qqi,k

pqj,l qa1,l1 qar,lr

qqi,k

qqi,k pqj,kpqj,k

Figure 5.6: The differential d_^

5.6). The term d_M : CHpY q Ñ CMpfW qrns counts mixed curves connecting

Reeb orbits to critical points of fW (see Figure 5.5).

The term d_^ can be written as follows:

d_^ qqi,k ¸qjPCritpfΣq

¸l,r

¸pqa1 ,...,qar q

¸APH2pΣq

¸AiPH2pXq

δdpAq,l°lαk . cpk; l, l1, . . . , lrq .

. GWΣA,r2

W ufΣpqiq,W

sfΣpqjq,W

sfΣpqa1q, . . . ,W

sfΣpqarq

.

. GWX,ΣA1,0,pl1q

H;W u

fΣpqa1q

. . .GWX,Σ

Ar,0,plrq

H;W u

fΣpqarq

pqj,l@c1pY Ñ Σq,W u

fΣpqiq XW s

fΣpqjq

D pqj,kwhere r indexes the number of augmentations and dpAq xc1pY Ñ Σq, Ay. The

Kronecker deltas ensure that we consider meromorphic sections of line bundles with

the correct difference of number of zeros and poles. The coefficients cpk; l, l1, . . . , lrq

were described in Section 5.1.2. The last term in the formula accounts for (Morse–

Bott) gradient flow lines in Y connecting qqi,k with pqj,k.Remark 5.4. The assumption that fΣ is a perfect Morse function simplifies our com-

putations a bit. Otherwise we would need to include additional terms in the formula,

namely terms d__ and d^^ corresponding to rigid gradient flow lines in Σ (these

terms would also appear in the contact homology differential of Y ).

Remark 5.5. As will be seen in Chapter 6, some contributions to d_^ consist of

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 66

terms with more than one end corresponding to the same qi P CritpfΣq. This is the

case, for instance, when the relevant Gromov–Witten invariants count curves in class

0 P H2pΣ,Zq (we will see this, for example, in the terms dqmk1 2pmk . . . in

the differential of symplectic homology of T S2). For these curves, the perturbation

terms in the Cauchy–Riemann equation on Σ are crucial. They imply that one such

configuration does not project to a single point in Σ, or, put differently, that it is not

a cover of a trivial cylinder. This is what allows us to achieve transversality with a

single Morse function fΣ, instead of needing distinct functions for different punctures

in the domains of our pseudo-holomorphic curves.

To describe the terms d_M , denote first CritpfW q tx1, . . . xm1u. Then,

d_M qqi,k k .¸

xjPCritpfW q

¸APH2pXq

δ#pΣXAq,k . GWX,ΣA,1,pkq

W ufWpxjq;W

ufΣpqiq

xj

The presence of the coefficient k was justified in Lemma 5.4. The only relevant

homology classes A are those in the image of the map H2pXzΣq Ñ H2pXq. The

Kronecker delta forces the pseudo-holomorphic curves in X to intersect Σ at only one

point, with order of tangency k. For a schematic representation, recall Figure 5.5.

There is a similar description of the pair-of-pants product. Products involving only

non-constant orbits are given not just by rigid pseudo-holomorphic pairs-of-pants in

R Y , but also by certain broken configurations.

We begin with a description of (possibly augmented) pairs-of-pants in R Y (see

Figure 5.7). Since CP 1zt0, 1,8u has no automorphisms, we can only fix one marker

on a pseudo-holomorphic map CP 1zt0, 1,8u Ñ R Y , using the fact that J is S1-

invariant (contrary to the case of holomorphic cylinders, in which we could fix two

markers). As a consequence, in the product we don’t need to consider analogues of

the coefficients cpk; l, l1, . . . , lrq that appeared in the differential (recall Definition 5.1).

Suppose that we are interested in the product of qqi1,k1 by pqi2,k2 , for example. Then,

a rigid mapCP 1zt0, 1,8u

ztaugmentation puncturesu Ñ R Y , lifting a pseudo-

holomorphic sphere that contributes to a pk3q-point Gromov–Witten number of Σ,

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 67

pqi2,k2

qqj,l qa1,l1 qar,lr

qqi1,k1 pqi2,k2

qqj,k

qqi1,k1

Figure 5.7: Augmented and non-augmented pairs-of-pants contributing to the product

would give

qqi1,k1 pqi2,k2 ¸

qjPCritpfΣq

¸l,r

¸pqa1 ,...,qar q

¸APH2pΣq

¸AiPH2pXq

δdpAq,l°lαk1k2 .

. GWΣA,r3

W ufΣpqi1q,W

ufΣpqi2q,W

sfΣpqjq,W

sfΣpqa1q, . . . ,W

sfΣpqarq

.

. GWX,ΣA1,0,pl1q

H;W u

fΣpqa1q

. . .GWX,Σ

Ar,0,plrq

H;W u

fΣpqarq

qqj,l . . .

where dpAq xc1pY Ñ Σq, Ay. As in the case of the differential, the Kronecker deltas

keep track of zeros and poles of meromorphic sections of line bundles over CP 1. There

are similar contributions in the case when we multiply two generators in yCH, and

the output is in yCH (in which case the marker is fixed on the output orbit).

In the case when there are no augmentation planes, which often happens for degree

reasons, we get

qqi1,k1 pqi2,k2 ¸

qjPCritpfΣq

¸l

¸APH2pΣq

δdpAq,lk1k2 .

. GWΣA,3

W ufΣpqi1q,W

ufΣpqi2q,W

sfΣpqjq

qqj,l . . .

There are also broken configurations contributing to the product. These are repre-

sented in Figure 5.8. They have several components: a cylinder in RY that connects

to another cylinder in W , with one removable singularity. At this point, it connects

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 68

qqi,k

qqi,k

pqj,l

qqj,l

Figure 5.8: Broken pair-of-pants

to a gradient flow line of fW that escapes to infinity and continues as a vertical line in

RY . This vertical line intersects a pair-of-pants at a removable singularity. In prin-

ciple, both holomorphic components in R Y could be something other than (covers

of) non-trivial cylinders. But we will focus on the case when they are, which is the

one relevant for our applications in Chapter 6. Note that in Figure 5.8 we drew the

pair-of-pants with removable singularity in a slightly slanted manner. This is because

the pair-of-pants has a stable domain, so our perturbation scheme from Section 4.2

implies that we get a lift of a perturbation of a constant map to Σ, in other words

a perturbation of a cover of a trivial cylinder. These broken configurations can be

counted as follows:

qqi,k qqj,l ¸qjPCritpfΣq

¸l

¸APH2pΣq

δ#pΣXAq,k . GWX,ΣA,1,piq

W ufWpqjq;W

ufΣpqiq

qqj,l . . .

The pseudo-holomorphic curves contributing to the relative Gromov–Witten number

describe the pseudo-holomorphic planes in W , which are actually cylinders with a

removable singularity, and the stable manifold for fW contains the gradient flow

lines going from the removable singularities to infinity in W . The Kronecker delta

selects pseudo-holomorphic planes with only one puncture, asymptotic to a Reeb

orbit of period k (or, equivalently, pseudo-holomorphic spheres in X with only one

intersection with Σ, of order k).

Remark 5.6. At this point, we should make a comment similar to that of Remark

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 69

pqi2,k2

xj

pqi2,k2

pqi1,k1

pqi1,k1

Figure 5.9: Constant orbit contributing to the product of two non-constant orbits

5.5. The fact that the pair-of-pants in R Y with removable singularity is only a

perturbation of a cover of a trivial cylinder is crucial to achieve transversality of the

evaluation maps at the punctures, for a single Morse function fΣ.

The product of two non-constant orbits can also have a contribution from constant

orbits (illustrated in Figure 5.9). The relevant part of these configurations is contained

in W , where there is no J-preserving S1-action. Therefore, no markers can be fixed.

We get

pqi1,k1 pqi2,k2 ¸

xjPCritpfW q

¸APH2pXq

δ#pΣXAq,k1k2 .

. GWX,ΣA,1,pk1,k2q

W ufWpxjq;W

ufΣpqi1q,W

ufΣpqi2q

xj . . .

The product of critical points should coincide with the Morse chain level prod-

uct. Usually, to define this operation, one needs more than one Morse function, for

transversality reasons. This can be avoided at the expense of replacing gradient flow

trees with gradient flow lines of one fixed Morse function, connected to a central

(perturbed) pseudo-holomorphic curve.

These formulas will be used in Chapter 6, to compute the ring structure on sym-

plectic homology groups of spheres.

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CHAPTER 5. RELATION WITH GROMOV–WITTEN NUMBERS 70

5.3.2 Floer homology

The split Floer homology of an S-shaped Hamiltonian in X has a similar description.

Let again fW : W Ñ R, fΣ : Σ Ñ R and fNΣ : NΣ Ñ R be Morse functions.

According to 3.1, the chain complex for split Floer homology of X is

HFpXq CHcvx

pY q ` yCHcvx

pY q ` CMpfW qrns`

` CHcve

pY q ` yCHcve

pY q ` CMpfNΣqrns

where the superscripts cvx and cve refer to the periodic orbits in the region where the

Hamiltonian is convex and concave, respectively. The degree of a concave generator

is 1 more than that of the corresponding convex generator. This complex is generated

over the Novikov ring Zrt, t1s. The Novikov variable t counts the number of positive

punctures on split Floer differentials, capped by fibers of NΣ Ñ Σ.

One can now write a 6 6 matrix representing the split Floer differential, whose

entries are explicitly described by absolute and relative Gromov–Witten numbers, as

well as some Morse-theoretic information. There is a description of the product that is

similar to the one given above for symplectic homology. An important difference when

the manifold is closed (and thus not exact) is that there might be product terms that

involve only critical points and that contain (broken) pseudo-holomorphic spheres

whose energy is not small (which means that they intersect Σ). This is because the

ring structure on Floer homology is isomorphic to quantum cohomology.

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Chapter 6

The example of cotangent bundles

of spheres

We will restrict our attention to pairs pX,Σq pQn, Qn1q, where Qn is the n-

dimensional complex projective quadric. QnzQn1 is the n-dimensional affine complex

quadric. When Qn is equipped with the restriction of the Fubini-Study symplectic

form that generates H2pCP n1;Zq, then QnzQn1 is symplectomorphic to the unit

cotangent bundle of Sn. Therefore, W , the completion of QnzQn1, is symplectomor-

phic to T Sn (see Exercise 6.20 in [MS98]). We will use the results of the previous

chapters to compute the symplectic homology rings SHpTSnq, for n ¡ 1.

As we will see in Proposition 6.1, Qn is a monotone manifold, with λQn 1n, if

n ¡ 1, and λQ1 12. Also, Qn1 PDpωq, if n ¡ 1, so K 1.

6.1 T S2

The one and two dimensional quadrics Q1, Q2 are isomorphic to CP 1 and CP 1CP 1,

respectively, as Kahler manifolds. Q1 includes into Q2 as the diagonal embedding

∆ : CP 1ãÑ CP 1 CP 1. In this case, Y is diffeomorphic to RP 3, and the Reeb flow

corresponds to the flow along the fibers of the bundle S1 Ñ RP 3 Ñ S2, of Chern

class 2 (see Remark 3.2 above and Proposition 6.1, below).

71

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 72

6.1.1 Relevant Gromov–Witten numbers

Let us now compile some Gromov–Witten numbers that will be relevant for our

computations. We will use the integrable complex structures in CP 1 and CP 1CP 1,

with respect to which the moduli spaces of holomorphic spheres are transverse (see

Lemma 3.3.1 and Example 3.3.6 in [MS04]). Call L the generator of H2pCP 1;Zq,corresponding to the complex orientation. Using the divisor equation, we have

GWCP 1

L,2 ppt, ptq 1

#pptX LqGWCP 1

L,3 ppt, pt, ptq 1.

This is because there is a unique holomorphic curve in class L, mapping 0, 1 and 8

to three generic points in CP 1. Since (unperturbed) pseudo-holomorphic curves in

homology class 0 P H2pCP 1;Zq are constant, we have

GWCP 1

0L,3

pt,CP 1,CP 1

1.

We also need some relative Gromov–Witten numbers of the pair pCP 1CP 1,∆q.

Denote by L1 and L2 the generators of H2pCP 1CP 1;Zq corresponding to holomor-

phic spheres on each of the factors. Then

GWCP 1CP 1,∆Li,0,p1q

pH; ptq GWCP 1CP 1

Li,1pptq 1

for i 1, 2 (the point constraint is in ∆, not in CP 1 CP 1). This is because if one

fixes a point p P CP 1CP 1, then there is exactly one holomorphic sphere in class Li

that goes through p.

We also have

GWCP 1CP 1,∆Li,1,p1q

pLj; ptq GWCP 1CP 1

Li,2pLj, ptq 1 δi,j.

for i, j 1, 2. This means, for instance, that there is a unique vertical sphere in

CP 1 CP 1 intersecting a horizontal sphere and a generic point in CP 1 CP 1, but

that there is no horizontal sphere intersecting another horizontal sphere and a generic

point in CP 1CP 1. This is because two different horizontal spheres do not intersect.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 73

The divisor equation also implies that

GWCP 1CP 1

Li,3ppt, Lj, Lkq #pLi X LkqGWCP 1CP 1

Li,2ppt, Ljq p1 δi,kqp1 δi,jq

which will end up being useful in the study of the symplectic homology of T S3.

Important are also the numbers

GWCP 1CP 1,∆Li,1,p1q

ppt; ∆q GWCP 1CP 1

Li,2ppt,∆q GWCP 1CP 1

Li,2ppt, L1 L2q

GWCP 1CP 1

Li,2ppt, L1q GWCP 1CP 1

Li,2ppt, L2q 1

for i 1, 2. This expresses the fact that, if we fix a generic point p P CP 1CP 1, then

there is a unique holomorphic sphere in class Li that goes through p and intersects

∆.

Finally, we have the following.

Lemma 6.1. 1. GWCP 1CP 1,∆L1L2,1,p2q

ppt; ptq 1;

2. GWCP 1CP 1,∆L1L2,1,p1,1q

ppt; pt, ptq GWCP 1CP 1

L1L2,3ppt, pt, ptq 1;

3. GWCP 1CP 1,∆2Li,1,p1,1q

ppt; pt, ptq GWCP 1CP 1

2Li,3ppt, pt, ptq 0, for i 1, 2.

Proof. Even though there are effective ways of expressing relative Gromov–Witten

numbers in terms of absolute numbers, we will compute these explicitly, using the

fact that the integrable complex structure is generic enough.

For the first one, it is enough to show that, for instance, there is a unique holo-

morphic map f : CP 1 Ñ CP 1 CP 1, such that rf s L1 L2 P H2pCP 1 CP 1;Zq,fp8q p8, 1q, fp0q p0, 0q, and such that f intersects the diagonal ∆ CP 1CP 1

in a non-transverse way. Since

fp8q fr1; 0s

p8, 1q

r1; 0s, r1; 1s

,

we can write in homogeneous coordinates

frz; 1s

raz b; 1s, rz c; z ds

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 74

which we will abbreviate as

fpzq

az b,

z c

z d

.

Since fp0q p0, 0q, we get b c 0, and so fpzq az, zpz dq

. Now, for the

tangency condition, note that

f 1pzq

a,

d

pz dq2

and so f 1p0q

a, 1d

. So f is tangent to the diagonal at p0, 0q precisely when

a 1d. Therefore, our space of maps f can be identified with the space of a P C.

Taking a quotient by the group of automorphisms of the domain pCP 1, t0,8uq, which

is also C, we get GWCP 1CP 1,∆L1L2,1,p2q

ppt; ptq 1, as wanted.

Now, to show that GWCP 1CP 1

L1L2,3ppt, pt, ptq 1, it is enough to show that, for

instance, there is a unique holomorphic map f : CP 1 Ñ CP 1 CP 1 such that

rf s L1 L2, fp8q p8, 1q, fp0q p0, 0q and fp1q p3, 4q. As we already saw,

the first two point constraints imply that fpzq az, zpz dq

. Now,

fp1q

a,

1

1 d

p3, 4q

implies that a 3 and d 34. So, f is uniquely specified, which shows that the

required Gromov–Witten number is 1.

Finally, the fact that GWCP 1CP 1

2Li,3ppt, pt, ptq 0, for i 1, 2, just encodes the fact

holomorphic curves in classes 2Li P H2pCP 1 CP 1;Zq are covers of either vertical

or horizontal spheres, and therefore cannot go through three generic points in CP 1

CP 1.

6.1.2 The group SHpTS2q

Σ CP 1 admits a perfect Morse function with two critical points. Call the minimum

m and the maximum M . W T S2 has a Morse function that grows at infinity and

has two critical points, both located on the zero section. Call the minimum e and

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 75

the saddle point c. Let this be the function fW of Section 5.3.1. The split symplectic

homology chain complex for a J-shaped Hamiltonian in R RP 3 is therefore

SCpTS2q Z

Ae, c, qmk, pmk,|Mk,xMk

Ewhere we take all integers k ¡ 0.

We can apply the results in the previous chapter to compute the differential.

dqmk1 d_^ qmk1 d_M qmk1 d_^ qmk1

cpk 1; k, 1q . GWCP 1

0L,3ppt,CP 1,CP 1q . GWCP 1CP 1,∆L1,0,p1q

pH; ptq

cpk 1; k, 1q . GWCP 1

0L,3ppt,CP 1,CP 1q . GWCP 1CP 1,∆L2,0,p1q

pH; ptq pmk

cpk 1; k 1q . GWCP 1

L,2 ppt, ptqxMk1

2 pmk 2 xMk1

d|Mk d_^|Mk d_M|Mk d_^|Mk

cpk; k 1, 1q . GWCP 1

0L,3ppt,CP 1,CP 1q . GWCP 1CP 1,∆L1,0,p1q

pH; ptq

cpk; k 1, 1q . GWCP 1

0L,3ppt,CP 1,CP 1q . GWCP 1CP 1,∆L2,0,p1q

pH; ptq xMk1

c1pRP 3 Ñ CP 1qpCP 1q pmk

2 xMk1 2 pmk

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 76

dqm2 d_^ qm2 d_M qm2

cp2; 1, 1q . GWCP 1

0L,3ppt,CP 1,CP 1q . GWCP 1CP 1,∆L1,0,p1q

pH; ptq

cp2; 1, 1q . GWCP 1

0L,3ppt,CP 1,CP 1q . GWCP 1CP 1,∆L2,0,p1q

pH; ptq pm1

GWCP 1CP 1,∆L1L2,1,p2q

ppt; ptq e

2 pm1 2 e

d|M1 d_^|M1 d_M|M1

c1pRP 3 Ñ CP 1qpCP 1q pm1

GWCP 1CP 1,∆L1,1,p1q

ppt; ∆q GWCP 1CP 1,∆L2,1,p1q

ppt; ∆qe

2 pm1 2 e

dqm1 d_^ qm1 d_M qm1 d_M qm1

GWCP 1CP 1,∆L1,1,p1q

W ufWpcq; pt

GWCP 1CP 1,∆

L2,1,p1q

W ufWpcq; pt

c

GWCP 1CP 1,∆L1,1,p1q

pS2; ptq GWCP 1CP 1,∆L2,1,p1q

pS2; ptqc

GWCP 1CP 1,∆L1,1,p1q

pL1 L2; ptq GWCP 1CP 1,∆L2,1,p1q

pL1 L2; ptqc

GWCP 1CP 1,∆L1,1,p1q

pL1; ptq GWCP 1CP 1,∆L1,1,p1q

pL2; ptq

GWCP 1CP 1,∆L2,1,p1q

pL1; ptq GWCP 1CP 1,∆L2,1,p1q

pL2; ptqc

p0 1 1 0q c 0

where we assume that S2 DS2ãÑ CP 1 CP 1 is oriented so as to define the

homology class p1,1q P H2pCP 1CP 1;Zq (the opposite choice of orientation would

also give a vanishing result).

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 77

Summing up, we have $''''&''''%dqmk1 2 pmk 2 xMk1

d|Mk 2pmk 2 xMk1

dqm2 2 pm1 2 e

d|M1 2 pm1 2 e

for k ¥ 2. Therefore,

SHpTS2;Zq Z

Ac, qm1, e,|Mk qmk1,xMk

E` Z2

Ae pm1,xMk pmk1

Etaking all k ¥ 1.

We can compare these results with the computations of Cohen-Jones-Yan, which

we recalled in Section 2.3. The indices on SHpTS2q can be computed using Lemma

3.4 and the fact that K 1 and λQ2 12. The following tables show how our

computations match the ones from algebraic topology. For the free part, we get

SHdpTS2q c qm1 e |Mk qmk1

xMk

HdpLS2q a b 1 bvk vk

d 0 1 2 2k 1 2k 2

for k ¥ 1. The choice of signs is motivated from the study of ring structure, which

will be described below. For the Z2-torsion part:

SHdpTS2q e pm1

xMk pmk1

HdpLS2q av avk1

d 2 2k 2

for k ¥ 1.

6.1.3 The ring SHpTS2q

We now compute the pair-of-pants product on SHpTS2q. To get the same result

as in [CJY04], we need to show that

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 78

qm1

qm1

xMk

|Mk

Figure 6.1: Broken pair-of-pants on T S2

1. a annihilates everything except for 1 and for the fact that a vk avk;

2. e 1 is the unit;

3. b2 b pbvkq pbvkq pbvlq 0;

4. b vk bvk;

5. vk vl vkl;

6. pbvkq vl bvkl;

7. pavkq pavlq 0;

8. pavkq vl avkl.

The product of two orbits in CH is zero, since one cannot fix two markers on a

pair-of-pants. This is the reason behind (3).

In the following, 3-point absolute Gromov–Witten numbers correspond to pseudo-

holomorphic pairs-of-pants in R RP 3, whereas 2-point relative Gromov–Witten

numbers correspond to either broken pairs-of-pants (as in the example of Figure 6.1)

or to critical points contributing to the product of non-constant orbits, as explained

in Section 5.3.1. The broken configurations are counted with a negative sign.

The product contains the pieces illustrated in Figure 6.1, and analogous ones with

mk in place of Mk. In order to apply the formulas from Section 5.3.1, which describe

them in terms of Gromov–Witten numbers, we need to determine W ufWppq for a generic

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 79

point p P ∆. Such an unstable manifold defines a cycle aL1bL2 P H2pCP 1CP 1;Zqthat intersects Σ precisely at p. This implies that

paL1 bL2q . pL1 L2q a b 1. (6.1)

On the other hand, H is a deformation of a Morse–Bott function that grows radially

on the fibers of T S2. Since the point x P Σ corresponds to a simple orbit of the

geodesic flow on S2, the Morse–Bott manifold of p is given by one hemisphere of the

zero section, glued to a copy of S1 r0,8q along the meridian in the zero section

that corresponds to the orbit of the geodesic flow. The first factor on S1 r0,8q

goes around that meridian and the second factor grows radially on the fibers. Such

Morse–Bott manifold perturbs to a disk that intersects the zero section only once,

say at the minimum e. Since the zero section defines L1 L2 P H2pCP 1 CP 1;Zq(for a certain choice of orientation), we have

paL1 bL2q . pL1 L2q a b 1. (6.2)

From (6.1) and (6.2), we conclude that pa, bq p0, 1q, and that aL1 bL2 L2.

We will not focus on the product formulas involving critical points, but check a

direct consequence of (1), namely that b pavq 0:

b pavq pqm1q pe pm1q

qm1

GWCP 1CP 1,∆L1,1,p1q

pL2; ptq GWCP 1CP 1,∆L2,1,p1q

pL2; ptqqm1

qm1 qm1 0

as wanted. For (4), observe that

b vk pqm1q xMk

GWCP 1

0L,3ppt,CP 1,CP 1q qmk1

GWCP 1CP 1,∆L1,1,p1q

pL2; ptq GWCP 1CP 1,∆L2,1,p1q

pL2; ptq |Mk

qmk1 |Mk bvk.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 80

For (5), we have

vk vl xMk xMl GWCP 1

0L,3pCP 1,CP 1, ptqxMkl

xMkl vkl.

For (6):

pbvkq vl |Mk qmk1

xMl

GWCP 1

0L,3pCP 1,CP 1, ptq|Mkl GWCP 1

0L,3ppt,CP 1,CP 1q qmkl1

|Mkl qmkl1 bvkl

For (7), there are three cases to consider: when k l 1

pavq pavq pe pm1q pe pm1q

e e e pm1 pm1 e pm1 pm1

e pm1 pm1 GWCP 1CP 1,∆L1L2,1,p1,1q

ppt; pt, ptq e

2 e 2 pm1 0

When k ¡ 1:

pavkq pavq pxMk1 pmkq pe pm1q

xMk1 e xMk1 pm1 pmk e pmk pml

xMk1 GWCP 1

0L,3pCP 1, pt,CP 1q pmk pmk GWCP 1

L,3 ppt, pt, ptqxMk1

2 xMk1 2 pmk 0

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 81

Finally, when k, l ¡ 1:

pavkq pavlq pxMk1 pmkq pxMl1 pmlq

xMk1 xMl1 xMk1 pml pmk xMl1 pmk pm1

GWCP 1

0L,3pCP 1,CP 1, ptqxMkl2 GWCP 1

0L,3pCP 1, pt,CP 1q pmkl1

GWCP 1

0L,3ppt,CP 1,CP 1q pmkl1 GWCP 1

L,3 ppt, pt, ptqxMkl2

2 xMkl2 2 pmkl1 0

For (8), we need to check two cases: when k 1

pavq vl pe pm1q xMl

xMl GWCP 1

0L,3ppt,CP 1,CP 1q pml1

xMl pml1 avl1

and, when k ¡ 1,

pavkq vl pxMk1 pmkq xMl

GWCP 1

0L,3pCP 1,CP 1, ptqxMkl1 GWCP 1

0L,3ppt,CP 1,CP 1q pmkl

xMkl1 pmkl avkl

6.2 T Sn

For higher n, the topology of Qn is more involved, so we need more preliminary work

before we can compute SHpTSnq.

6.2.1 The topology of QN

We will need perfect Morse functions on smooth complex projective quadrics QN °Nk0 z

2k 0

( CPN1. We will use the following result (see Proposition 2.4.22 in

[CG10]):

Proposition. Let X ãÑ CPN be a smooth complex projective variety and assume

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 82

that there is an algebraic action of C on CPN which preserves X and has finitely

many fixed points in X. Then X admits a perfect Morse function F , whose critical

points are the fixed points of the action. The gradient flow lines of F with respect to

the Kahler metric are the orbits of the R-action on X, given by R C.

The action of S1 C on X turns out to be Hamiltonian, for the induced Kahler

form, and F is the corresponding moment map. The critical submanifolds are complex

submanifods, so the indices of all critical points are even and F is perfect. The

statement about Morse trajectories follows from the assumption that the action is

holomorphic:

x∇F, .y dF ω p., XF q ω pJ., JXF q ω pJXF , J.q

xJXF , .y

BJ

B

, .

F

BB

Br

, .

F

ùñ ∇F

B

Br

Now, to construct a perfect Morse function on QN , one just needs to show that it

admits a C-action with finitely many critical points, and which extends to CPN1.

We will consider separatly the cases of N even and N odd. Before we begin, note

that, if we take homogeneous coordinates z0, . . . , zM on CPM , then under the change

of variables pzk, zlq ÞÑ pu, vq pzk izl, zk izlq, the expression z2k z2

l becomes uv.

When N 2m is even: Take coordinates z0, . . . , z2m1 on CP 2m1, and Q2m °nk0 z

2k 0

(. Change variables pz2k, z2k1q ÞÑ puk, vkq, k P t0, . . . ,mu, as above, so

that Q2m °m

k0 ukvk 0(. Now, let C act on CP 2m1 by

λ . ru0; v0;u1; v1; . . . ;um; vms ru0; v0;λ1u1;λv1; . . . ;λmum;λmvms, λ P C

This action clearly preserves Q2m, and its 2m 2 critical points are, all in Q2m:

r1; 0; . . . ; 0s, r0; 1; 0; . . . ; 0s, . . . , r0; . . . ; 0; 1s

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 83

We can apply that proposition and conclude that there is a perfect Morse function

on Q2m.

When N 2m 1 is odd: take coordinates z0, . . . , z2m on CP 2m, and Q2m1 °nk0 z

2k 0

(. Change variables pz2k1, z2kq ÞÑ puk, vkq, k P t1, . . . ,mu, as above, so

that Q2m1 z2

0 °mk1 ukvk 0

(. Now, let C act on CP 2m by

λ . rz0;u1; v1; . . . ;um; vms rz0;λ1u1;λv1; . . . ;λmum;λmvms, λ P C

As above, this action clearly preserves Q2m1, and its 2m 1 critical points are:

r1; 0; . . . ; 0s, r0; 1; 0; . . . ; 0s, . . . , r0; . . . ; 0; 1s

The first one is not in Q2m1, so the quadric has only 2m critical points. The propo-

sition implies again the existence of a perfect Morse function on Q2m.

We will now write down explicitly the stable and unstable manifolds for these

Morse functions, and compute the relevant intersections. In particular, we will de-

scribe the two dimensional spaces of gradient flow lines, and how the middle dimen-

sional classes intersect, when N is even. Both pieces of information will be used in

the symplectic homology computations.

When N 2m is even: given r P R

r . ru0; v0;u1; v1; . . . ;um; vms ru0; v0; r1u1; rv1; . . . ; rmum; rmvms

so

A1 : W upr1; 0; . . . ; 0sq

"x P Qn

limtÑ0

r . x r1; 0; . . . ; 0s

*

ru0; 0; 0; v1; . . . ; 0; vms

(

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 84

and

W spr1; 0; . . . ; 0sq

"x P Qn

limrÑ8

r . x r1; 0; . . . ; 0s

* ru0; 0;u1; 0; . . . ;um; 0s

(.

This shows that r1; 0; . . . ; 0s is a critical point of index N . Call it p1N . Similarly,

A2 : W upr0; 1; . . . ; 0sq r0; v0; 0; v1; . . . ; 0; vms

(and

W spr0; 1; . . . ; 0sq r0; v0;u1; 0; . . . ;um; 0s

(so r0; 1; . . . ; 0s is another critical point of index N . Call it p2

N . For the remaining

critical points, we have the following:

W upr0; . . . ; 0; 1loooomoooon2i1

; 0; . . .sq ru0; v0;u1; v1; . . . ;ui; vi; 0; vi1; . . . ; 0; vms

(and

W spr0; . . . ; 0; 1loooomoooon2i1

; 0; . . .sq r0; . . . ; 0;ui; 0;ui1; 0; . . . ;um; 0s

(so r0; . . . ; 0; 1loooomoooon

2i1

; 0; . . .s is a critical point of index N 2i. Call it pN2i. Also

W upr0; . . . ; 0; 1loooomoooon2i2

; 0; . . .sq r0; . . . ; 0; vi; 0; vi1; . . . ; 0; vms

(and

W spr0; . . . ; 0; 1loooomoooon2i2

; 0; . . .sq ru0; . . . ; vi;ui1; 0; . . . ;um; 0s

(therefore r0; . . . ; 0; 1loooomoooon

2i2

; 0; . . .s is a critical point of index N2i. Call it pN2i. Note that

the closures of the unstable manifolds of these points consist of projective hyperplanes

of complex dimension m i, for i ¡ 0. On the other hand, the W uppN2iq are

hyperplane sections QN X CPmi1.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 85

We thus have one critical point of every even index, except for index N , with two

critical points. Let us now determine the trajectories that connect critical points of

index difference 2:

W uppN2iq XW sppN2i2q r0; . . . ; 0; vi; 0; vi1; . . . ; 0; vms

(X

X ru0; . . . ; vi1;ui2; 0; . . . ;um; 0s

(

r0; . . . ; 0; vi; 0; vi1; 0; . . . ; 0s

( CP 1

for 0 ¤ i m, taking p2N when i 0. Also,

W upp1Nq XW sppN2q

ru0; 0; 0; v1; . . . ; 0; vms

(X

X ru0; v0;u1; v1;u2; 0; . . . ;um; 0s

(

ru0; 0; 0; v1; 0; . . . ; 0s

( CP 1.

Similarly,

W uppN2iq XW sppN2i2q ru0; v0;u1; v1; . . . ;ui; vi; 0; vi1; . . . ; 0; vms

(X

X r0; . . . ; 0;ui1; 0;ui; 0; . . . ;um; 0s

(

r0; . . . ; 0;ui1; 0;ui; 0; . . . ; 0s

( CP 1

for 0 i ¤ m, taking p1N when i 1. Also,

W uppN2q XW spp2Nq

ru0; v0;u1; v1; 0; v2; . . . ; 0; vms

(X

X r0; v0;u1; 0; . . . ;um; 0s

(

r0; v0;u1; 0; . . . ; 0s

( CP 1.

We have concluded that if ind p ind q 2, then W uppq XW spqq CP 1, the

generator of H2pQN ;Zq with complex orientation.

Now, we determine Ai . Aj, the (homology) intersection products. Recall that we

are still assuming N 2m. We will have to consider two cases separately: m odd

and m even:

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 86

• m odd: It will be useful to introduce the following family of complex submani-

folds of QN , for s P r0, 1s:

A2s :

rsv1; v0;sv0; v1; . . . ; svm; vm1;svm1; vms

(Note that A2

0 A2 and A21

rv1; v0;v0; v1; . . . ; vm; vm1;vm1; vms

(. On

homology,

A1 . A2 A1 . A21

ru0; 0; 0; v1; . . . ; 0; vms

(X

X rv1; v0;v0; v1; . . . ; vm; vm1;vm1; vms

(

r1; 0; 0; 1; 0; . . . ; 0s

( pt

Also

A2 . A2 A2 . A21

r0; v0; . . . ; 0; vms

(X

X rv1; v0;v0; v1; . . . ; vm; vm1;vm1; vms

( H

Similarly, one can show that A1 . A1 H.

Remark 6.1. We need these perturbations because, even though the Morse func-

tions that we chose on the QN are perfect, they are not Morse–Smale. So, we

need to make an additional slight perturbation to get the desired perfect Morse–

Smale functions fQN .

• m even: Take now

A2s :

r0; v0; sv2; v1;sv1; v2; . . . ; svm; vm1;svm1; vms

(Again, A2

0 A2. Now, A21

r0; v0; v2; v1;v1; v2; . . . ; vm; vm1;vm1; vms

(.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 87

On homology,

A1 . A2 A1 . A21

ru0; 0; 0; v1; . . . ; 0; vms

(X

X r0; v0; v2; v1;v1; v2; . . . ; vm; vm1;vm1; vms

( H

Also,

A2 . A2 A2 . A21

r0; v0; . . . ; 0; vms

(X

X r0; v0; v2; v1;v1; v2; . . . ; vm; vm1;vm1; vms

(

r0; 1; 0; . . . ; 0s

( pt

Similarly, one can show that A1 . A1 pt.

Another result we will need is that B : A1 A2 is non-primitive (which means

that it is Poincare dual to a multiple of the Kahler class ω; see more about primitive

cohomology at the beginning of Section 6.2.2), and that C : A1A2 is primitive. It

is the case that Qk QN is Poincare dual to ωNk. The fact that C is primitive is

equivalent to C . Qm 0. Also, proving that PDpBq ωm PDpQmq amounts to

showing that B . Ai Qm . Ai, for i 1, 2.

Qm QN X CPm1 inside CPN1

QN X ru0; v0;u1; 0; . . . ;um; 0s

(

ru0; 0;u1; 0; . . . ;um; 0s

(Y r0; v0;u1; 0; . . . ;um; 0s

(

2CPm

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 88

so

Qm . A1 ru0; 0;u1; 0; . . . ;um; 0s

(Y r0; v0;u1; 0; . . . ;um; 0s

(X

X ru0; 0; 0; v1; . . . ; 0; vms

( r1; 0; . . . ; 0s

( pt

and

Qm . A2 ru0; 0;u1; 0; . . . ;um; 0s

(Y r0; v0;u1; 0; . . . ;um; 0s

(X

X r0; v0; 0; v1; . . . ; 0; vms

( r0; 1; 0; . . . ; 0s

( pt.

We can now see that

C . Qm pA1 A2q . Qm 0

so C is primitive, and

B . A1 pA1 A2q . A1 pt Qm . A1

B . A2 pA1 A2q . A2 pt Qm . A2

so B PDpωmq and B Qm P HNpQN ;Zq.

Let us now consider the case when N 2m 1 is odd: given r P R

r . rz0;u1; v1; . . . ;um; vms rz0; r1u1; rv1; . . . ; rmum; rmvms

As before

W upr0; . . . ; 0; 1loooomoooon2i

; 0; . . .sq rz0;u1; v1; . . . ;ui; vi; 0; vi1; . . . ; 0; vms

(and

W spr0; . . . ; 0; 1loooomoooon2i

; 0; . . .sq r0; . . . ; 0;ui; 0;ui1; 0; . . . ;um; 0s

(

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 89

therefore r0; . . . ; 0; 1loooomoooon2i

; 0; . . .s is a critical point of index N 2i 1. Call it pN2i1.

Also

W upr0; . . . ; 0; 1loooomoooon2i1

; 0; . . .sq r0; . . . ; 0; vi; 0; vi1; . . . ; 0; vms

(and

W spr0; . . . ; 0; 1loooomoooon2i1

; 0; . . .sq rz0; . . . ; vi;ui1; 0; . . . ;um; 0s

(so r0; . . . ; 0; 1loooomoooon

2i1

; 0; . . .s is a critical point of index N 2i 1. Call it pN2i1. Note that

the unstable manifolds of these points consist of planes of complex dimension m i,

for i ¡ 0. On the other hand, the W uppN2i1q are hyperplane sections.

As above, we can describe the 2-dimensional spaces of connecting trajectories:

W uppN2i1q XW sppN2i1q r0; . . . ; 0; vi; 0; vi1; . . . ; 0; vms

(X

X rz0; . . . ; vi1;ui2; 0; . . . ;um; 0s

(

r0; . . . ; 0; vi; 0; vi1; 0; . . . ; 0s

( CP 1

for 0 i m.

W uppN1q XW sppN1q rz0;u1; v1; 0; v2; . . . ; 0; vms

(X

X rz0;u1; v1;u2; 0; . . . ;um; 0s

(

rz0;u1; v1; 0; . . . ; 0s

( Q1 2CP 1

W uppN2i1q XW sppN2i3q rz0;u1; v1; . . . ;ui; vi; 0; vi1; . . . ; 0; vms

(X

X r0; . . . ; 0;ui1; 0;ui; 0; . . . ;um; 0s

(

r0; . . . ; 0;ui1; 0;ui; 0; . . . ; 0s

( CP 1

for 1 i ¤ m. We have concluded that if ind p ind q 2, then W uppq XW spqq

CP 1, the positive generator of H2pQN ;Zq, except if ind p N 1. In that case,

W uppq XW spqq 2CP 1 P H2pQN ;Zq.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 90

The last result we will need about the topology of quadrics is a description of

relevant Chern classes. We need to fix some notation, first. Denote by NQN the

normal bundle for the inclusion QN ãÑ QN1. Recall that Q1 is biholomorphic to

CP 1 and denote by Q1 the generator of H2pQ1;Zq with the complex orientation.

Also, Q2 is biholomorphic to CP 1 CP 1; denote by Li, i P t1, 2u, the generators of

H2pQ2;Zq given by the coordinate spheres, with complex orientation. These Li are

homologous under the inclusion Q2 ãÑ QN , for N ¡ 2, and give a generator L of

H2pQN ;Zq.

Proposition 6.1. The following holds about the relevant first Chern classes:

1. xc1pTQ1q, Q1y 2 and xc1pNQ1q, Q1y 2;

2. xc1pTQ2q, Liy 2 and xc1pNQ2q, Liy 1, for i 1, 2;

3. xc1pTQNq, Ly N and xc1pNQNq, Ly 1, for N ¥ 3.

This is a consequence of the additivity of the first Chern class, applied to the

inclusions QN ãÑ CPN1 and QN ãÑ QN1. The formulas for Chern classes of tangent

bundles imply that the QN are monotone, with λQN 1N , if N ¡ 1, and λQ1 12.

The formulas for the Chern classes of normal bundles imply that QN1 PDpωq in

QN , if N ¡ 1, so K 1.

6.2.2 Gromov–Witten numbers of QN

We will extract the relevant Gromov–Witten numbers of complex projective quadrics

from known computations of their quantum cohomology rings. Note that QN

CPN1 is an example of a smooth complete intersection, which is a projective variety

C of dimension N cut out by r polynomials (of degrees d1, . . . , dr) inside CPNr.

If N ¥ 3, then the Lefschetz hyperplane theorem implies that H2pC;Zq Z. The

additivity of the first Chern class implies that c1pCq pN r 1°diqrωs, where

rωs P H2pC;Zq is the hyperplane class (which we will, for ease of notation, not be

careful to distinguish from the Kahler form ω). In particular, if N r 1°di ¡ 0,

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 91

then C is monotone. One can actually show that

HkpC;Zq

$&%Z for 0 ¤ k ¤ 2N even

0 for 0 ¤ k ¤ 2N odd

for k N . Let

HNpC;Qq0 :

$&%HNpC;Qq if N odd

kerpωN2|.q : HNpC;Qq Ñ Q

if N even

where px|yq ³Cx Y y is the Poincare pairing. Call HNpC;Qq0 the primitive coho-

mology of C. If N is even, then it turns out that HNpC;Qq QxωN2y`HNpC;Qq0.

For more on the topology of complete intersections, see Chapter 5 of [Dim92].

Denote the quantum product by , quantum powers by xk and usual cup powers

by xk. We will use the following result (see [Bea95]).

Theorem (Beauville). Let C CPNr be a smooth complete intersection of degree

pd1, . . . , drq and dimension N ¥ 3, with N ¥ 2°pdi 1q 1. Let d : d1 . . . dr and

δ :°pdi1q. The quantum cohomology algebra QHpC;Qq is the algebra generated

by the hyperplane class ω and the primitive cohomology HNpC;Qq0, with the relations:

ωpN1q dd11 . . . ddrr ω

δt ω x 0 x y px|yq1

d

ωN dd1

1 . . . ddrr ωpδ1qt

for x, y P HNpC,Qq0.

We apply this result to the case of a quadric C QN , which is a hypersurface of

degree 2. We get the following.

Corollary. The quantum cohomology algebra QHpQN ,Qq is the algebra generated by

the hyperplane class ω and the primitive cohomology HNpQN ,Qq0, with the relations:

ωpN1q 4ωt ω x 0 x y px|yq1

2

ωN 4t

for x, y P HNpQN ,Qq0.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 92

Recall that

x y ¸

APH2pQN q

px yqA txc1pTQN q,AyN ,

and that, according to Proposition 6.1, xc1pTQNq, Ly N , for N ¥ 3. Therefore, by

degree reasons, ω ωk ωk1 for 0 ¤ k ¤ N 2. Inductively, we get

ωk ωk

for 0 ¤ k ¤ N 1. For similar degree reasons,

ωN ω ωN1 ωN l0tL and ω ωN l1ωt

L

for some l0, l1 P Z. This implies that

ωpN1q ω ωN ω pωN l0tLq pl0 l1qωt

L.

By the Corollary above, we conclude that l0 l1 4. Below, we will argue that

l0 l1 2.

The quantum product on cohomology contains the information about genus 0,

3-point Gromov–Witten numbers. The relation is given by the formula

GWCA,3px, y, zq

px yqA|z

for x, y, z P HpCq. We will use Poincare duality to write Gromov–Witten numbers

with respect to homology: GWCA,3pPDCpxq,PDCpyq,PDCpzqq GWC

A,3px, y, zq.

Let us now compute some Gromov–Witten numbers of QN . We will make exten-

sive use of the associativity and (graded) commutativity of the quantum product. We

begin with GWQNL,3 ppt, L,Hq, where pt P QN is a point, L QN is a copy of CP 1 and

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 93

H QN X CPN is a hyperplane section.

GWQNL,3 ppt, L,Hq GWQN

L,3

1

2ωN ,

1

2ωN1, ω

1

4

pω ωNqL|ω

N1

1

4

»QN

l1ωN

l12

1

4

pω ωN1qL|ω

N

1

4

»QN

l0ωN

l02

Therefore, l0 l1 2, since we have seen already that l0 l1 4, and

GWQNL,3 ppt, L,Hq 1.

In QN , a holomorphic sphere u of class L intersects a generic hyperplane section H

at a unique point. This Gromov–Witten invariant tells us that there is a unique such

u intersecting a generic point and line. Using the divisor equation, we also get

GWQNL,2 ppt, Lq

1

#pLXHqGWQN

L,3 ppt, L,Hq 1.

Let us compute now

GWQNL,3 ppt, pt, QNq GWQN

L,3

1

2ωN ,

1

2ωN , 1

1

4

pωN 1qL|ω

N

1

4

»QN

0 0.

This is the expected answer: since generically there is a unique line through a point

and a line in QN , there should be no line through two generic points in QN .

Now

GWQN2L,3ppt, pt, ptq GWQN

2L,3

1

2ωN ,

1

2ωN ,

1

2ωN

1

8

pωN ωNq2L|ω

N

1

8

»QN

4ωN 1.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 94

We have used the fact that

ωN ωN pωN 2qLq2 ω2N 4ωNqL 4q2L

4ωqL ωN1 4ωNqL 4q2L 4q2L

Finally, if N is even and C PDQN pP q, where P P HNpQN ,Qq0, then

GWQNL,3 ppt, C, Cq GWQN

L,3

1

2ωN , P, P

1

2

pP P qL|ω

N

1

2

»QN

»QN

P Y P1

2pl0 4q

ωN

»QN

P Y P.

This integral can be computed explicitly: if P PDpCq PDpA1 A2q, as above,

then »QN

P Y P C . C pA1 A2q . pA1 A2q

A1 . A1 2A1 . A2 A2

#2 if m odd

2 if m even

so

GWQNL,3 ppt, C, Cq

#2 if m odd

2 if m even

On the other hand, for B A1 A2 PDpωmq,

GWQNL,3 ppt, B,Bq GWQN

L,3

1

2ωN , ωm, ωm

1

2

pωm ωmqL|ω

N

1

2

pωNqL|ω

N

1

2

pωN l0q

LqL|ωNl02

»QN

ωN 2

Also, since P is primitive,

GWQNL,3 pp,B,Cq GWQN

L,3

1

2ωN , ωm, P

1

2

pωm P qL|ω

N 0

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 95

One can now compute the remaining numbers:

GWQNL,3 ppt,A

1, A1q GWQNL,3

pt,

1

2pB Cq,

1

2pB Cq

1

4

GWQN

L,3 ppt, B,Bq 2 GWQNL,3 ppt, B, Cq GWQN

L,3 ppt, C, Cq

#1 if m odd

0 if m even

GWQNL,3 ppt,A

2, A2q GWQNL,3

pt,

1

2pB Cq,

1

2pB Cq

1

4

GWQN

L,3 ppt, B,Bq 2 GWQNL,3 ppt, B, Cq GWQN

L,3 ppt, C, Cq

#1 if m odd

0 if m even

GWQNL,3 ppt, A

1, A2q GWQNL,3

pt,

1

2pB Cq,

1

2pB Cq

1

4rGWQN

L,3 ppt, B,Bq GWQNL,3 ppt, C, Cqs

#0 if m odd

1 if m even

so

GWQNL,3 pA

i, Aj, ptq

#δi,j if m odd

1 δi,j if m eveni, j P t1, 2u

To deal with the torsion terms of symplectic homology of T SN1, when N

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 96

2m 1 is odd, we will also need

GWQNL,3

W uppN1q,pt,W

sppN1q GWQN

L,3 pCPm1, pt, Qmq

GWQNL,3

1

2ωm,

1

2ωN , ωm1

1

4

pωm ωm1qL|ω

N

1

4

pωNqL|ω

N

1

4

pωN l0q

LqL|ωNl04

»QN

ωN 1

For completeness, let us also write down the Gromov–Witten numbers in homology

class 0L, which correspond to the intersection product that was computed above:

GWQN0L,3ppt,QN , QNq #pptXQN XQNq 1

GWQN0L,3pA

i, Aj, QNq #pAi X Aj XQNq

#1 δi,j if m odd

δi,j if m eveni, j P t1, 2u

Finally, we will also need some relative Gromov–Witten numbers of pQN , QN1q,

namely

GWQN ,QN1

L,1,p1q ppt;Lq GWQNL,2 ppt, Lq 1

and

GWQN ,QN1

L,1,p1q pL; ptq GWQNL,2 pL, ptq 1.

6.2.3 The group SHpTSnq

The results of the previous sections can be used to compute SHpTSnq, for n ¡ 2.

As in the case of T S2, we use a Morse function on T Sn that grows at infinity

and has two critical points e (minimum) and c (saddle), to define the function fB of

Section 3.2. We begin with the case of even n. From the discussion above, we get the

chain complex

SCpTSnq Z xe, c, qqi,k, pqi,ky

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 97

where we take all even integers 0 ¤ i ¤ 2n 2 and all integers k ¡ 0. For the

differential, we have the following:

dqq0,l1 d_^qq0,l1 d_Mqq0,l1 d_^qq0,l1

cpl 1; lq . GWQn1

L,2 pL, ptq pq2n4,l pq2n4,l

for l ¥ 1.

dqq2,l1 d_^qq2,l1 d_Mqq2,l1 d_^qq2,l1

xc1pY Ñ Qn1q,CP 1y pq0,l1

cpl 1; lq . GWQn1

L,2 pL, ptq pq2n2,l pq0,l1 pq2n2,l

for l ¥ 1.

dqq2k,l d_^qq2k,l d_Mqq2k,l d_^qq2k,l

xc1pY Ñ Qn1q,CP 1y pq2k2,l pq2k2,l

for 2 ¤ k ¤ n, k n2, l ¥ 1.

dqqn,l d_^qqn,l d_Mqqn,l d_^qqn,l xc1pY Ñ Qn1q, 2CP 1y pqn2,l 2 pqn2,l

for l ¥ 1.

dqq2,1 d_^qq2,1 d_Mqq2,1

xc1pY Ñ Qn1q,CP 1y pq0,1 GWQn,Qn1

L,1,p1q ppt;Lq e pq0,1 e.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 98

Summing up, the differential in SHpTSnq, for n ¡ 2 even, is$'''''''''''''''''''''&'''''''''''''''''''''%

dqq0,l1 pq2n4,l

dqq2,l1 pq2n2,l pq0,l1

dqq4,l pq2,l

...

dqqn2,l pqn4,l

dqqn,l 2 pqn2,l

dqqn2,l pqn,l...

dqq2n2,l pq2n4,l

dqq2,1 pq0,1 e

for l ¥ 1. On the remaining generators, the differential vanishes.

The case of odd n is slightly different. Not surprisingly, the difference occurs near

the middle dimensional homology classes of the divisor Qn1. We begin with the

case n ¡ 3, for ease of notation (the case n 3 will be described below). The chain

complex is

SCpTSnq Z

@e, c, qqi,k, pqi,k, qq1

n1,k, pq1n1,k, qq2

n1,k, pq2n1,k

Dwhere we take all even integers 0 ¤ i ¤ 2n 2, i n 1, and all integers k ¡ 0. For

the differential, we now have

dqqin1,l d_^qqin1,l d_Mqqin1,l d_^qqin1,l

xc1pY Ñ Qn1q,CP 1y pqn3,l pqn3,l

for i 0, 1 and l ¥ 1.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 99

dqqn1,l d_^qqn1,l d_Mqqn1,l d_^qqn1,l

xc1pY Ñ Qn1q,CP 1y pq1n1,l xc1pY Ñ Qn1q,CP 1y pq2

n1,l

pq1n1,l pq2

n1,l

for l ¥ 1.

Summing up, when n ¡ 3 is odd, the differential is$'''''''''''''''''''''''&'''''''''''''''''''''''%

dqq0,l1 pq2n4,l

dqq2,l1 pq2n2,l pq0,l1

dqq4,l pq2,l

...

dqq1n1,l pqn3,l

dqq2n1,l pqn3,l

dqqn1,l pq1n1,l pq2

n1,l

dqqn3,l pqn1,l

...

dqq2n2,l pq2n4,l

dqq2,1 pq0,1 e

for l ¥ 1. On the remaining generators, the differential vanishes.

We then get, for n ¡ 2 even,

SHpTSnq Z xc, qq0,1, e, qq0,l1 qq2n2,l, pq0,l1y ` Z2Z xpqn2,ly

and, for n ¡ 3 odd,

SHpTSnq Z

@c, qq0,1, e, qq1

n1,l qq2n1,l, pq1

n1,l, qq0,l1 qq2n2,l, pq0,l1

DWe can again compare this with the Cohen–Jones–Yan result, in Section 2.3. The

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 100

indices are computed using Lemma 3.4. For n ¡ 2 even, the free part is

SHdpTSnq c qq0,1 e qq2n2,k qq0,k1 pq0,k1

HdpLSnq a b 1 bvk vk

d 0 n 1 n pn 1qp2k 1q pn 1qp2k 1q 1

and for the Z2Z torsion we have

SHdpTSnq pqn2,k

HdpLSnq avk

d pn 1q2k

If n ¡ 3 is odd, then there is no torsion, and the free part is given by

SHdpTSnq c qq0,1 e pqq 1

n1,k qq 2n1,kq

HdpLSnq a au 1 au2k

d 0 n 1 n pn 1q2k

pq 1n1,k pqq2n2,k qq0,k1q pq0,k1

u2k1 au2k1 u2k

pn 1q2k 1 pn 1qp2k 1q pn 1qp2k 1q 1

The signs in the table above are as follows:

au2k p1qgpk,mqpqq1n1,k qq2

n1,kq and u2k1 p1qF pk,mq pq1n1,k,

au2k1 p1qGpk,mqpqq2n2,k1 qq0,kq, u2k p1qfpk,mq pq0,k1

where f, F, g,G : N2 Ñ Z2 are such that

• if m is odd, then fm 1, Gm 0 and Fm gm 0 (we could also choose

Fm gm 1).

• if m is even, then fmprq Gmprq r 1 and Fmprq gmprq r (we could

also choose Fmprq gmprq r 1);

Finally, the case n 3 can be dealt with as above, but the differential has a

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 101

slightly different lookx. The chain complex is

SCpTS3q Z

@e, c, qqi,k, pqi,k, qq1

2,k, pq12,k, qq2

2,k, pq22,k

Dwhere we take i P t0, 4u, and all integers k ¡ 0. The differential is$''''''''''&''''''''''%

dqq0,l1 pq12,l pq2

2,l

dqq12,l1 pq4,l pq0,l1

dqq22,l1 pq4,l pq0,l1

dqq4,l pq12,l pq2

2,l

dqq12,1 pq0,1 e

dqq22,1 pq0,1 e

for l ¥ 1. On the remaining generators, the differential vanishes. Therefore,

SHpTS3q Z

@c, qq0,1, e, qq1

2,l qq22,l, pq1

2,l, qq0,l1 qq4,l, pq0,l1

DComparing with the Cohen–Jones–Yan result, we get

SHdpTS3q c qq0,1 e qq 1

2,k qq 22,k pq 1

2,k qq4,k qq0,k1 pq0,k1

HdpLS3q a au 1 au2k u2k1 au2k1 u2k

d 0 2 3 4k 4k 1 4k 2 4k 3

6.2.4 The ring SHpTSnq

Let us now compute the pair-of-pants product. We start with the case of even n. To

get a result matching [CJY04], we need to show the following:

1. a kills everything except for 1 and for the fact that a vk avk;

2. e 1 is the unit;

3. b2 b pbvkq pbvkq pbvlq 0;

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 102

qq0,1

qq0,1

pqi,k

qqi,k

Figure 6.2: Broken pair-of-pants on T Sn

4. b vk bvk;

5. vk vl vkl;

6. pbvkq vl bvkl;

7. avk vl avkl.

We will not focus on the equations that involve constant orbits. As in the case of

T S2, the product of two orbits in CH is zero, since one cannot fix two markers on

a pair-of-pants. This implies (3).

As in the case of T S2, we use the description of the product given in Section 5.3.1.

The 3-point absolute Gromov–Witten numbers correspond to holomorphic pairs-of-

pants in R Y , whereas 2-point relative Gromov–Witten numbers correspond to

broken pairs-of-pants, which are counted with a negative sign. Figure 6.2 depicts the

latter. In this case, given generic p P Qn1, we have rW ufWppqs L P H2pQn;Zq.

For (4), observe that

b vk pqq0,1q ppq0,k1q

GWQn1

L,3 ppt, pt, Qn1q qq0,k1 GWQn1

2L,3 ppt, pt, ptq qq2n2,k

GWQN ,QN1

L,1,p1q pL; ptq qq0,k1

0 qq0,k1 qq2n2,k qq0,k1 bvk.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 103

For (5), we have

vk vl ppq0,k1q ppq0,l1q

GWQn1

L,3 ppt, pt, Qn1q pq0,kl1 GWQn1

2L,3 ppt, pt, ptq pq2n2,kl

0 pq0,kl1 pq0,kl1 vkl.

For (6):

pbvkq vl pqq2n2,k qq0,k1q ppq0,l1q

GWQn1

0L,3 pQn1, pt, Qn1q qq0,kl1 GWQn1

L,3 pQn1, pt, ptq qq2n2,kl

GWQn1

L,3 ppt, pt, Qn1q qq0,kl1 GWQn1

2L,3 ppt, pt, ptq qq2n2,kl

qq0,kl1 0 qq2n2,kl 0 qq0,kl1 qq2n2,kl bvkl.

For (7):

pavkq vl pqn2,k ppq0,l1q GWQn1

L,3

W uppn2q, pt,W

sppn2q pqn2,kl . . .

GWQn1

L,3 pCP n21, pt, Qn2q pqn2,kl pqn2,kl pqn2,kl avkl

(remember this is for the Z2Z torsion).

Now for the case of odd n. Write m : n12

. We need to show:

1. e is the unit;

2. a2 a paukq paukq paulq 0;

3. a uk auk (special case: k 1);

4. uk ul ukl;

5. paukq ul aukl, k ¡ 1;

6. pauq uk auk1.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 104

We will once again not focus on the formulas that involve critical points. As

before, we expect that the product of two orbits in CH should vanish, and thus get

the last identity in (2).

For (4), there are several cases to consider:

• k 2r, l 2s even:

u2r u2s p1qfpr,mq pq0,r1

p1qfps,mq pq0,s1

p1qfpr,mqfps,mq

GWQn1

L,3 ppt, pt, Qn1q pq0,rs1

GWQn1

2L,3 ppt, pt, ptq pq2n2,rs

0 pq0,rs1 p1qfpr,mqfps,mq pq2n2,rs

p1qfpr,mqfps,mq1 pq0,rs1 p1qfprs,mq pq0,rs1 u2r2s

because fpr,mq fps,mq 1 fpr s,mq.

• k 2r 1, l 2s 1 odd:

u2r1 u2s1 p1qF pr.mq pq1

n1,r

p1qF ps,mq pq1

n1,s

p1qF pr,mqF ps,mq

GWQn1

0L,3 pA1, A1, Qn1q pq0,rs

GWQn1

L,3 pA1, A1, ptq pq2n2,rs1

p1qF pr,mqF ps,mqpA1 . A1q pq0,rs

GWQn1

L,3 pA1, A1, ptq pq2n2,rs1

$&%p1qF pr,mqF ps,mq0 pq0,rs pq2n2,rs1

if m odd

p1qF pr,mqF ps,mqpq0,rs 0 pq2n2,rs1

if m even

p1qF pr,mqF ps,mqm pq0,rs p1qfprs1,mq pq0,rs u2r2s2

because F pr,mq F ps,mq m fpr s 1,mq.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 105

• k 2r even, l 2s 1 odd: it is useful to recall that in HmpQn1;Zq

Ai . Aj

$&%1 δi,j if m odd

δi,j if m even

u2r u2s1 p1qfpr,mq pq0,r1

p1qF ps,mq pq1

n1,s

$'''''''&'''''''%

p1qfpr,mqF ps,mq

GWQn1

L,3 ppt, A1, A1q pq2n1,rs

GWQn1

L,3 ppt, A1, A2q pq1n1,rs

if m odd

p1qfpr,mqF ps,mq

GWQn1

L,3 ppt, A1, A1q pq1n1,rs

GWQn1

L,3 ppt, A1, A2q pq2n1,rs

if m even

$&%p1qfpr,mqF ps,mqpq2n1,rs 0 pq1

n1,rs

if m odd

p1qfpr,mqF ps,mq0 pq1

n1,rs pq2n1,rs

if m even

p1qfpr,mqF ps,mq pq2n1,rs p1qfpr,mqF ps,mq1 pq1

n1,rs

p1qF prs,mq pq1n1,rs u2r2s1

because fpr,mq F ps,mq 1 F pr s,mq.

We now consider equation (5) above. Again, there are several cases to consider:

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 106

• k 2r, l 2s even:

pau2rq u2s p1qgpr,mqqq1n1,r qq2

n1,r

p1qfps,mq pq0,s1

$'''''''''''''''''''''''&'''''''''''''''''''''''%

p1qgpr,mqfps,mq

GWQn1

L,3 pA1, pt, A1q qq2n1,rs

GWQn1

L,3 pA1, pt, A2q qq1n1,rs

p1qgpr,mqfps,mq1

GWQn1

L,3 pA2, pt, A1q qq2n1,rs if m odd

GWQn1

L,3 pA2, pt, A2q qq1n1,rs

p1qgpr,mqfps,mq

GWQn1

L,3 pA1, pt, A1q qq1n1,rs

GWQn1

L,3 pA1, pt, A2q qq2n1,rs

p1qgpr,mqfps,mq1

GWQn1

L,3 pA2, pt, A1q qq1n1,rs if m even

GWQn1

L,3 pA2, pt, A2q qq2n1,rs

$''''''''''&''''''''''%

p1qgpr,mqfps,mqqq2n1,rs 0 qq1

n1,rs

p1qgpr,mqfps,mq10 qq2

n1,rs qq1n1,rs

if m odd

p1qgpr,mqfps,mq0 qq1

n1,rs qq2n1,rs

p1qgpr,mqfps,mq1qq1n1,rs 0 qq2

n1,rs

if m even

p1qgpr,mqfps,mq1pqq1n1,rs qq2

n1,rsq

p1qgprs,mqpqq1n1,rs qq2

n1,rsq au2r2s

because gpr,mq fps,mq 1 gpr s,mq.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 107

• k 2r 1, l 2s 1 odd:

pau2r1q u2s1 p1qGpr,mqqq2n2,r1 qq0,r

p1qF ps,mq pq1

n1,s

$'''''''''''''''''''''''&'''''''''''''''''''''''%

p1qGpr,mqF ps,mq

GWQn1

0L,3 pQn1, A1, A1q qq2

n1,rs1

GWQn1

0L,3 pQn1, A1, A2q qq1

n1,rs1

p1qGpr,mqF ps,mq1

GWQn1

L,3 ppt, A1, A1q qq2n1,rs1 if m odd

GWQn1

L,3 ppt, A1, A2q qq1n1,rs1

p1qGpr,mqF ps,mq

GWQn1

0L,3 pQn1, A1, A1q qq1

n1,rs1

GWQn1

0L,3 pQn1, A1, A2q qq2

n1,rs1

p1qGpr,mqF ps,mq1

GWQn1

L,3 ppt, A1, A1q qq1n1,rs1 if m even

GWQn1

L,3 ppt, A1, A2q qq2n1,rs1

$''''''''''&''''''''''%

p1qGpr,mqF ps,mq0 qq2

n1,rs1 qq1n1,rs1

p1qGpr,mqF ps,mq1qq2n1,rs1 0 qq1

n1,rs1

if m odd

p1qGpr,mqF ps,mqqq1n1,rs1 0 qq2

n1,rs1

p1qGpr,mqF ps,mq10 qq1

n1,rs1 qq2n1,rs1

if m even

p1qGpr,mqF ps,mqqq1n1,rs1 qq2

n1,rs1

p1qgprs1,mqqq1n1,rs1 qq2

n1,rs1

au2r2s2

because Gpr,mq F ps,mq gpr s 1,mq.

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 108

• k 2r even, l 2s 1 odd:

pau2rq u2s1 p1qgpr,mqqq1n1,r qq2

n1,r

p1qF ps,mq pq1

n1,s

p1qgpr,mqF ps,mq

GWQn1

0L,3 pA1, A1, Qn1q qq0,rs

GWQn1

L,3 pA1, A1, ptq qq2n2,rs1

p1qgpr,mqF ps,mq1

GWQn1

0L,3 pA2, A1, Qn1q qq0,rs

GWQn1

L,3 pA2, A1, ptq qq2n2,rs1

p1qgpr,mqF ps,mq

$''''''&''''''%

0 qq0,rs qq2n2,rs1

qq0,rs 0 qq2n2,rs1 if m odd

qq0,rs 0 qq2n2,rs1

0 qq0,rs qq2n2,rs1 if m even

p1qgpr,mqF ps,mqm1qq2n2,rs1 qq0,rs

p1qGprs,mqqq2n2,rs1 qq0,rs

au2r2s1

because gpr,mq F ps,mq m 1 Gpr s,mq.

• k 2r 1 odd, l 2s even:

pau2r1q u2s p1qGpr,mqqq2n2,r1 qq0,r

p1qfps,mq pq0,s1

p1qGpr,mqfps,mq

GWQn1

0L,3 pQn1, pt, Qn1q qq0,rs

GWQn1

L,3 pQn1, pt, ptq qq2n2,rs1

p1qGpr,mqfps,mq1

GWQn1

L,3 ppt, pt, Qn1q qq0,rs

GWQn1

2L,3 ppt, pt, ptq qq2n2,rs1

p1qGpr,mqfps,mqqq0,rs 0 qq2n2,rs1

p1qGpr,mqfps,mq10 qq0,rs qq2n2,rs1

p1qGpr,mqfps,mq1qq2n2,rs1 qq0,rs

p1qGprs,mqqq2n2,rs1 qq0,rs

au2r2s1

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CHAPTER 6. THE EXAMPLE OF COTANGENT BUNDLES OF SPHERES 109

because Gpr,mq fps,mq 1 Gpr s,mq.

Finally, for (6) there are 2 cases:

• k 2r even:

pauq u2r pqq0,1q p1qfpr,mq pq0,r1

p1qfpr,mq1

GWQn1

L,3 ppt, pt, Qn1q qq0,r1 GWQn1

2L,3 ppt, pt, ptq qq2n2,r

GWQN ,QN1

L,1,p1q pL; ptq qq0,r1

p1qfpr,mq10 qq0,r1 qq2n2,r qq0,r1

p1qGpr1,mqqq2n2,r qq0,r1

au2r1

because fpr,mq 1 Gpr 1,mq.

• k 2r 1 odd:

pauq u2r1 pqq0,1q p1qF pr,mq pq1n1,r

$''''''''''''''''&''''''''''''''''%

p1qF pr,mq1

GWQn1

L,3 ppt, A1, A1q qq2n1,r

GWQn1

L,3 ppt, A1, A2q qq1n1,r if m odd

GWQN ,QN1

L,1,p1q pL; ptq qq1n1,r

p1qF pr,mq1

GWQn1

L,3 ppt, A1, A1q qq1n1,r

GWQn1

L,3 ppt, A1, A2q qq2n1,r if m even

GWQN ,QN1

L,1,p1q pL; ptq qq1n1,r

$&%p1qF pr,mq1qq2n1,r 0 qq1

n1,r qq1n1,r

if m odd

p1qF pr,mq10 qq1

n1,r qq2n1,r qq1

n1,r

if m even

p1qF pr,mqqq1n1,r qq2

n1,r

p1qgpr,mq

qq1n1,r qq2

n1,r

au2r

because F pr,mq gpr,mq. We conclude that our description of the ring struc-

ture on SHpTSnq matches that of HpLS

nq, as computed in [CJY04].

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Luıs Miguel Pereira de Matos Geraldes Diogo

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

(Yakov Eliashberg) Principal Adviser

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

(Eleny Ionel)

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

(Søren Galatius)

Approved for the University Committee on Graduate Studies