Higher rank local systems in Lagrangian Floer theory

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by  Momchil Konstantinov

Higher rank local systems in Lagrangian Floer theory
Momchil Konstantinov
7
1 Abstract
0
WeextendFloertheoryformonotoneLagrangianstoallowcoefficientsinlocalsystems
2
of arbitrary rank. Unlike the rank 1 case, this is often obstructed by Maslov 2 discs. We

r studyexactlywhattheobstructionisanddefinesomenaturalunobstructedsubcomplexes.
a
ToillustratetheseconstructionswedosomeexplicitcalculationsfortheChiangLagrangian
M
L ⊆ CP3. For example, we equip L with a particular rank 2 local system W over F
∆ ∆ 2
for which the resulting Floer complex CF∗(W,W) is unobstructed despite the presence
3
of Maslov 2 discs. We compute that the cohomology HF∗(W,W) is non-zero and deduce

] that L∆ cannot be disjoined from RP3 by a Hamiltonian isotopy.
G
S
Contents
.
h
t
a 1 Introduction 2
m
2 Floer Cohomology and Local Systems 5
[
2.1 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2
2.1.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
v
2.1.2 Local Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4
2 2.2 The Obstruction Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6 2.3 Definition, Obstruction and Invariance . . . . . . . . . . . . . . . . . . . . . . . 11
3 2.4 The Pearl Complex and the Obstruction Revisited . . . . . . . . . . . . . . . . 17
0
2.5 Honest Subcomplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
.
1
0 3 The Monotone Fukaya Category 26
7 3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1
3.2 Split-Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
:
v 3.2.1 Hochschild cohomology and the closed-open string map . . . . . . . . . 30
i 3.2.2 Hochschild homology and the open-closed string map . . . . . . . . . . 31
X
3.2.3 The bimodule P (E) and the evaluation map H(µ) . . . . . . . . . . . 32
r W
a 3.2.4 The coproduct map ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.5 Proof of the split-generation criterion . . . . . . . . . . . . . . . . . . . 36
4 Application to the Chiang Lagrangian 39
4.1 Topology of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
∆
4.2 Computation of Floer Cohomology with Local Coefficients . . . . . . . . . . . . 43
4.2.1 Morse Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.2 Contributions from holomorphic discs . . . . . . . . . . . . . . . . . . . 47
4.3 Proof of Theorem 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Some additional calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Appendix A: Perturbation of Morse Data 58
1
Appendix B: Representations of the binary dihedral group of order 12 over F 61
2
1 Introduction
Lagrangian Floer cohomology is a very powerful tool for studying the topology and relative
position of Lagrangian submanifolds in a symplectic manifold. The even more elaborate ma-
chinery of Fukaya categories encodes an immense amount of information about them. These
invariantsarealldefinedby“counting”(perturbed)puncturedpseudoholomorphicdiscswhose
boundary components are restricted to lie on a collection of Lagrangians. One can enrich this
structureevenfurtherbyrecordingsomeinformationaboutthehomotopyclassesofpathsthat
these boundaries trace. One way to accomplish this is by using local systems of coefficients for
the theory.1
Rank1localsystemsareaclassicaltoolinLagrangianFloertheoryandarewidelyapplied,
specificallyinthecontextofMirrorSymmetry. Anaturalquestiontoasktheniswhetherhigher
ranklocalsystemswouldyieldusefulinvariants. Inhispaper[Dam12],Damianwasthefirstto
realise that in the monotone setting, it is not always possible to define Floer cohomology with
coefficients in a local system of rank higher than 1 since Maslov 2 discs with boundary on the
Lagrangianmayobstructd2 =0. Infact,concentratingonthisobstructionforthelocalsystem
arising from the universal cover, he derives many interesting restrictions on the topology of
monotone Lagrangians in linear symplectic spaces, including a proof of the monotone Audin
conjecture for Lagrangians with contractible universal cover ([Dam12]). From another point
of view, a systematic treatment of high rank local systems in the exact case (when no such
obstructionsarise)wasdonebyAbouzaidin[Abo12a],whereheconstructsanextendedFukaya
category which contains Lagrangians with higher rank local systems. The powerful results of
thatpaperthenfollowbyanapplicationof(amodifiedversionof)thesplit-generationcriterion,
again due to Abouzaid ([Abo10]). The goal of the present paper is to combine the above two
approaches in their simplest forms: we study exactly how Maslov 2 discs obstruct d2 = 0 in
the monotone case and give an application of Abouzaid’s criterion to an example in which this
obstruction vanishes despite the existence of Maslov 2 disc bubbles.
Regarding the first point, recall that in the absence of local systems, the differential on
the Floer complex CF∗(L0,L1) of a monotone pair (see 2.1.1 below for a quick review of
monotonicity) squares to zero, if and only if
m (L0)=m (L1), (1)
0 0
where these are the so-called obstruction numbers of the Lagrangians, defined as algebraic
counts of Maslov 2 discs passing through a generic point. Exactly the same condition car-
ries over when one considers local systems of rank 1, except that in the algebraic count each
disc is weighted by the monodromy of the local system around its boundary. Applying the
same proof to the case when the Lagrangians are equipped with local systems Ej → Lj of
higher rank, one arrives at the notion of an obstruction section m (Lj,Ej): Lj → End(Ej)
0
and a generalised form of condition (1) - see equation (6) below. When this condition is sat-
isfied (for example when m (Lj,Ej) = 0 for j ∈ {0,1}), one obtains a well-defined invariant
0
HF∗((L0,E0),(L1,E1)). Thisisnotsurprisingandiswell-knowntoexpertsbut,toourknowl-
edge, is not explicitly written anywhere, so we devote sections 2.2 and 2.3 to reviewing these
facts.
1Anotherrelatedapproachtorecordingsimilarhomotopydataistocallupontechniquesfromthetopology
of loop spaces. This idea was pioneered by Viterbo in [Vit97] and has been extensively explored. For a small
sampleseee.g. [BC07a],[Fuk06],[Abo12b].
2
In section 2.4 we examine how the same obstruction appears in the setting of a compu-
tationally powerful theory of Biran and Cornea - the pearl complex [BC07b]. This machinery
allows one to efficiently calculate Floer cohomology of a Lagrangian with itself and applies
equallywellwhenhigherranklocalsystemscomeintoplay. Section2.5isdevotedtotheobser-
vationthatthepossiblyobstructedFloercomplexCF∗((L0,E0),(L1,E1))containsamaximal
unobstructed subcomplex CF∗((L0,E0),(L1,E1)) which we call the central Floer complex. In
thecasewhenL0 =L1 =LandE0 =E1 =E thereisafurtherunobstructedsubcomplex: the
monodromy Floer complex CF∗ (E). We observe that when E is the local system arising
mon reg
from the universal cover of L, the differential of CF∗ (E ) can be seen as a deformation of
mon reg
the differential on the complex C∗(L;E ) of singular cochains with coefficients in the local
conj
system E , where E arises from the conjugation action of π (L) on the group algebra
conj conj 1
F [π (L)].
2 1
Next,insection3weexplainhowtoenlargethemonotoneFukayacategoryoverF soasto
2
includeLagrangiansequippedwithlocalsystemsofrankhigherthan1. Thismimics[Abo12a].
In section 4 we illustrate all of the above techniques in some explicit computations for
the Chiang Lagrangian L ⊆ CP3. This Lagrangian is defined as the manifold of equilateral
∆
trianglesinscribedequatoriallyintheunitsphereS2 ⊆R3anditembedsnaturallyinCP3under
the identification CP3 ∼= Sym3(CP1). The fact that this embedding is Lagrangian was first
noticedbyRiverChiangin[Chi04]. TheFloertheoryofL wascomputedbyEvansandLekili
∆
in [EL14] where they prove that HF∗(L ,L ) is non-zero if and only if one works over a field
∆ ∆
ofcharacteristic5. Further, byequippingtheCliffordtorusT andL withsuitableF −local
Cl ∆ 5
systems α , β of rank 1, they also obtain HF∗((T ,α ),(L ,β )) (cid:54)= 0. As a consequence,
ζ ζ Cl ζ ∆ ζ
L cannotbedisplacedfromitself,orfromtheCliffordtorusbyaHamiltonianisotopy(infact
∆
EvansandLekilishowsomethingmuchstronger: (L ,β )generatesitssummandoftheFukaya
∆ ζ
category over F and by varying β one can place it in any of the four summands; see [EL14,
5 ζ
Section 8]). One can then turn to studying the relationship between L and another classical
∆
Lagrangian in CP3, namely RP3. One easily checks that when they are in their standard
positions, they intersect cleanly in two disjoint circles and it is then a natural question to
ask whether they can be displaced by a Hamiltonian isotopy. A standard obstruction to such
displaceability would be the non-vanishing of the Floer cohomology HF(L ,RP3;F) for some
∆
field F. However, the obstruction numbers of the two Lagrangians are m (RP3) = 0 (since
0
RP3 has minimal Maslov number 4) and m (L ) = ±3 (see [EL14]) and so it follows from
0 ∆
(1) that the Floer cohomology HF∗(L ,RP3;F) is well-defined only when char(F)=3. Now,
∆
by the Auroux-Kontsevich-Seidel criterion (see e.g. [She13, Lemma 2.7]) one sees that when
char(F) = 3 both HF∗(RP3,RP3;F) and HF∗(L ,L ;F) must vanish (m = 0 is not an
∆ ∆ 0
eigenvalueofquantummultiplicationbyc (CP3)incharacteristic3). SinceHF∗(L ,RP3;F)is
1 ∆
a(respectivelyleftandright)moduleovertheserings,itmustalsovanish. Thus,standardFloer
cohomologywithfieldcoefficientscannotbeusedtoaddressthequestionofnon-displaceability.
In the present work we apply the idea of Floer theory with coefficients in a local system to this
situation. We establish the following result:
Theorem 1.1. There exists an F −local system W →L of rank 2 such that m (L ,W)=0
2 ∆ 0 ∆
and HF∗((L ,W),(L ,W))(cid:54)=0.
∆ ∆
The main corollary which we derive from this is:
Corollary 1.2. The Chiang Lagrangian L cannot be displaced from RP3 by a Hamiltonian
∆
isotopy.
Theorem 1.1 is proved in section 4.2 by an explicit calculation using the pearl complex.
Our starting point are the results from [EL14]. For the numerical calculations we use Wolfram
3
Mathematica (the code can be found on the author’s home page). To establish Corollary 1.2
one enlarges the monotone Fukaya category of CP3 in order to include Lagrangians with finite
rank local systems and in particular (L ,W). In this enlarged category, (L ,W) and RP3
∆ ∆
(withnolocalsystem)lieinthesamesummand. AresultbyTonkonog([Ton15,Corollary1.2])
statesthatRP3 split-generatesthissummandandthuswemusthaveHF∗((L ,W),RP3)(cid:54)=0,
∆
since by Theorem 1.1 we know that (L ,W) is an essential object in this category.
∆
Finally, in section 4.4 we compute the cohomology of the central and monodromy Floer
complexes of L with coefficients in some other F local systems. In particular we find that
∆ 2
in many cases one can obtain a non-zero Floer invariant which does not agree with the corre-
sponding Morse invariant. Specifically we make the following
Observation 1.3. IfW istherank12localsystemcomingfromtheuniversalcoverS3 →L
reg ∆
and W is the local system induced by the conjugation action of π (L ) on F [π (L )] one
conj 1 ∆ 2 1 ∆
has
0(cid:54)=HF∗ (W )(cid:54)=H∗(L ;W ).
mon reg ∆ conj
That is, the “quantum corrections” to the differential on C∗(L ;W ) are non-trivial but
∆ conj
they do not kill the cohomology.
In other words, when one uses local coefficients, the “narrow–wide” dichotomy, introduced by
Biran-Cornea in [BC09], does not hold.
To deduce the above results we only need to work over the field F and do not require
2
any grading on Lagrangians. It is reasonable to assume that the results below should hold in
the more general setting when one considers graded spin Lagrangians in order to define graded
invariants over fields of arbitrary characteristic (of course, then a substantial amount of signs
would need to be introduced). For simplicity, we restrict all our discussions to the setting
modulo 2.
Acknowledgements
First and foremost I would like to thank my supervisor Jonny Evans for suggesting the local
systems approach to the problem of displacing L form RP3 and for his many patient ex-
∆
planations and general encouragement. My understanding of holomorphic curves has greatly
benefited also from the excellent teaching of Chris Wendl. I thank also Yankı Lekili and my
fellow students at the LSGNT for numerous useful conversations. Special thanks to Agustin
Moreno, Tobias Sodoge, Emily Maw and Antonio Cauchi for their help and for being such a
faithful audience.
WolframMathematicawasusedtoproducetheplotsinFigures12and13andfornumerical
calculations. Most other plots were produced using the tikz-3dplot package of Jeff Hein.
This work was supported by the Engineering and Physical Sciences Research Council
[EP/L015234/1] as the author is part of the EPSRC Centre for Doctoral Training in Geome-
try and Number Theory (The London School of Geometry and Number Theory), University
College London.
4
2 Floer Cohomology and Local Systems
2.1 Terminology and Notation
2.1.1 Monotonicity
Let (M,ω) be a symplectic manifold and L ⊆ M – a Lagrangian submanifold. Recall that
monotonicity concerns the following standard maps:
• the symplectic area homomorphism:
(cid:90)
I : π (M)→R; I (A):= ω ,
ω 2 ω
A
• the Chern homomorphism:
(cid:90)
I : π (M)→Z; I (A):= c (TM),
c1 2 c1 1
A
• the Maslov homomorphism: I : π (M,L) → Z, defined by evaluation of the Maslov
µ,L 2
class µ∈H2(M,L;Z).
We call the symplectic manifold M monotone, if there exists a constant λ > 0 such that for
every A ∈ π (M) one has I (A) = 2λI (A). We then call a Lagrangian submanifold L ⊆ M
2 ω c1
monotone if one has I (A)=λI (A) for every A∈π (M,L), where I : π (M,L)→R is
ω,L µ,L 2 ω,L 2
the map induced by I .
ω
Remark 2.1. OneusuallycallsaLagrangianmonotonewheneverI andI arepositivelypro-
ω µ,L
portional. AstandardresultbyViterbo([Vit87])assertsthatthepullbackmapj∗: H2(M,L;Z)→
H2(M;Z) satisfies j∗µ = 2c (TM) and this implies, firstly, that such Lagrangians can exist
1
only in monotone symplectic manifolds and, secondly (at least when I doesn’t vanish identi-
ω
cally, i.e. M is not symplectically aspherical), that the constants of proportionality are related
by a factor of 2. This shows that with our definition above we don’t lose any generality in the
non-aspherical case. Note further that if M is monotone and L ⊆ M is any Lagrangian with
H1(L;Z) = 0 (e.g. if H (L;Z) is finite) then, since the map j∗ is injective in this case, the
1
Lagrangian L must also be monotone.
TheconceptofmonotonicityalsoextendstopairsofLagrangiansubmanifolds(cf. [Poz99]
3.3.2). Given two Lagrangians L0, L1 in M, the area and Maslov homomorphisms can also be
evaluatedon(homotopyclassesof)continuousmapsu: S1×[0,1]→M withu(S1×{0})⊆L0
and u(S1×{1})⊆L1. We denote these extensions by I and I respectively. Then
ω,L0,L1 µ,L0,L1
we call (L0,L1) a monotone pair of Lagrangians if each of them is monotone and further
I = λI . It is not hard to see that if (L0,L1) is a monotone pair, then so is
ω,L0,L1 µ,L0,L1
(ψ(L0),L1)foranyHamiltoniandiffeomorphismψ. AnotherusefulfactisthatifL0 andL1 are
monotone Lagrangians and for at least one j ∈{0,1} the image of π (Lj) in π (M) under the
1 1
mapinducedbyinclusionistrivial, then(L0,L1)isamonotonepair(see[Poz99]Remark3.3.2
or [Oh93], Proposition 2.7). Finally, if L is monotone, then the pair (L,L) is always monotone
([Oh93], Proposition 2.10).
2.1.2 Local Systems
In this paper, we will be concerned with compact, connected, monotone Lagrangians equipped
withlocalsystemsofvectorspacesoverthefieldoftwoelementsF . Forussuchalocalsystem
2
5
would just be a functor E: Π1L → VectF2, where Π1L is the fundamental groupoid of the
Lagrangian L. More concretely, it is an assignment of an F −vector space E for each point
2 x
x ∈ L and an isomorphism P : E → E for each homotopy class γ of paths in L with
γ s(γ) t(γ)
source s(γ) and target t(γ), in a manner which is compatible with concatenation of paths. As
is customary, we call these isomorphisms parallel transport maps.
We will sometimes write E → L to denote such a local system, the notation being a
shorthand for the map
(cid:97)
E → L
x
x
v ∈E (cid:55)→ x.
x
Similarly,byasectionσ: L→E wewillmeanasectionofthismap. Wewillcallsuchasection
parallel if for every path γ on L one has P (σ(s(γ)))=σ(t(γ)).
γ
Notethat,sinceLispath-connected,theinclusionBπ (L,x)Opp (cid:44)→Π L(whereBπ (L,x)
1 1 1
denotes the category with one object and morphisms in bijection with π (L,x) whose compo-
1
sition follows the group law in π (L,x)) induces an equivalence of categories and so we get an
1
equivalence
Fun(Π1L,VectF2)(cid:39)Fun(Bπ1(L,x)Opp,VectF2). (2)
Note that our conventions are such that concatenation of paths will be written from left to
right, while compositions of maps, as usual, from right to left. Since this can cause headaches
in explicit computations, let us spell-out concretely how the above equivalence plays out in
practice. Giventwopointsx,y ∈LwewriteΠ L(x,y):=Hom (x,y)forthesetofhomotopy
1 Π1L
classes of paths connecting x to y. To go from left to right in (2), one can associate to each
local system E →L, a right representation of the fundamental group π (L,x), by considering
1
the action of Π L(x,x) ∼= π (L,x)Opp on the fibre E . To go the other way, since L admits a
1 1 x
universalcoverp: L(cid:101) →L,wecanusethefollowingconstruction. Supposewehavealinearright
action of π1(L,x) on some F2−vector space V. Then we consider the quotient V ×π1(L,x)L(cid:101) =
(V ×L(cid:101))/π1(L,x), where the right action on L(cid:101) is by inverse deck transformations (to identify
the group of deck transformations with π (L,x) we have chosen a fixed basepoint x˜∈p−1(x)).
1
ThisquotientcomesequippedwithaprojectiontoL=L(cid:101)/π1(L,x)andforanyy ∈Lwedefine
E tobethefibreofthismapovery. Theparalleltransportmapalonganypathγ ∈Π L(y,z)
y 1
is then defined to be the one induced by
V ×p−1(y) → V ×p−1(z)
(v,y˜) (cid:55)→ (v,t(γ )),
(cid:101)y˜
where γ(cid:101)y˜ is any lift of the path γ to L(cid:101), satisfying s(γ(cid:101)y˜) = y˜. This map is equivariant with
respect to the right π (L,x)−action and so descends to a well-defined map P : E → E on
1 γ y z
the quotients.
Wording and notation: Note that above we viewed the path γ slightly ambiguously
as both a morphism in the fundamental groupoid and (when we mentioned lifting) an actual
continuous map γ: [a,b] → L. This ambiguous use of the word path persists throughout this
work since eliminating it would cause notational clutter. Hopefully the meaning is clear from
thecontext. Giventwopathsγ andδ inLwitht(γ)=s(δ),theirconcatenationwillbewritten
as γ ·δ ∈ Π L(s(γ),t(δ)). Since sometimes we will use two different local systems Ej → L,
1
j ∈ {0,1} on the same space, we shall write P : Ej → Ej to distinguish between the
j,γ s(γ) t(γ)
parallel transport maps.
6
2.2 The Obstruction Section
Given a monotone pair of Lagrangians (L0,L1) equipped with local systems E0 → L0 and
E1 → L1, we define in section 2.3 below a Floer-theoretic invariant HF∗((L0,E0),(L1,E1)).
However, the existence of this invariant is often obstructed by Maslov 2 disc bubbles and this
is captured by the obstruction sections m (Ej): Lj →End(Ej). This claim is made precise in
0
Theorem 2.13 below. In order to describe the obstruction section, we concentrate on a single
monotone Lagrangian L⊆M, equipped with a local system E →L. We begin by making our
setup precise and establishing some notation.
Let (M,ω) be a monotone symplectic manifold and let L⊆M be a monotone Lagrangian
submanifold with minimal Maslov number N ≥2. Let E →L be a local system of F -vector
L 2
spaces. Let J(M,ω) denote the space of ω-compatible almost complex structures on M, that
is, the space of sections J of End(TM) which satisfy J2 = −Id and such that g (·,·) :=
J
ω(·,J·) is a Riemannian metric on M. We denote by D the closed unit disc in C. Given
J ∈J(M,ω), we will be concerned with J-holomorphic discs with boundary on L, i.e. smooth
maps u: (D,∂D)→(M,L), which satisfy the Cauchy-Riemann equation
du+J(u)◦du◦i=0. (3)
Such a disc is called simple if there exists an open and dense subset S ⊆ D2 such that for all
z ∈ S one has u−1(u(z)) = {z} and du(z) (cid:54)= 0. Now let us introduce the following pieces of
notation.
• For any class C ∈π (M,L) and any k ∈Z we set
2
M(cid:102)C(L;J) := {u∈C∞((D,∂D),(M,L)) : du+J(u)◦du◦i=0, [u]=C},
(cid:91)
M(cid:102)(k,L;J) := M(cid:102)C(L;J),
C∈π2(M,L)
µ(C)=k
MC(L;J) := M(cid:102)C(L;J)/G,
M(k,L;J) := M(cid:102)(k,L;J)/G,
whereG=PSL(2,R)isthereparametrisationgroupofthediscactingbyprecomposition.
We will write qG: M(cid:102)C(L;J)→MC(L;J) for the quotient map.
• We further set
MC0,1(L;J) := M(cid:102)C(L;J)×GS1,
M0,1(k,L;J) := M(cid:102)(k,L;J)×GS1,
where we view S1 = ∂D and an element φ ∈ G acts by φ·(u,z) = (u◦φ−1,φ(z)). We
shall denote the corresponding quotient map again by q .
G
• The above moduli spaces come with natural evaluation maps,
e(cid:101)v: M(cid:102)C(L;J)×S1 →L, e(cid:101)v(u,z):=u(z),
which clearly descend to maps ev: MC (L;J)→L.
0,1
• Foranypointp∈LwethenwriteMC (p,L;J)andM (p,k,L;J)forthesetev−1({p}),
0,1 0,1
where the evaluation map is restricted to the appropriate domain and we set
M(cid:102)C (p,L;J) := q−1(MC (p,L;J))⊆M(cid:102)C(L;J)×S1
0,1 G 0,1
M(cid:102)0,1(p,k,L;J) := qG−1(M0,1(p,k,L;J))⊆M(cid:102)(k,L;J)×S1.
7
• We shall decorate any of the above sets with a superscript ∗ to denote the subset, con-
(cid:110) (cid:111)
sisting of simple discs. For example M(cid:102)C,∗(L;J) := u∈M(cid:102)C(L;J) : u is simple and
MC0,,1∗(L;J):=M(cid:102)C,∗(L;J)×GS1.
All these spaces are equipped with the C∞–topology. When there is no danger of confusion we
shall sometimes simplywrite u∈M (p,k,L;J) forthe equivalenceclass [u,z]=q (u,z) and
0,1 G
∂u∈π (L,p) for the homotopy class of the loop s(cid:55)→u(cid:0)ze2πis(cid:1).
1
By standard transversality arguments (see [MS12], Chapter 3) it follows that there exists
a Baire subset J (L) ⊆ J(M,ω) such that for all J ∈ J (L) and any class C ∈ π (M,L),
reg reg 2
the space M(cid:102)C,∗(L;J) has the structure of a smooth manifold of dimension n+µ(C) and the
evaluation map e(cid:101)v: M(cid:102)C,∗(L;J)×S1 → L is smooth. Since the reparametrisation action on
M(cid:102)C,∗(L;J) is free and proper, one has that MC(L;J) is a smooth manifold of dimension
n+µ(C)−3 and the quotient map q is everywhere a submersion (in particular the map
G
ev: MC,∗(L;J) → L is also smooth). Further transversality arguments (i.e. the Lagrangian
0,1
boundry analogue of [MS12, Proposition 3.4.2]) show that for any smooth map of manifolds
F: X →L, there exists a Baire subset J (L|F)⊆J (L) such that for every J ∈J (L|F)
reg reg reg
the maps F: X → L and ev: M∗ (k,L;J) → L are everywhere transverse. When X is
0,1
a submanifold of L and F is the inclusion map we shall write simply J (L|X). Results by
reg
Kwon-OhandLazzarini([KO00,Laz00])yieldthatwhenLismonotoneonehasM(N ,L;J)=
L
M∗(N ,L;J) and a further application of Gromov compactness ensures that the manifold
L
M(N ,L;J) is actually compact. In particular if N ≥ 2 then M (2,L;J) is a compact
L L 0,1
manifold (possibly empty) of dimension dim(M(cid:102)(2,L;J)×S1)−dim(G)= n+2+1−3 =n.
Therefore for any p ∈ L and Jp ∈ J (L|p) the manifold M (p,2,L;Jp) consists of a finite
reg 0,1
number of points. We are now ready to define the obstruction section.
Definition 2.2. Let E be an F -local system on a monotone Lagrangian submanifold L ⊆
2
(M,ω) with N ≥ 2. The obstruction section for E is a section of the local system End(E),
L
definedasfollows. Foreverypointp∈LwechooseanalmostcomplexstructureJp ∈J (L|p)
reg
and set
(cid:88)
m (p,E;Jp):= P ∈End(E ).
0 ∂u p
u∈M0,1(p,2,L;Jp)
The obstruction section is then
m (E): L → End(E)
0
p (cid:55)→ m (p,E;Jp).
0
Remark 2.3. Note that when E is trivial and of rank one m (p,E;Jp) is the (local) F −degree
0 2
of the map ev: M (2,L;Jp)→L.
0,1
As stated, the obstruction section appears to depend on the choices of almost complex
structures Jp. This is not the case, as the following proposition shows.
Proposition 2.4. The following invariance properties hold:
i) For any p∈L and J,J(cid:48) ∈J (L|p) one has m (p,E;J)=m (p,E;J(cid:48));
reg 0 0
ii) m (E) is a parallel section of End(E),
0
iii) if ψ: M →M is any symplectomorphism, then for every point p∈L, one has
m (E)(p)=m (ψ E)(ψ(p)).
0 0 ∗
8
In the remaining part of this section we prove this proposition. We will make use of the
following lemma.
Lemma 2.5. Let C = C0((D,∂D),(M,L)) equipped with the compact-open topology. Let
γ: [0,1] → L be a continuous path and let C := {(t,u,z) ∈ [0,1]×C ×S1 : γ(t) = u(z)}.
γ
Further let ν: [0,1] → C be a continuous path and write ν(s) = (t(s),u ,z ). Then the loops
γ s s
δ : [0,1]→L, δ (s):=u (z e2πis) and δ : [0,1]→L,
ν(0) ν(0) 0 0 ν(1)

γ(t(3s)), s∈[0,1/3]

δ (s):= u (cid:0)z e2πi(3s−1)(cid:1), s∈[1/3,2/3]
ν(1) 1 1
γ(t(3−3s)), s∈[2/3,1]
are homotopic based at u (z ).
0 0
Proof. An explicit homotopy is given by H: [0,1]×[0,1]→L,

γ(t(cid:16)(3s)), s∈(cid:17)[0,r/3], r ∈[0,1]
H(s,r)= ur zre2πi33−s−2rr , s∈[r/3,1−r/3], r ∈[0,1]
γ(t(3−3s)), s∈[1−r/3,1], r ∈[0,1]
Continuity of H follows from that of ν and of the evaluation map C×S1 →L.
To establish part i) of proposition 2.4 we need to consider a homotopy of almost complex
structures,interpolatingbetweenJ andJ(cid:48). Standardtransversalityandcompactnessarguments
imply the following.
Theorem2.6. SupposeLismonotonewithN ≥2andletp∈LandJ,J(cid:48) ∈J (L|p). Then
L reg
there exists a Baire subset J (J,J(cid:48))⊆C∞([0,1],J(M,ω)) such that for every Jˆ∈J (J,J(cid:48))
reg reg
one has Jˆ(0)=J, Jˆ(1)=J(cid:48) and if we set
M(cid:102)0,1(p,2,L;Jˆ) := {(λ,u,z)∈[0,1]×C∞((D,∂D),(M,L))×S1 : du+Jˆ(λ)◦du◦i=0,
µ([u])=2,u(z)=p}and
M0,1(p,2,L;Jˆ) := M(cid:102)0,1(p,2,L;Jˆ)/G,
thenM(cid:102)0,1(p,2,L;Jˆ)isasmooth4-dimensionalmanifoldwithboundary. Further,M0,1(p,2,L;Jˆ)
is a compact 1-dimensional manifold with boundary
∂M (p,2,L;Jˆ)={0}×M (p,2,L;J) (cid:97){1}×M (p,2,L;J(cid:48))
0,1 0,1 0,1
and the quotient map qG: M(cid:102)0,1(p,2,L;Jˆ)→M0,1(p,2,L;Jˆ) is everywhere a submersion.
From this theorem it follows that the elements of ∂M (p,2,L;Jˆ) are naturally paired
0,1
up as opposite endpoints of closed intervals. Let (λ,[u,z]) and (λ(cid:48),[u(cid:48),z(cid:48)]) be such a pair with
λ ≤ λ(cid:48) (note that λ, λ(cid:48) ∈ {0,1}) and let ν¯: [0,1] → M (p,2,L;Jˆ) be any parametrisation
0,1
of the interval which connects them. Choose a lift ν: [0,1] → M(cid:102)0,1(p,2,L;Jˆ) of ν¯. Since
M(cid:102)0,1(p,2,L;Jˆ) embeds continuously into Cp (this is notation from Lemma 2.5, where we let γ
be the constant path at p), we can apply Lemma 2.5 to obtain P =P . We then have
∂u ∂u(cid:48)
(cid:88)
m (p,E;J)+m (p,E;J(cid:48))= P = 0,
0 0 ∂u
(λ,[u,z])∈∂M0,1(p,2,L;Jˆ)
9
because every term in the sum appears an even number of times. This proves part i) of
Proposition 2.4. We shall henceforth use the notation m (E)(p) as in Definition 2.2.
0
Wenowmoveontoprovingpartii). Letp, q ∈Landletγ: [0,1]→Lbeanysmoothpath
with γ(0) = p, γ(1) = q. Then, by what we explained above about achieving transversality of
the evaluation map with any other map, there exists a Baire subset J (L|γ) ⊆ J (L|p)∩
reg reg
J (L|q) such that for every J ∈J (L|γ), the space
reg reg
(cid:110) (cid:111)
M(cid:102)0,1(γ,2,L;J):= (s,u,z)∈[0,1]×M(cid:102)(2,L;J)×S1 : u(z)=γ(s)
is a smooth 4-dimensional manifold with boundary. Further, the manifold M (γ,2,L;J) :=
0,1
M(cid:102)0,1(γ,2,L;J)/G is a 1-dimensional compact manifold with boundary
(cid:97)
∂M (γ,2,L;J)={0}×M (p,2,L;J) {1}×M (q,2,L;J).
0,1 0,1 0,1
(cid:96)
Thus again the elements of M (p,2,L;J) M (q,2,L;J) are naturally paired up as
0,1 0,1
endpoints of intervals. Let N be the number of such intervals and choose parametrisations
ν¯1,...,ν¯N: [0,1]→M0,1(γ,2,L;J)andcorrespondingliftsν1,...,νN: [0,1]→M(cid:102)0,1(γ,2,L;J)
with ν (s) = (ti(s),ui,zi) such that ti(0) ≤ ti(1) ∀i (recall ti(0),ti(1) ∈ {0,1}). Since
i s s
M(cid:102)0,1(γ,2,L;J) embeds continuously in Cγ then by applying Lemma 2.5 to νi, we obtain that
P =P for all 1≤i≤N. (4)
∂ui0 δνi(1)
Let N , N ∈{1,...,N +1} be such that ti(0)=0, ti(1)=0 for all 1≤i≤N −1, ti(0)=0,
1 2 1
ti(1)=1 for all N ≤i≤N −1 and ti(0)=1, ti(1)=1 for all N ≤i≤N. Note then that if
1 2 2
1 ≤ i ≤ N −1, the loop δ is based at p and lies in the homotopy class ∂ui ∈ π (L,p); if
1 νi(1) 1 1
N ≤i≤N −1, then δ is again based at p and lies in the class γ·∂ui ·γ−1 ∈π (L,p); if
1 2 νi(1) 1 1
N ≤i≤N, then δ is based at q and lies in the class ∂ui ∈π (L,q). We thus have
2 νi(1) 1 1
N(cid:88)1−1(cid:16) (cid:17) N(cid:88)2−1
m (E)(p)+P−1◦m (E)(q)◦P = P +P + P
0 γ 0 γ ∂ui ∂ui ∂ui
0 1 0
i=1 i=N1
(cid:32)N(cid:88)2−1 (cid:88)N (cid:16) (cid:17)(cid:33)
+ P−1◦ P + P +P ◦P
γ ∂ui ∂ui ∂ui γ
1 0 1
i=N1 i=N2−1
N(cid:88)1−1(cid:16) (cid:17) N(cid:88)2−1(cid:16) (cid:17)
= P +P + P +P−1◦P ◦P
∂ui0 δνi(1) ∂ui0 γ ∂ui1 γ
i=1 i=N1
(cid:32) N (cid:33)
(cid:88) (cid:16) (cid:17)
+ P−1◦ P +P ◦P
γ ∂ui0 δνi(1) γ
i=N2
N(cid:88)2−1(cid:16) (cid:17)
= 0+ P +P +P−1◦0◦P
∂ui0 δνi(1) γ γ
i=N1
= 0.
This concludes the proof of part ii) of Proposition 2.4.
Finally,partiii)isaneasyconsequenceofi). Indeed,weknowthatwearefreetochooseJ ∈
J (L|p) to compute m (E)(p) and J(cid:48) ∈J (ψ(L)|ψ(p)) to compute m (ψ E)(ψ(p)). So let
reg 0 reg 0 ∗
J beanyelementofJ (L|p)andsetJ(cid:48) =ψ J. Then,almosttautologically,wehavethatJ(cid:48) ∈
reg ∗
J (ψ(L)|ψ(p))(compatibilitywithωisensuredbythefacttheψisasymplectomorphism). It
reg
isthenclearthatm (ψ(p),ψ E;ψ J)=m (p,E;J)andthiscompletestheproofofProposition
0 ∗ ∗ 0
2.4.
10