GLOBAL FUKAYA CATEGORY II: SINGULAR CONNECTIONS, …

Direct link to author’s version November 5, 2021GLOBAL FUKAYA CATEGORY II: SINGULAR CONNECTIONS,

QUANTUM OBSTRUCTION THEORY AND OTHERAPPLICATIONS

YASHA SAVELYEV

Abstract. We perform computations and give applications of the theory de-veloped in Part I of the homotopy natural classifying map

BHam(M,ω) → (S, NFuk(M,ω)),

with the right hand side the space (∞-groupoid) of infinity categories in thecomponent of the A∞ nerve of the Fukaya category of a monotone symplecticmanifold (M,ω). For example, we find new curvature phenomena for PU(2)or Ham(S2) smooth and singular connections on principal bundles over S4.Even for PU(2), these phenomena are invisible to Chern-Weil theory, and areinaccessible to known Yang-Mills theory techniques. So this can be under-stood as one application of Floer theory and the theory of ∞-categories inbasic differential geometry. On the way we also construct what we call quan-tum Maslov classes, which are higher degree variants of the relative Seidelmorphism, this also leads to applications in Hofer geometry of the space ofLagrangian equators in S2.

1. Introduction

Let Ham(M,ω) denote the group of Hamiltonian symplectomorphisms of a sym-plectic manifold (M,ω), understood as a Frechet Lie group, with its C∞ topology.A Hamiltonian bundle is a smooth fiber bundle

M → P → X,

with structure group Ham(M,ω). Given such a bundle P with M monotone, inPart I [20] we have constructed a classifying map

fP : X → (S, NFuk(M))

where S denotes the space of ∞-categories, concretely quasi-categories, in the com-ponent of the A∞ nerve NFuk(M) of the Fukaya category of M , with this nervehaving the structure of an ∞-category. This extends to the universal level, so thatthere is a universal classifying map:

BHam(M,ω) → (S, NFuk(M)).

From here on we just refer to [20] as Part I.The construction also induces a kind simplicial fibration over X called (co)-

Cartesian fibration, with fiber modelled on NFuk(M). We called this the globalFukaya category Fuk∞(P ) of P . As first computational step, we show that forP a non-trivial Hamiltonian S2 fibration over S4, the maximal Kan sub-fibration

2000 Mathematics Subject Classification. 53D37, 55U35, 53C21.Key words and phrases. Fukaya category, action of the group of Hamiltonian symplectomor-

phisms, infinity categories, singular connections, curvature constraints.1

http://yashamon.github.io/web2/papers/fukayaII.pdf

2 YASHA SAVELYEV

of Fuk∞(P ), which is just a combinatorial analogue of a Serre fibration, is non-trivial. In particular Fuk∞(P ) is non-trivial as a (co)-Cartesian fibration and sofP is homotopically non-trivial. This gives in particular:

Theorem 1.1. The natural homomorphism as constructed in Part I,

Z = π4(BHam(S2), id)k−→ π4(S, NFuk(S2)),

is injective.

None of the homotopy groups of S are known, so this is in a sense an applicationof geometry to algebraic topology. Morally, such an application is possible becausegeometry forces a priori A∞-associativity of certain structures, which may thenhave formal consequences in algebraic topology.

The calculation is performed by constructing perturbation data in such a waythat we are reduced to calculation of a certain higher product in a certain Fukayatype A∞ category. Note that this is an actual chain level calculation. To performthis calculation we relate it to the computation of a certain quantum Maslov, whichare certain higher dimensional analogues of the relative Seidel element in [9]. Thecalculation of this quantum Maslov class uses a regularization technique based on“virtual Morse theory” for the Hofer length functional [18].

It is likely that k is surjective. Surjectivity is in a sense the statement thatup to equivalence there are no exotic (co)-Cartesian fibrations over S4, with fiberequivalent to N(Fuk(S2)) - they all come from Hamiltonian S2 fibrations, via theglobal Fukaya category.

1.1. Application in basic Riemannian geometry. As one less expected appli-cation, we can use the computation of Theorem 1.1 to obtain lower bounds forcurvature of certain types singular connections.

Definition 1.2. Let G → P → X be a principal G bundle, where G is a FrechetLie group. A singular G-connection on P is a closed subset C ⊂ X, and asmooth Ehresmann G-connection A on P |X−C .

The above definition is basic, as one often puts additional conditions, see forinstance [7], [23].

1.1.1. A non-metric measure of curvature. Let G as above be a Frechet Lie group,we denote by lieG its Lie algebra and let

n : lieG → R

be an Ad invariant Finsler norm. For a principal G-bundle P over a Riemannsurface (S, j), and given a G connection A on P define a 2-form αA on S by:

αA(v, jv) = n(RA(v, jv)),

where RA is the curvature 2-form of A, withRA(v, w) ∈ lieAutPz,

for z ∈ S, v, w ∈ TzS, Pz fiber over z, and AutPz ' G the group of G-torsorautomorphisms of Pz, where ' means non-canonical group isomorphism.

And define

(1.1) arean(A) =

∫S

αA.

GLOBAL FUKAYA CATEGORY II 3

In the case A is singular with singular set C, αA is defined on S − C so we define

arean(A) =

∫S−C

αA,

with the right hand side now being an extended integral. This area is a non-metricmeasurement meaning that no Riemannian metric on S is needed.

It is possible to extend the functional above to a functional on the space C ofG-connections on principal G bundles P → ∆n. It may seem that ∆n has noconnection to Riemann surfaces but in fact there is an intriguing connection byway of existence of certain axiomatized systems of maps:

u : Sd → ∆n,

where Sd denotes the universal curve over Rd - the moduli space of complex struc-tures on the disk with d + 1 punctures on the boundary, and S

d is Sd with nodalpoints of the fibers removed. Such a chosen system is referred to as U . This wasdeveloped in Part I and plays a principal role there.

There is then a functional:

areaU : C → R≥0,(1.2)

defined with respect to a choice of U . When n = 2 it is just the area functional aspreviously defined.

1.1.2. Abstract resolutions of singular connections. Avoiding generality, supposethat A is a singular G-connection on a principal G-bundle P → Sn, with a singlesingularity x0. We will show that it is possible to control the curvature of thesingular connection A if we impose a certain structure on the singularity of A.The simplest way to do this is to ask for existence of a certain kind of abstractresolution. First a simplicial G-connection on D on P , as defined in Section 13.1,is basically a functorial assignment of a smooth G connection DΣ on Σ∗P for eachsmooth

Σ : ∆n → Sn.

Definition 1.3. For A, P as above a simplicial resolution of A is a simplicial Gconnection Ares on P , with the following properties. Let Σ0 : ∆n → Sn representπn(S

n, x0), thenΣ∗

0|interior∆nA = AresΣ0

|interior∆n .

In the following theorem G = PU(2), n = 4 and the norm n on liePU(2) will betaken to be the operator norm normalized so that the Finsler length of the shortestone parameter subgroup from id to −id is 1

2 . We will omit n in notation. We alsoimpose an additional constraint on Ares, so that the curvature at “∞” is boundedby a threshold, which means the following. Let

Σ∞ : ∆4 → x0

be the constant map, suppose that:

areaU AresΣ∞

< 1/2,

and suppose for simplicity that AresΣ∞

is trivial along the edges of ∆4, later on thiscondition is generalized, see Proposition 13.4. We say in this case that Ares is asub-quantum resolution. The following is proved in Section 13.

4 YASHA SAVELYEV

Theorem 1.4. Let P → S4 be a non-trivial principal PU(2) bundle. Let A bea singular PU(2)-connection on P with a single singularity at x0. Then for anysub-quantum resolution Ares of A and for any U as above

areaU (AresΣ0

) ≥ 1/2.

The theorem has certain extensions to Hamiltonian singular connections A, un-derstanding P as a principal Ham(S2) bundle, Section 13.

Here is one basic class of examples.

Example 1.5. Let P be as above, and A′ be an ordinary smooth PU(2) connectionon P . Express S4 as a union of sub-balls D4

± ⊂ S4, intersecting only in theboundary. Suppose that that we have the property that areaU (A′) |D4

−< 1

2 . Let A bethe singular connection on P obtained as the push-forward of A′ by the bundle mapq : P → P over the singular smooth map q : S4 → S4 taking D4

− to a single point∞ ∈ S4, with q|interior D4

−an immersion. Then A has a sub-quantum resolution

Ares essentially by construction, and we will not elaborate. In this case, the theoremabove simply yields that areaU (A′

D4+) ≥ 1/2.

Let us summarize the above example as the following basic differential geometricresult. It can be formally understood as a corollary of Theorem 1.4, as partiallyexplained above, but it is more elementary to see it as a corollary of Theorem 13.4,which appears later.

Corollary 1.6 (Of Theorem 1.4 and of Theorem 13.4). Let P be a non-trivialPU(2) (or Ham(S2)) bundle P → S4, let D4

± be as above and let A be a smoothPU(2) or Ham(S2) connection on P . Suppose that areaU (A) |D4

−< 1

2 , thenareaU (A) |D4

+≥ 1

2 .

The proof of even the above corollary traverses the entirety of the theory here.As this is a very elementary result we may hope for a simpler argument. It isquite non-obvious how to do this even for PU(2). In particular the computationof the quantum Maslov class, in Section 10, alone is insufficient, as we also had todo some kind of algebraic topological gluing of the Floer theory data. In the caseof PU(2), one idea might be to replace Floer theory used here by the technicallysimpler mathematical Yang-Mills theory over surfaces [2]. If we want to mimic theargument presented in this paper, then we should first extend Yang-Mills theoryto work with G-bundles over surfaces with corners and holonomy constraints overboundary. This might be possible, but beyond this things are unclear, since, asmentioned, we also use certain abstract algebraic topology to glue the data, and itis not clear how this would work for Yang-Mills theory.

We may use the same idea as in the example above to “push forward” simpli-cial, (not just smooth) connections to singular connections with more complicatedsingularities, in such a way that we again by construction would have sub-quantumresolutions. In this case Theorem 1.4 no longer has an elementary interpretationas in corollary above.Physical Interpretation. There are possible physical interpretations for singular con-nections, as appearing in the context here. A PU(2) = PSU(2) connection A onP in physical terms represents a Yang-Mills field on the space-time S4. When


the space-time has a black hole singularity, the fields solving the Einstein-Yang-Mills equations (mathematically connections as above) likewise develop singular-ities. There is a wealth of physics literature on this subject, and I don’t knowwhat has the highest priority, but here is one reference [14]. As quantum gravityis often related to simplicial ideas, it is not inconceivable that the mathematicalsub-quantum resolution condition above also has a physical interpretation.

At this point the reader may be curious why Theorem 1.1 has something todo with Theorem 1.4. We cannot give the full story, but the idea is that thesimplicial fibration Fuk∞(P ) only sees the principal bundle P (and its curvature)by the behavior of certain holomorphic curves. When one has the sub-quantumcondition on the curvature of Ares

Σ∞, certain holomorphic curves are ruled out so that

from the view point of Fuk∞(P ), AresΣ∞

is the trivial connection, (its curvature isundetectable) but Fuk∞(P ) is non-trivial as a fibration so that the aforementionedholomorphic curves and consequently curvature must appear elsewhere.

1.2. First quantum obstruction and smooth invariants. It is very temptingto use the theory of the global Fukaya category to find new invariants of smoothmanifolds. One such invariant is already discussed in Part I, as the homotopy classof the classifying map X → S of the projectivized, complexified tangent bundle of asmooth manifold X. This by itself is not a very practical invariant, but we may tryto extract more manageable invariants from this. We present here a constructionof an integer invariant which is based on our theory. This is probably just thebeginning of the story for invariants of smooth manifolds based on Floer-Fukayatheory.

Let P → X be a Hamiltonian M -bundle, as previously. Let

Fuk∞(P ) → X

be the associated (co)-Cartesian fibration, and let

K(P ) → X

be its maximal Kan sub-fibration as in Lemma 3.2, which we may understand asjust a usual Serre topological fibration. Define

q-obs(P ) ∈ N t ∞,

to be the degree of the first obstruction to a section of K(P ). That is q-obs(P ) isthe smallest integer n such that there is no section of K(P ) over the n skeleton ofX. over the n − 1 skeleton of X does not extend to a section over the n-skeleton.When no such n exists we set q-obs(P ) = ∞.

Theorem 1.7. Let S2 → P → S4 be a non-trivial Hamiltonian fibration then:

q-obs(P ) = 4.

Indeed the proof of Theorem 1.1 can be understood as showing that the associ-ated obstruction class in

H4(S4, π3(NFuk(S2)))

is non-trivial.

6 YASHA SAVELYEV

1.2.1. First quantum obstruction as a manifold invariant. Let X be a smooth man-ifold, and let P (X) denote the fiber-wise projectivization of TX⊗C. We then define

q-obs(X) := q-obs(P (X)) ∈ N t ∞,

which is then an invariant of the smooth manifold X. Either this invariant isexpressible in terms of classical invariants, which would be fascinating since theconstruction is in terms pseudo-holomorphic curves or this invariant is new, thatis not expressible in classical terms, which would also be interesting. There areof course gauge theory based invariants of smooth (3,4)-folds, like Donaldson andSeiberg-Witten invariants. I do not see any connections of the above to theseinvariants at the moment, even in dimension 4. It should be noted that this “firstquantum obstruction” invariant is only sensitive to the tangent bundle, whereasfor example Donaldson invariants can see finer aspects of the smooth structure. Infact the “quantum Novikov conjecture” of Part I would immediately imply that thefirst quantum obstruction is only a topological invariant of X. So that the abovementioned conjecture is a sense very pessimistic, as its negation means new smoothstructure sensitive invariants of manifolds.

1.3. Hamiltonian rigidity vs flexibility. By way of the calculation we also ob-tain an application in Hofer geometry. It can be understood as a relative analogueof a result in [19]. Let Lag(M,L0) denote the space of oriented Lagrangian sub-manifolds of a symplectic manifold (M,ω), Hamiltonian isotopic to L0, we mayalso just write Lag(M). Let ΩL0

Lag(M) denote the space of based smooth loopsin Lag(M), constant near end points, and let Ωtaut

L0Lag(M) ⊂ ΩL0

Lag(M) be thesubspace of loops taut concordant to the constant loop at L0. The notion of tautconcordance is defined in more generality in Definition 6.5. In the case above, twoloops

p1, p2 ∈ ΩL0Lag(M)

are said to be taut concordant if the following holds. We have a Lagrangian sub-fibration

L ⊂ Cyl ×M, Cyl = S1 × [0, 1],

so that L over the boundary circles corresponds, in the natural sense, to the pairp1, p2, so that there is a Hamiltonian connection A on M × [0, 1] preserving L, andso that the coupling form ΩA of A vanishes on L, see Section 6.1 for the definitionof coupling forms.

Note that of course Lag(S2) is homotopy equivalent to Lageq(S2) ' S2 whereLageq(S2) denotes the space of oriented equators in S2. Moreover, there is anembedding

i : ΩS2 → ΩtautL0

Lag(S2),

this is because two loops p1, p2 ∈ ΩLageq(S2) ' ΩS2 are taut concordant iff theyare homotopic in Lageq(S2), see Lemma 10.4.

Theorem 1.8. Let L0 ⊂ S2 be the equator. And let

f : S2 → ΩtautL0

Lag(S2),

represent i∗g, for g the generator of

π2(Ω(S2)) ' π3(S

2)) ' Z,


and i as above. Then we have identity for the systole with respect to L+:min

f ′,[f ′]=[f ]maxs∈S2

L+(f ′(s)) = 1/2 · area(S2, ω),

where L+ denotes the positive Hofer length functional, as defined in Section 10.1.1.The minimum is attained on a cycle of equators in S2.

Even though everything is now smooth, this is not obvious. For by contrastif we measure a related quantity of the “girth” (infimum of the diameter of arepresentative) of the generator [g] of π2Lag(S

2), as in [15], then there is an upperbound for this girth which is smaller then the lower bound for girth consideredin the subspace of Lag(S2) consisting of equators. So moving from equators togeneral oriented S1 Lagrangians in S2 we may reduce the girth to less than theclassically expected girth. By “classical” we mean for the classical objects: greatcircles. Indeed, it may be that girth of the generator

[g] ∈ π2Lag(S2)

is actually 0. (This would rather astonishing however.) On the other hand, ourtheorem says that this kind of non-classical squeezing does not happen at all forthe systole we consider. In other words whereas our systole exhibits Hamiltonianrigidity, the girth in [15] while closely related, exhibits flexibility.

Theorem 1.8 is proved in section 12. On the way in Section 9.1 we construct thequantum Maslov classes. We show their non-triviality in Section 10. The Sections12, 9.1, 10 are mostly logically independent of the ∞-categorical and even the A∞setup and may be read independently. Theorems 1.1, 1.7 are proved in Section 4.2,they are basic consequences of the main technical lemma.

2. Acknowledgements

I am grateful to RIMS institute at Kyoto university and Kaoru Ono for the in-vitation, financial assistance and a number of discussions which took place there.Much thanks also ICMAT Madrid and Fran Presas for providing financial assis-tance, and a lovely research environment during my stay there.

Contents

1. Introduction 11.1. Application in basic Riemannian geometry 21.2. First quantum obstruction and smooth invariants 51.3. Hamiltonian rigidity vs flexibility 62. Acknowledgements 73. Outline 84. Qualitative description of the perturbation data 104.1. Extending Dpt to higher dimensional simplices 104.2. The main lemma and immediate consequences 125. Towards the proof of Lemma 4.5 136. Hamiltonian fibrations and taut structures, holomorphic sections and

area bounds 146.1. Coupling forms 146.2. Hamiltonian structures on fibrations 156.3. Area of fibrations 18

8 YASHA SAVELYEV

6.4. Gluing Hamiltonian structures 207. Construction of small data 248. The product µ4

Σ4−(γ1, . . . , γ4) and the quantum Maslov classes 25

8.1. Constructing suitable Ar 268.2. Restructuring the data Ar 288.3. Computing [ev(Hn, A0)] 309. Quantum Maslov classes 319.1. Definition of the quantum Maslov classes 3210. Computation of the quantum Maslov class Ψ(a) 3210.1. Morse theory for the Hofer length functional 3211. Finishing up the proof of Lemma 4.5 3712. Proof of Theorem 1.8 3813. Singular and simplicial connections and curvature bounds 3913.1. Simplicial connections 39Appendix A. Homotopy groups of Kan complexes 41Appendix B. On the Maslov number 43B.1. Dimension formula for moduli space of sections 44References 44

3. Outline

In what follows when we say Part I we shall mean [20]. We will mostly followthe notation setup in Part I. The reader may review the basics of simplicial sets,as used by us, in Section 3 of Part I. For a more detailed introduction, which alsoincludes some theory of quasi-categories, we recommend Riehl [16]. Here are somespecific summary points.

Notation 3.1. We use notation ∆n to denote the standard topological n-simplex.For the standard representable n-simplex as a simplicial set we use the notation∆n

• . When X is a smooth manifold X• will mean the simplicial set of singularsimplices of X. (Similarly for topological spaces.) That is X•([n]) is the set ofsmooth maps ∆n → X. Likewise if p : X → Y is a map of spaces p• : X• → Y•will mean the induced simplicial map. X• can also denote an abstract simplicial setwhen there is no possibility of confusion. We will denote abstract Kan complexesor quasi-categories by calligraphic letters e.g. X ,Y.

Let us briefly review what we do in Part I. Let M → Pp−→ X be a Hamiltonian

fibration. Denote by ∆(X) := ∆/X• the smooth simplex category of X, withobjects smooth maps Σ : ∆n → X and morphisms commutative diagrams:

∆n ∆m

X,

Σ0

mor

Σ1

where mor : ∆n → ∆m is a simplicial map, that is a linear map taking vertices tovertices, preserving the order. The distinction with Simp(X) is that in ∆(X) weallow mor to be degenerate.


Then given certain auxiliary perturbation data D, which in particular involvesa choice of a natural system of maps from the universal curves to ∆n and certainchoices of Hamiltonian connections, we construct in Part I a functor

F : ∆(X) → A∞ − Cat,

where A∞ − Cat denotes the category of A∞ categories. The properties of thisfunctor are such that we may algebraically, naturally construct from this a functor

Funit : ∆(X) → A∞ − Catunit,

with A∞−Catunit denoting the category of unital A∞ categories, by taking unitalreplacements. In what follows we rename F by F raw and Funit by F , as this willvisually simplify notation.

We then defineFuk∞(P ) = colim∆(X) NF,

which is shown to be an ∞-category whose equivalence class (under concordance) isindependent of all choices. This also has the structure of a co-Cartesian fibration:

NFuk(M) → Fuk∞(P ) → X•,

where NFuk(M) is the A∞ nerve of the Fukaya category of M and X•. The notionof co-Cartesian fibration corresponds to, in categorical language, a relaxation of thenotion of Serre/Kan fibrations. Indeed what will do here is extract from the abovedata a Kan fibration and work with that, since then we can just use standard toolsof topology. To this end we have the following elementary lemma.

Lemma 3.2. Suppose we have a (co)-Cartesian fibration p : Y → X , where Xis a Kan complex. Let K(Y) denote the maximal Kan sub-complex of Y thenp : K(Y) → X is a Kan fibration.

The proof is given in Appendix A. In particular by the above lemma K(P ) :=K(Fuk∞(P )) is a Kan fibration over X•.

Definition 3.3. We say that a Kan fibration or a (co)-Cartesian fibration P overa Kan complex X is non-trivial if it is not null-concordant. Here P is null-concordant means that there is a Kan respectively (co)-Cartesian fibration

Y → X ×∆1•,

whose pull-back by i0 : X → X ×∆1• is trivial and by i1 : X → X ×∆1

• is P. Herethe two maps i0, i1 correspond to the two vertex inclusions ∆0

• → ∆1•.

Theorem 3.4. Suppose that P → S4 is a non-trivial Hamiltonian S2 fibration thenp• : K(P ) → S4

• does not admit a section. In particular K(P ) is a non-trivial Kanfibration over S4

• and so Fuk∞(P ) is a non-trivial (co)-Cartesian fibration over S4• .

This is the main technical result of the paper. Although in a sense we justare just deducing existence of a certain holomorphic curve, for this deduction weneed a global compatibility condition involving multiple moduli spaces, involvedin multiple local datum’s of Fukaya categories, so that this computation will notstraightforward.

The proof will be aided by constructing suitable perturbation data, and willbe split into a number of sections. As previously indicated the arguments arequiet general, however for concreteness and to simplify an already rather complexframework we focus on a special case.

10 YASHA SAVELYEV

4. Qualitative description of the perturbation data

A bit of possibly non-standard terminology: we say that A is a model for B insome category, with weak equivalences, if there is a morphism mod : A → B whichis a weak-equivalence. The map mod will be called a modelling map. In our contextthe modeling map mod always turns out to be a monomorphism.

Let Fuk(S2) denote the Z2-graded A∞ category over Q, with objects orientedspin Lagrangian submanifolds Hamiltonian isotopic to the equator. Our particularconstruction of Fuk(M) is presented in Part I.

Denote by Fukeq(S2) ⊂ Fuk(S2) the full sub-category obtained by restrictingour objects to be equators in S2. We take our perturbation data Dpt for constructionof Fuk(S2) so that the following is satisfied. All the connections A(L,L′) for L,L′ ∈Fukeq(S2) are PU(2)-connections. For L intersecting L′ transversally the PU(2)connection A(L,L′) is the trivial flat connection. For L = L′ the correspondingconnection is generated by an autonomous Hamiltonian.

The associated cohomological Donaldson-Fukaya category DF (S2) is equivalentas a linear category over Q to FH(L0, L0) (considered as a linear category withone object) for L0 ∈ Fuk(S2).

It is easily verified that a morphism (1-edge) f is an isomorphism in the nerveNFuk(S2), see Part I for definitions, if and only if it corresponds, under the nerveconstruction N , to a morphism in Fuk(S2) that induces an isomorphism in DF (S2).Such a morphism will be called a c-isomorphism.

Consequently the maximal Kan subcomplex K(S2) of NFuk(S2) is characterizedas the maximal subcomplex of this nerve with 1-simplices the images by N of c-isomorphisms in Fuk(S2).

Remark 4.1. It would be interesting (and likely elementary) to identify the homo-topy type of K(S2).

4.1. Extending Dpt to higher dimensional simplices. Let us model D4• and

S3• as follows. Take the standard representable 3-simplex ∆3

•, and the standardrepresentable 0-simplex ∆0

•. Then collapse all faces of ∆3• to a point, that is take

the colimit of the following diagram:

(4.1)

∆0•

∆2• ∆2

• ∆2• ∆2

•

∆3•

i0 i1i2

i3

Here ij are the inclusion maps of the non-degenerate 2-faces. This gives a simplicialset S3,mod

• modelling the simplicial set S3• , in other words there is a natural a weak-

equivalenceS3,mod• → S3

• .

Now take the cone on S3,mod• , denoted by C(S3,mod

• ), and collapse the one non-degenerate 1-edge. The resulting simplicial set D4,mod

• is our model for D4•, it may


be identified with a subcomplex of D4• so that the inclusion map mod : D4,mod

• → D4•

induces a weak homotopy equivalence of pairs(4.2) (D4,mod

• , S3,mod• ) → (D4

•, S3•).

We set b0 ∈ D4• to be the vertex which is the image by mod of the unique 0-vertex

in D4,mod• .

Suppose we have a commutative diagram:

D4 S4 D4

S3

h+ h−

i i

where i : S3 → D4 is the natural boundary inclusion, s.t. h± : D4 → S4 aresmooth, with their images covering S4, s.t.

h+(D4) ∩ h−(D

4)

is contained in the image E ofh± i : S3 → S4,

and so that h± takes b0 to x0. For example we may just let h− to represent thegenerator of π4(S

4, x0) and for h+ to be constant map to x0. We call such a pairh± a complementary pair.

We setD± := h±(D

4,mod• ) ⊂ S4

•

and we set Σ± ∈ S4• to be the image by h± of the sole non-degenerate 4-simplex of

D4,mod• . We also set

∂D± := h±(∂D4,mod• ),

where ∂D4,mod• ) is the image of the natural inclusion S3,mod

• → D4,mod• .

Fix a Hamiltonian frame for the fiber Px0of P over x0, in other words a dif-

feomorphism S2 → Px0that is in the maximal atlas of trivializations of P as a

Ham(S2) structure group bundle. In particular this allows us to identify Fuk(S2)with F raw(x0). Denote by x0,• the image of the map

∆0• → S4,mod

• ,

induced by the inclusion of the 0-simplex x0. Recall from Part I that given pertur-bation data for a non-degenerate simplex, we assigned extended perturbation datafor all degeneracies of this simplex. So our data Dpt for x0 induces perturbationdata for all degeneracies x0, that is for all simplices of x0,•, this data will again bedenoted by Dpt.

Fix an object L0 ∈ Fukeq(S2) ⊂ F raw(x0). Denote by γ ∈ homF raw(x0)(L0, L0)the generator of FH1(L0, L0), i.e. the fundamental chain, so that it corresponds tothe identity in DF (L0, L0). This γ is uniquely determined by our conditions andcorresponds to a single geometric section. Denote by Li

0 the image of L0 by theembedding

F raw(x0) → F raw(Σ+)

corresponding to the i’th vertex inclusion into ∆4, i = 0, . . . , 4.Let mi be the edge between i− 1, i vertices and set

mi := Σ+ mi.

12 YASHA SAVELYEV

Let Σ0i denote the 0-simplex obtained by restriction of Σ4 to the i’th vertex. Note

that each mi is degenerate by construction, so we have a induced morphismsF raw(pr) : F raw(mi) → F raw(x0)

corresponding (F raw is a functor) to the degeneracy morphism in ∆(S4):pr : mi → Σ0

i .

Finally, for each Li−10 , Li

0 we have a c-isomorphism

γi : Li−10 → Li

0

in F raw(mi) ⊂ F raw(Σ+) which corresponds to γ, meaning that the fully-faithfulprojection F raw(pr) takes γi to γ. We will denote by γi,j the analogous morphismsLi0 → Lj

0.

Notation 4.2. Let us abbreviate from now the morphism spaces homF raw(Σ±) byhomΣ± , and µd

F raw(Σ±) by µdΣ±

.

Definition 4.3. We call perturbation data D for P small if it is extends the dataDpt as above, and if with respect to D

(4.3) µdΣ+

(γ1, . . . , γd) = 0, for 2 < d < 4,

where (γ1, . . . , γd) is a composable chain, and each γi is of the form γi,j as above.

We will see further on how to construct such small data, assume for now that itis constructed.

Let fJ, corresponding to an n-simplex, be as in the definition of the A∞ nervein Part I, where J is a subset of [n] = 0, . . . , n.

Lemma 4.4. Let D be small as above, then there is a a 4-simplex σ ∈ NF raw(Σ+)with faces determined by the conditions:

• fJ = 0, for J any subset of [4] with at least 3 elements.• fi−1,i = γi for γi as before.

Proof. This follows by (4.3) and by the identity µ2Σ+

(γ, γ) = γ.

If we take our unital replacements so that γ corresponds to the unit then σinduces a section of K(P+) → D+, where K(P±) will be shorthand for K(P )restricted over D±.

Leti :

(K(P+)|∂D+ := p−1

• (∂D+))→ K(P−),

be the natural inclusion map. Setsec = i σ h+|∂D4,mod

•.

4.2. The main lemma and immediate consequences.

Lemma 4.5. Suppose that P is a non-trivial Hamiltonian fibration and D smalldata for P as above, then sec as above does not extend to a section of K(P−).

The proof of this lemma, which will be broken up in parts, will follow shortly.

Proof of Theorem 1.7. Clearly q-obs(P ) ≥ 4, since the 3-skeleton of S4 is trivial.By Lemma 4.5 above K(P ) does not have a section over the 4-cell representing thegenerator of π4(S

4).


Remark 4.6. When P is obtained by clutching with a generator of π3(PU(2)), andwhen h± are embeddings, the class [sec] in π3(K(P−)) ' π3(K(S2)) can be thoughtof as “quantum” analogue of the class of the classical Hopf map.

Proof of Theorem 3.4. It might be helpful to first review Appendix A before readingthe following. If we take any small perturbation data D for P , then the first partfollows immediately by Lemma 4.5. So K(P ) is non-trivial as a Kan fibration.This then implies that Fuk∞(P ) is non-trivial as a (co)-Cartesian fibration, whichmeans specifically that its classifying map

fP : S4 → (S, NFuk(S2))

is not null-homotopic.To see this, suppose otherwise that we have a null-homotopy H of fP , then this

gives a (co)-Cartesian fibration

P → S4• × I•,

restricting to Fuk∞(P ) over S4• × 0• and to NFuk(S2) × S4

• over the other endS4• ×1•. Here 0•, respectively 1• are notation for the simplicial sets ij,• : ∆0

• → ∆1•,

j = 0, 1, where ij,• are induced by the pair of boundary point inclusions.Now take the maximal Kan sub-fibration of P, then by Lemma 3.2 we obtain a

trivialization of K(P ) which is a contradiction.

Proof of Theorem 1.1. Theorem 3.4 implies that the group homomorphismZ ' π4(BHam(S2, id) −→ π4(S, NFuk(S2)),

has vanishing kernel, so that the result follows.

5. Towards the proof of Lemma 4.5

We will denote by L0,• the image of the map ∆0• → K(P−), induced by the

inclusion of L0 into K(S2) as a 0-simplex. Suppose that sec extends to a sectionof K(P−), so we have map

e : D4,mod• → K(P−)

extending sec over ∂D4,mod• . We may assume that e lies over h−, meaning

p• e = h−.

Since it can be homotoped to have this property. To see this, first take a relativehomotopy of

p• e : (D4,mod• , ∂D4,mod

• ) → (D−, ∂D−)

to h−, using that we have a homotopy equivalence of pairs (4.2), and then lift thehomotopy to a relative homotopy upstairs using the defining lifting property of Kanfibrations.

And so we have a 4-simplexT = e(Σ4) ∈ K(P−)

projecting to Σ− ∈ D− by p•. Since T is in the image of e, all but one 3-faces ofT are totally degenerate with image in L0,•, and with this exceptional 3-face beingthe sole non-degenerate 3-face of sec, (of sec(∂D4,mod

• )).Then if mi,j , γi,j are as in the previous section but corresponding now to Σ−

rather then Σ+, by the boundary condition on e, the edges of T correspond under

14 YASHA SAVELYEV

the nerve construction to the γi,j , since this is the corresponding condition for theedges of sec.

Lemma 5.1. For D small as above, and for the unital replacement F of F raw asabove, the simplex T exists if and only if µ4

Σ−(γ1, . . . , γ4) is exact.

Proof. The following argument will be over F2 as opposed to Q as the signs will notmatter. Recall that we take the unital replacement so that γ ∈ homF raw(Px0

)(L0, L0)corresponds to the unit in the unital replacement.

Now if T ∈ K(P−) as above exists, then it corresponds under unital replacementto a 4-simplex T ′ ∈ NF raw(Σ−) satisfying the following, by the nerve construction,condition on its 4-face f[4] ∈ homΣ−(L

00, L

40):

(5.1) µ1Σ−

f[4] =∑

1<i<4

f[4]−i +∑s

∑(J1,...Js)∈decomps

µsΣ−

(fJ1, . . . , fJs

).

By our conditions on the boundary of T , by the condition on the unital replacement,and by the conditions in Lemma 4.4, we must have fJ = 0, for every proper subsetJ ⊂ [4], in some length s decomposition of [4], unless J = i, j in which casefi,j = γi,j . Given this (5.1) holds if and only if µ4

Σ−(γ1, . . . , γ4) is exact.

We are going to show that for a certain small D0, µ4Σ−

(γ1, . . . , γ4) does not vanishin homology, which will finish the argument. However the calculation will requiresignificant setup.

6. Hamiltonian fibrations and taut structures, holomorphic sectionsand area bounds

We collect here some preliminaries on moduli spaces of holomorphic sections offibrations with Lagrangian boundary constraints, and the closely related curvaturebounds. There is a likely new discussion involving taut Hamiltonian structures, butmuch of this material has previously appeared elsewhere, perhaps in less generality.We will eventually need all that is presented in this section, but the reader mayonly skim on the first reading.

6.1. Coupling forms. We refer the reader to [11, Chapter 6] for more details onwhat follows. A Hamiltonian fibration is a smooth fiber bundle

M → P → X,

with structure group Ham(M,ω) with its C∞ Frechet topology. A Hamiltonianconnection is just an Ehresmann connection for a Hamiltonian fibration.

A coupling form, originally appearing in [6], for a Hamiltonian fibration M →P

p−→ X, is a closed 2-form Ω on P whose restriction to fibers coincides with ω andwhich has the property: ∫

M

Ωn+1 = 0 ∈ Ω2(X),

with integration being integration over the fiber operation. Such a 2-form de-termines a Hamiltonian connection AΩ, by declaring horizontal spaces to be Ω-orthogonal spaces to the vertical tangent spaces. A coupling form generating agiven connection A is unique. A Hamiltonian connection A in turn determinesa coupling form ΩA as follows. First we ask that ΩA induces the connection A


as above. This determines ΩA up to values on A-horizontal lifts v, w ∈ TpP ofv, w ∈ TxX. We specify these values by the formula

(6.1) ΩA(v, w) = RA(v, w)(p),

where RA is the lie algebra valued curvature 2-form of A. Specifically, for each x,RA|x is a 2-form valued in C∞

norm(p−1(x)) - the space of 0-mean normalized smoothfunctions on p−1(x).

6.2. Hamiltonian structures on fibrations. Let S be a Riemann surface withboundary, with punctures in the boundary, and a fixed structure of strip charts atends, positive or negative, as in Part I. Let M → S

pr−→ S be a Hamiltonian fiberbundle, with model fiber monotone symplectic manifold (M,ω), with distinguishedHamiltonian bundle trivializations

[0, 1]× (0,∞)×M → S

at the positive ends, and with distinguished Hamiltonian bundle trivializations

[0, 1]× (−∞, 0)×M → S,

at the negative ends. These are collectively called called strip charts. Given thestructure of such bundle trivializations we say that S has end structure.

Definition 6.1. LetL ⊂ (S|∂S = pr−1(∂S)) → ∂S

be a Lagrangian sub-bundle, with model fiber an object, in the sense of Part I, (inparticular a spin oriented Lagrangian submanifold) so that L is a constant sub-bundle in the strip chart trivializations above. We say in this case that L respectsthe end structure. In the strip chart coordinates at the end ei, let Lj

i be the fibersof L over

j × t, j = 0, 1, by assumption t independent.

For L as above, we say that a Hamiltonian connection A on S is compatiblewith the connections Ai on [0, 1] ×M for each end ei, if in the strip coordinatechart at the ei end, A is flat and R-translation invariant and has the form Ai whereAi denotes its R-translation invariant extension Ai to (0,±∞)×R×M , dependingon whether the end is positive or negative. We say that a Hamiltonian connectionA on S as above, is L-exact if A preserves L (this means that the horizontal spacesof A are tangent to L).

For A compatible with Ai as above, a family jz of fiber wise ω-compatiblealmost complex structures on S will be said to respect the end structure if ateach end ei, in the strip coordinate chart above, the family jz is R-translationinvariant and is admissible with respect to Ai, in the sense of Part I, Definition 5.3.The data Θ = (S, S,L,A, jz), with A compatible with Ai, jz, respecting theend structure, will be called Hamiltonian structure.

We will normally suppress jz in the notation and elsewhere for simplicity, asit will be purely in the background in what follows, (we do not need to manipulateit explicitly).

Definition 6.2. Let (S, S,L,A) be a Hamiltonian structure, we say that a smoothsection σ of S → S is asymptotically flat if at each end ei of S, σ C1-converges

16 YASHA SAVELYEV

to an A-flat section. Specifically, in the strip coordinates for a positive end, thismeans that there is a A-flat section

σ : [0, 1]× (0,∞) → [0, 1]× (0,∞)×M,

so that for every ε > 0 there is a t > 0 so thatdC1(σ, σ|[0,1]×[t,∞)) < ε.

Similarly for a negative end. Given a pair of asymptotically flat sections σ1, σ2,with boundary in L, we say that they have the same relative class if they areasymptotic to the same flat sections at each end, and are homologous, relativeboundary condition and relative the asymptotic constraints at the ends. The set ofrelative classes will be denoted by Hsec

2 (S,L).

6.2.1. Families of Hamiltonian structures.

Definition 6.3. A family Hamiltonian structure or henceforth just Hamilton-ian structure, consists of the following:

(1) A smooth, connected, compact, oriented manifold K with boundary, (orcorners).

(2) For each r ∈ K a Hamiltonian structure (Sr, Sr,Lr,Ar), so that there aresmooth fibrations

S → S p1−→ K, S → S p−→ K,

and Sr, Sr correspond to the fibers of the first and second fibration,respectively, where the second fibration has fiber a Riemann surface, so thatSr = p−1r. The first fibration is a fibration whose fibers p−1

1 (r) arethemselves the total spaces of a smooth Hamiltonian fibration Sr → Sr, sothat the structure group of S p1−→ K can be reduced to smooth Hamiltonianbundle maps of the fiber. To elaborate further, let

M → S → S

be a Hamiltonian M -fibration over a Riemann surface S. Let Aut denotethe group of Hamiltonian M -bundle automorphisms of S. Then S p1−→ K isthe associated bundle P ×Aut S for some principal Aut bundle P over K.

(3) The chartsei,r : [0, 1]× (0,∞)×M → Sr,

for the positive ends, fit into a Hamiltonian M -bundle diffeomorphism ontothe image:

(6.2) ei : [0, 1]× (0,∞)×K ×M → S,similarly for the negative ends.

(4) We then have an induced smooth r-family of connections e∗i,rAr on [0, 1]×(0,±∞)×M , and an induced smooth r-family of Lagrangian subfibrationse−1

i,rLr over ∂[0, 1]× (0,±∞). We ask that

∀r : e−1i,rLr = 0 × (0,±∞)× L0

i ∪ 1 × (0,±∞)× L1i ,

where Lji are as in Definition 6.1 and we ask that

∀r : e∗i,rAr = Ai

for Ai,Ai as previously.


(5) There is a Hamiltonian connection A on S → S that extends all the con-nections Ar (in the natural sense), and preserving L := ∪rLr.

We will write Sr, Sr,Lr,ArK for this data, K may be omitted from notationwhen it is implicit.

In the notation above, if there exists a Hamiltonian connection A on S → S asin the last point, so that ΩA vanishes on L, we will say that Sr, Sr,Lr,ArK is ahyper taut Hamiltonian structure.

6.2.2. Moduli spaces of sections of Hamiltonian structures. Let Θ = (S, S,L,A) bea Hamiltonian structure. For a section σ of S define its vertical L2 energy by

e(σ) :=

∫S

|πvert dσ|2,

πvert : T S → T vertS

is the A-projection, for T vertS the vertical tangent bundle of S, that is the kernel ofthe projection T S → TS. Define M(Θ) to be the Gromov-Floer compactification ofthe space of J(A)-holomorphic sections σ of S, with finite vertical L2 energy, (alsocalled Floer energy), and with boundary on L. Note that for any JA-holomorphicσ we have an identity:

e(σ) =

∫S

σ∗ΩA,

and ΩA vanishes on L by the condition that A preserves L, so that the stan-dard energy controls apply, to deduce the standard Gromov-Floer compactificationstructure.

More generally, if Θr = Sr, Sr,Lr,ArK is a Hamiltonian structure, let

M(Θr)

be the Gromov-Floer compactification of the space of pairs (σ, r), r ∈ K with σ aJ(Ar)-holomorphic, finite vertical L2 energy section of Sr, with boundary on Lr.

We also denote byM(Θr, A) ⊂ M(Θr)

the subset corresponding to relative class A curves, where the latter is as definedabove.

Let Θr = (Sr, Sr,Lr,Ar) be a Hamiltonian structure, then for each end ei ofSr we have a Floer chain complex

CF (L0i , L

1i ,Ai, jz),

(independent of r by part 4 of Definition 6.3) generated by Ai-flat sections of[0, 1]×M , with boundary on L0

0, L01. This is defined as in Section 6.1 of Part I. We

may also abbreviate CF (L0i , L

1i ,Ai, jz) by just CF (L0

i , L1i ).

Definition 6.4. We say that Θr is A-regular if the pairs (Ai, jz) are regularso that the Floer chain complexes CF (Ai, jz) are defined, and if M(Θr, A) isregular, (transversely cut out). And Θr is regular if it is A-regular for all A.We say that it is A-admissible if there are no elements

(σ, r) ∈ M(Θr, A),

for r in a neighborhood of the boundary of K.

18 YASHA SAVELYEV

Definition 6.5. Given a pair Θir = S0

r , S0r ,L0

r,A0rK, i = 1, 2, of Hamiltonian

structures we say that they are concordant if there is a Hamiltonian structureTr = Tr, Tr,L′

r,A′rK×[0,1],

with an oriented diffeomorphism (in the natural sense, preserving all structure)

S0r , S

0r ,L0

r,A0rKop t S1

r , S1r ,L1

r,A1rK → Tr, Tr,L′

r,A′rK×∂I ,

where op denotes the opposite orientation for K.

Definition 6.6. We say that a Hamiltonian structure Θr = Sr, Sr,Lr,ArKis taut if for any pair r1, r2 ∈ K, Θr1 is concordant to Θr2 by a concordanceTτ , Tτ ,L′

τ ,A′τ[0,1] which is a hyper taut Hamiltonian structure.

Definition 6.7. Given an A-admissible pair Θir, i = 1, 2, of Hamiltonian struc-

tures, we say that they are A-admissibly concordant if there is an A-admissibleHamiltonian structure

Tr, Tr,L′r,A′

rK×[0,1],

which furnishes a concordance. If this concordance is in addition a taut Hamiltonianstructure, then we say that these pairs are A-admissibly taut concordant.

Lemma 6.8. Let Θr = Sr, Sr,Lr,Ar be A-regular and A-admissible, with Sr

having one distinguished negative end e0, and let γ0 be the asymptotic constraint ate0 of the class A. For Lj

0, j = 0, 1 as above. DefineevA = ev(Θr, A) := #M(Θr, A)γ0 ∈ CF (L0

0, L10),

where #M(Θr, A, γ0) means signed count of elements when the dimension is 0,and is otherwise set to be zero. Suppose that CF (L0

0, L10) is defined with respect to

the connection A0 and that this chain complex is perfect. Then evA is a cycle sinceCF (L0

0, L10) is perfect, and its homology class depends only on the A-admissible

concordance class of Θr.

Proof. Suppose we are given an A-admissible concordance (which we may assumeto be regular)

T = Tr, Tr,L′r,A′

rK×[0,1]

between Hamiltonian structures Θ0r, and Θ1

r. Then we get a one dimensionalcompact moduli space M(Tr, Tr,L′

r,A′r, A). By assumption on the perfection of

CF (L00, L

01), boundary contributions from Floer degenerations cancel out, so that

the boundary is:∂M(Tr, Tr,L′

r,A′r, A) = M(Θrop tM(Θr,

where op denotes opposite orientation. From which the result follows.

6.3. Area of fibrations.

Definition 6.9. For a Hamiltonian connection A on a bundle M → S → S, withS a Riemann surface, define

area(A) = infα∫S

α |ΩA + π∗(α) is nearly symplectic,(6.3)

where ΩA is the coupling form of A, α is a 2-form on S, and where ΩA + π∗(α)nearly symplectic means that

(ΩA + π∗(α))(v, jv) ≥ 0,


for v, jv horizontal lifts with respect to ΩA, of v, jv ∈ TzS, for all z ∈ S.

Note that area(A) could be infinite if there are no constraints on A at the ends.However, when the infimum above is finite it is attained on the 2-form determinedby:

(6.4) αA(v, jv) := |RA(v, jv)|+,

where v ∈ TzS, RA(v, w) as before identified with zero mean smooth function onthe fiber Sz over z and | · |+ is operator: |H|+ = maxSz

H.

Lemma 6.10. Let (S, S,L,A) be Hamiltonian data. For σ ∈ M(S, S,L,A) wehave

−∫S

σ∗ΩA ≤ area(A).

Proof. We have ∫S

σ∗(ΩA + π∗α) ≥ 0,

whenever ΩA + π∗(α) is nearly symplectic, by the defining properties of JA and byσ being JA-holomorphic. From which our conclusion follows.

Lemma 6.11. Let (St, St,Lt,At)[0,1] be a taut concordance. Let σj, j = 0, 1 beasymptotically flat sections of Sj in relative class A. Then

−∫S1

σ∗1ΩA1

= −∫S0

σ∗0ΩA0

,

whenever both integrals are finite. In particular, for a Hamiltonian structure(S, S,L,A)

∫Sσ∗ΩA depends only on the relative class of A, whenever the inte-

gral is finite.

Proof. By the hypothesis, there is a connection A on S, extending each At and sothat ΩA vanishes on L ⊂ S. The first part then follows by Stokes theorem. Hereare the details. For σj as above and for each end ei, cut off the part of the sectionσj lying over [0, 1] × (tδ1,δ2 ,∞) in the corresponding strip chart at the end. Heretδ1,δ2 is such that σ0|[0,1]×t is C1 δ1-close to σ1|[0,1]×t for all t > tδ1,δ2 and foreach end, and is such that∫

[0,1]×(tδ1,δ2,∞)

σ∗j |[0,1]×(tδ1,δ2

,∞)ΩAj< δ2, j = 1, 2,

for each end ei. Call the sections with the ends cut off as above by σδ1,δ2j , they are

sections over the compact surfaces Scutj , with ends correspondingly cut off. Then

by Stokes theorem, using that ΩA is closed and using the vanishing of ΩA on L:for each ε there exists δ1, δ2 such that∫

Scut1

(σδ1,δ21 )∗ΩA −

∫Scut0

(σδ1,δ20 )∗ΩA < ε,

and ∫Scutj

(σδ1,δ2j )∗ΩAj −

∫Sj

σ∗jΩAj < ε, j = 1, 2.

20 YASHA SAVELYEV

The last part of the lemma follows, as A preserving L immediately implies thatΩA vanishes on L, so that a constant concordance

(St, St,Lt,At)[0,1]is taut.

Definition 6.12. For σ a relative class A section of Θ = (S, S,L,A) let us call:

−∫S

σ∗ΩA,

the A-coupling area of σ, denoted by carea(Θ, σ), we may also write carea(Θ, A)for the same quantity. By the lemma above this is an invariant of the taut concor-dance class of Θ.

Definition 6.13. Given Hamiltonian structure Θ = (S, S,L,A) we will say thatΘ is A-small if

area(Θ) < carea(Θ, A).

Similarly, given a taut Hamiltonian structure Sr, Sr,Lr,ArK we say that it is isA-small near boundary if (Sr, Sr,Lr,Ar) is A-small for r in a neighborhood ofthe ∂K.

Lemma 6.14. Suppose that Θ = (S, S,L,A) is A-small then M(Θ, A) is empty.Or as a contrapositive, if M(Θ, A) is non-empty then:

carea(Θ, A) ≤ area(Θ).

Proof. This is just a reformulation of Lemma 6.10.

Lemma 6.15. Let Sr, Sr,Lr,ArK be a taut Hamiltonian structure with K con-nected, so that in particular, for each r, Θr = (Sr, Sr,Lr,Ar) is taut concor-dant to a fixed Θ. Suppose that Sr, Sr,Lr,ArK is A-small near boundary thenSr, Sr,Lr,ArK is A-admissible for all A such that carea(Θ, A) > 0.

Proof. Follows immediately by the lemma above.

6.4. Gluing Hamiltonian structures. A Hamiltonian connection A on [0, 1]×Mis determined by a choice of a function H : [0, 1]×M → R, normalized to have meanzero at each moment. The holonomy path of A is a path φA : [0, 1] → Ham(M,ω),generated by the Hamiltonian H. Given L0 ∈ Lag(M) we get a path φA : [0, 1] →Lag(M) starting at L0, defined by φA(t) = φA(t)(L0). We will say that these pathsare generated by A or by H, with the latter as above.

Let (S, S,L,A) be a Hamiltonian structure. Where at each end ei the corre-sponding connection Ai is determined by L± length κi, where L± is as in Section10.1.1.

Let D denote Riemann the surface which is topologically D2 − z0, z0 ∈ ∂D2,endowed with a choice of a strip chart at the end (positive or negative dependingon context). The complex structure j here is as induced from C under the assumedembedding D2 ⊂ C.

We may then cap off some of the open ends eini=0 of S by gluing at the endscopies of D with oppositely signed end. More explicitly, in the strip coordinatecharts at some, say positive, end ei of S, excise [0, 1] × (t,∞) for some t > 0,call the resulting surface S − ei. Likewise excise the negative end of D, call this


surface D− end. Then glue S− ei with D− end, along their new smooth boundarycomponents. Let us denote the capped off surface by S/. Since S is naturallytrivialized at the ends, we may similarly cap off Sr over the ei end by gluing witha bundle D ×M at the end obtaining a Hamiltonian M bundle S/ over S/.

More generally we have a certain gluing operation of Hamiltonian structures. Inthe case of “capping off” as above we glue Θ = (S, S,L,A) with the Hamiltonianstructure Θ′ = (D×M,D,L′,A′) at the ei end, provided A′ is compatible with theconnection Ai, in the sense of Section 6.2, and provided L is compatible with L′.The latter means that Lj

i = L′ji where these are Lagrangians corresponding to the

strip chart trivialization of L,L′ at the corresponding ends, as in Definition 6.1.Let us name the result of this capping off Θ#iΘ

′. The following is immediate:Lemma 6.16. Suppose that ΘrK, Θ′

rK with Θ′r = (D×M,D,L′

r,A′r) are taut

Hamiltonian structures. Then:Θr#iΘ

′rK

is taut, whenever the gluing operation is well defined, that is whenever we havecompatibility of connections and Lagrangian sub-fibrations at the corresponding end.Definition 6.17. Let π : R → [0, 1] denote the retraction map, sending (−∞, 0]to 0, and sending [1,∞) to 1. Fix a parametrization ζ : R → ∂D, which satisfiesζ(t) ∈ 0 × (0,±∞) for t ∈ (−∞, 0], and ζ(t) ∈ 1 × (0,±∞) for t ∈ t[1,∞),where we are using the coordinates of the strip chart e0 : [0, 1] × (0,±∞) → D.Given a smooth path

p : [0, 1] → Lag(M)

constant near 0, 1, let Lp ⊂ ∂D ×M denote the Lagrangian subfibration over ∂D,with fiber over r ∈ ∂D given by p π(r). We say that a Lagrangian subfibration Las above is determined by p if L = Lp, after a fixed choice of parametrization ofboundary of D by R.Lemma 6.18. Let p and Lp ⊂ ∂D ×M be as in Definition above with L±(p) = ρ,where p is some lift of p to Ham(M,ω), that is p(t) = p(t)(p(0)). Let A0 be aHamiltonian connection on [0, 1]×M , generated by a Hamiltonian H with L± lengthκ, constant near the end points. Then there is a Hamiltonian connection Ap

0 onD ×M , preserving Lp, and compatible with respect to A0, with area(A0) ≤ κ+ ρ.The construction is natural in the sense that (p,A0) 7→ A0 can be made into asmooth map (of Frechet spaces).Proof. Let q : [0, 1] → Ham(M,ω) be the holonomy path of A0, q(0) = id, gen-erated by H. Let p · q be the concatenation in diagramatic order, and H ′ be itsgenerating Hamiltonian.

Define a coupling form Ω′ on D2 ×M :Ω′ = ω − d(η(rad) ·H ′dθ),

for (rad, θ) the modified angular coordinates on D2, θ ∈ [0, 1], 0 ≤ rad ≤ 1, andη : [0, 1] → [0, 1] is a smooth function satisfying

0 ≤ η′(rad),

and

(6.5) η(rad) =

1 if 1− δ ≤ rad ≤ 1,

rad2 if rad ≤ 1− 2δ,

22 YASHA SAVELYEV

for a small δ > 0. By elementary calculationarea(A′) = L+(p · q) = L+(p) + L+(q),

where A′ is the connection induced by Ω′. Set arc = (1, θ), 0 ≤ θ ≤ 1/2. Letarcc denote the complement of arc in ∂D2. Fix a smooth embedding i : D2 → Dso that the image of the embedding contains ∂D − end where end is the image ofthe distinguished (say positive) end strip chart

[0, 1]× (0,∞) → D,

so that i(arc) ⊂ endc, and so that i(arcc) ⊂ end as illustrated in the Figure 1.

Figure 1.

Next fix a deformation retraction ret of D onto i(D2), so that in the strip chartabove ret, for r ≥ 1, is the composition i param pr, for

pr : [0, 1]× (0,∞) → [0, 1]

the projection and forparam : [0, 1] → arcc ⊂ D2

a diffeomorphism. Finally set Ω = ret∗Ω′ on D×S2, and set Ap0 to be the induced

Hamiltonian connection. As constructed A0 will be compatible with A0, whenthe end of D is positive. When the end is negative we take the reverse pathsp−1, q−1.

Using the above lemma, we may then put a L/-exact Hamiltonian connection,A/ on S/ (see Definition 6.2), with

(6.6) area(A/) ≤ area(A) + κi.

This connection is obtained by gluing with the Hamiltonian structure (D×M, Ap=const0 )

as obtained in the Lemma 6.18 above.

Lemma 6.19. Let L0 ⊂ M be a monotone Lagrangian submanifold with mono-tonicity constant const > 0: ω(A) = const · µ(A), µ the Maslov number. Let

Θ := Θr := Sr, Sr,Lr,ArKbe a Hamiltonian structure with K connected. And let Θ

/r = (S

/r , S

/r ,L/

r ,A/r) be

obtained from Θr by capping off each end ei, so that (6.6) is satisfied. Supposethat the Floer chain complex CF (L0

i , L1i ,Ai) is perfect for each i and that Ai is

generated by a time dependent Hamiltonian Hi with L± length κi.For a given A ∈ Hsec

2 (S,L), if

∀r : area(Ar) < carea(Θ/r , A

/)−∑i

κi,


where A/ is the capping off of A is described in the proof, then M(Θr, A) isempty. Moreover, if Lr is the trivial bundle with fiber L0 for each r and Ar islikewise trivial over the boundary of Sr then

∀r : carea(Θ/r , A

/) = −const ·Maslovvert(A/),

where Maslovvert is as in Appendix B.

Proof. Suppose otherwise that we have an element (σ0, r0) ∈ M(Θr, A). Supposefor the moment that M(Θr, A) is regular.

There is a PSS isomorphism

QH(L) → FH(L,L),

where the right hand side is defined using our construction in terms of flat sections,and the left hand side is interpreted for example as the Pearl complex complex ho-mology, [3]. We won’t give the full construction of this map, as it just reformulationof well known constructions, as presented in [3] for instance. But we can quicklysay how to get PSS([L]). Let

Θ− = (D ×M,D,L,A−),

be the Hamiltonian structure with e0 being a negative end, L trivial with fiber L(which is an object as before), and

A− := Ap=const0 ,

with right hand side as in Lemma 6.18, for p being the constant path. Suppose thatΘ− is regular. Define PSS([L]) as the homology class of the chain c determinedby:

〈c, γ〉 =∑A

#ev(Θ−, A),

where the sum is over all classes A ∈ Hsec2 (D × M,L), which have asymptotic

constraint γ, for γ a geometric generator of CF (L,L,A0). As the PSS map is anisomorphism in our monotone context, as is well known, and since CF (L0

i , L1i ,Ai)

is perfect for each i, the asymptotic constraint γi of σ0 at each ei end must be hitby the PSS map, (note that at the negative ends γi are interpreted in the dualcomplex CF (L0

i , L1i ,Ar

i ), where Ari is the reverse of Ai).

By gluing with the holomorphic sections of M ×D coming from the constructionof the PSS maps for each end ei, we obtain a J(A/

r0)-holomorphic, class A/ sectionσ/0 of Θ/

r0 .By Lemma 6.14:

carea(Θ/r0 , A

/) ≤ area(A/r0) ≤ area(Ar0) +

∑i

κi,

socarea(Θ/

r0 , A/)−

∑i

κi ≤ area(Ar0),

so that we contradict the hypothesis and in the case M(Θr, A) is regular we aredone with the first part of the lemma. When it is not regular instead of gluing justpre-glue to get a holomorphic building σ

/0 , and the conclusion follows by the same

argument.

24 YASHA SAVELYEV

To prove the last part of the lemma, note that in this case each Θ/r is taut

concordant toΘ0 = (D2 ×M,D2,L,Atr),

with L trivial with fiber L0, and for Atr the trivial connection. Andcarea(Θ0, ·) = −const ·Maslovvert(·)

as functionals on Hsec2 (D2 ×M,L) with const > 0. It follows by Lemma 6.11 that

carea(Θ/, σ/0) = carea(Θ0, σ

/0) = −const ·Maslovvert(σ

/0) = −const ·Maslov(A/).

7. Construction of small data

To forewarn, we use here notation and notions from Part I, especially fromSections 4, 5 in Part I. Let mi denote the edges of ∆4 as before.

Let mkk=dk=1 be a composable sequence in Π(∆n), which we recall means that

the target of mi−1 is the source of mi for each i. Recall from Part I that theperturbation data D in particular specifies for each n and for each such composablesequence certain maps

u(mk, n) : Ed → ∆n,

where Ed is the universal curve over Rd, and Ed denotes Ed with nodal points of

the fibers removed. Collective such a chosen collection of maps satisfying certainaxioms is denoted U . And we have already mentioned this in the introduction. Therestriction of u(mk, n) to the fiber Sr of E

r over r, is denoted by u(mk, n, r),which may also be abbreviated by ur.

D also specifies for each Σ : ∆n → X, each r, each composable chain (m1, . . . ,md)in Π(∆n), and for each chain of objects L′

0, . . . , L′d with L′

i ⊂ P |mi(1), i ≥ 1,L′0 ⊂ P |m1(0), a Hamiltonian connection on

(7.1) Sr := (Σ u(mk, n, r))∗P → Sr.

Here as before mi = Σmi. When n = 4, Σ = Σ+, and ∀i : L′i = L0 with the latter

as before we write these connections as A+r (mk), further abbreviated by A+

r asmk will be implicit in what follows.

Suppose that D extends Dpt from before. If Lr ⊂ Sr|∂Sr denotes the trivialLagrangian sub-bundle with fiber L0, then we obtain Hamiltonian structure (foreach composable d-chain mk) Θ+ = Θ+

r = Sr,Sr,Lr,A+r Rd

, which were dis-cussed in the previous section. By the properties of these connections, necessitatedby D, at each end ei of Sr, A+

r is compatible with the connection Ai = A(L0, L0),where A(L0, L0) is the connection on [0, 1] × S2 part of our Floer data D. ThenΘ+ is trivially taut since for each r Lr is naturally trivial and Ar is likewise trivialover ∂Sr, for each r, by assumed properties of these connections.

Set~ :=

1

2area(S2, ω).

Let κ denote the L± length of the holonomy path in Ham(S2) of A0 = A(L0, L0).We may suppose that(7.2) ∀r : area(A+

r ) < ~− 5κ,

is satisfied after taking κ to be sufficiently small. So that in particular A+r is small

for each r, with small as defined in the previous section.


Fix a complex structure j0 on M , and let Jr be the family of complex structureson Sr induced by (A+

r , j0).

Lemma 7.1. As in Part I, let

M = M(γ1, . . . , γd; γ0,Σ+, Jr, A),

denote the set of elements of M(Θ+, A) with asymptotic constraints γi at each eiend. Here each γk, k 6= 0, is of the form γi,j where this is as before. Then wheneverthe class A is such that M has virtual dimension 0, and d satisfies 2 < d ≤ 4, Mis empty.

Proof. LetΘ/ := (Θ+)/,

and A/ be as in Section 6.4. For a fixed r, by the Riemann-Roch (Appendix B) weget that the expected dimension of M(Θ/, A/) is

1 +Maslovvert(A/).

Consequently, when γ0 = γ, the expected dimension of M is:

(7.3) 1 +Maslovvert(A/)− 1 + (dimRd = d− 2).

We need the expected dimension of M to be 0, and d ≥ 3, so Maslovvert(A/) ≤ −1.But Maslovvert(A/) = −1 is impossible as the minimal positive Maslov number is2. Consequently, the result follows by Lemma 6.19 and by the property (7.2).

When γ0 is the Poincare dual to γ, we would get Maslovvert(A/) ≤ −2 so forthe same reason the conclusion follows.

So if we choose our data D so that the hypothesis of the lemma above aresatisfied, then with respect to D:

µ2Σ4

±(γi,j , γj,k) = γi,k(7.4)

µ3Σ4

±(γ1, . . . , γ3) = 0, for γi as above(7.5)

µ4Σ4

+(γ1, . . . , γ4) = 0.(7.6)

In particular this D is small.

8. The product µ4Σ4

−(γ1, . . . , γ4) and the quantum Maslov classes

The productµ4Σ4

−(γ1, . . . , γ4)

a priori depends on various choices, like the choices of h±, and then choice of dataD0. However by Lemma 6.8, so long as there is a homotopy of the choices, togetherwith a homotopy of associated perturbation data Dt, so that Dt is small for all t,the above product is t-invariant. Thus, in particular for the purpose of computationwe may take h+ to be the constant map to x0 and

h− : (D4, ∂D4) → (S4, x0)

the complementary map, that is representing the generator of π4(S4, x0) ' Z, we

further suppose that h− is an embedding in the interior of D4.

26 YASHA SAVELYEV

Let Σ− be the corresponding 4-simplex of S4• as before. We need to study the

moduli spaces

(8.1) M(γ1, . . . , γ4; γ0,Σ−, Ar, A),

where Ar now denotes the connections on

(8.2) Sr := (Σ− u(m1, . . . ,m4, 4, r))∗P → Sr,

part of some small data D0 as above. We abbreviate u(m1, . . . ,m4, 4, r) by ur inwhat follows.

By the dimension formula (7.3), since we need the expected dimension of (8.1)to be zero, the class A/ satisfies:

Maslovvert(A/) = −2,

and we must have γ0 = γ0,4, in other words the latter morphism corresponds tothe fundamental chain.

Notation 8.1. From now on, by slight abuse, A0 refers to various section classesof various Hamiltonian structures such that the associated class A

/0 satisfies:

Maslovvert(A/0) = −2.

8.1. Constructing suitable Ar. To get a handle on (8.1) we want to constructvery special small data D0.

A Hamiltonian S2 fibration over S4 is classified by an element

[g] ∈ π3(Ham(S2), id) ' π3(PU(2), id) ' Z.

Such an element determines a fibration Pg over S4 via the clutching construction:

Pg = D4− × S2 tD4

+ × S2 ∼,

with D4−, D4

+ being 2 different names for the standard closed 4-ball D4, and theequivalence relation ∼ is (d, x) ∼ g(d, x),

g : ∂D4− × S2 → ∂D4

+ × S2, g(d, x) = (d, g(d)−1(x)).

We suppose that x0 = h±(b0) that previously appeared, is a point common toD4

± ⊂ S4.From now on Pg will denote such a fibration for a non-trivial class [g]. Note

that the fiber of Pg over the base point x0 ∈ S3 ⊂ D4± (chosen for definition of

the homotopy group π3(Ham(S2), id)) has a distinguished, by the construction,identification with S2. Take A to be a connection on P ' Pg which is trivial in thedistinguished trivialization over D4

+. This gives connections

A′r := (ur)

∗A

on Sr,ur = Σ− ur.

By the last axiom for the system U introduced in Part I, we may choose ur sothat the family ur(Sr) induces a singular foliation of S4, that is smooth outsidex0, with x0 being the image by ur of the ends (images of ei) and the boundaryof each Sr, and so that each ur is an embedding on the complement of u−1

r (x0).Denote by E the subset S3 ⊂ S4 bounding D4

±. We may in addition suppose thateach ur intersects E transversally, again on the complement of u−1

r (x0).


By the above, the preimage by ur of E contains a smoothly embedded curve cras in Figure 2, and ur takes cr into E. This cr not uniquely determined but wemay fix a family r 7→ cr, with parametrizations

cr : R → Sr

so that cr maps (−∞, 0) diffeomorphically onto the image by e0 of 0×(−∞, 0), sothat likewise cr maps (1,∞) diffeomorphically onto the image by e0 of 1×(−∞, 0),and so that cr is a C0 continuous family. We set:

cr := ur cr.In Figure 2, the regions R± are the preimages by ur of D4

± ⊂ S4, and cr boundsR−. It follows that cr likewise induces a singular foliation of the equator E ' S3

Figure 2. The labels L0 indicate that the Lagrangian subbundleis constant with corresponding fiber L0. The curve cr bounds R−.

that is smooth outside x0.So each A′

r is flat in the region R+, in fact is trivial in the distinguished trivi-alization of Sr over R+, corresponding to the distinguished trivialization of P overD4

+. Likewise we have a distinguished trivialization of Sr over R−, correspondingto the distinguished trivialization of P over D4

−. In this latter trivialization let

φr : R → Ham(S2, ω)

be the holonomy path of A′r over cr. Then by construction,

φr|(−∞,0]t[1,∞) = id,

so that we may definef(r) ∈ ΩL0Lag(M,L0),

byf(r)(t) = φr(t)(L0), t ∈ [0, 1]

where the right hand side means apply an element of Ham(S2) to L0 to get a newLagrangian. We will say that f(r) is generated by A′

r.Note that by construction

(8.3) φr(t) = g(cr(t)), t ∈ [0, 1],

28 YASHA SAVELYEV

if we identify cr(t) with an element of S3.Let D2

0 ⊂ R4 be an embedded closed disk D2, not intersecting the boundary∂R4, so that ∂D2

0 is in the normal gluing neighborhood N of ∂R4, where N is asin Part I.

So we have a continuous map

f : D20 → ΩL0

Lag(S2).

And f(∂D20) = L0, with the right hand side denoting the constant loop at L0.

Then by construction and (8.3) in particular, f ' lag, where ' is a homotopyequivalence, and where

(8.4) lag : S2 → ΩL0Lag(S2)

is the compositionS2 g′

−→ ΩidPU(2) → ΩL0Lageq(S2),

for g′ naturally induced by g, and for the second map naturally induced by the map

PU(2) → Lageq(S2), φ 7→ φ(L0).

We then deform each A′r to a connection Ar, which is as follows. In the region

R+ Ar is still flat, but at each end ei, Ar is compatible with A(L0, L0), where thisis as in Section 6.2, and so that Ar is still trivial over the boundary of Sr.

Since Sr and A′r are trivial for r ∈ Rd −D2

0, with trivialization induced by thetrivialization of P+, and since the condition (7.2) holds, we may insure that

(8.5) area(Ar) < ~− 5κ,

for r in the complement of D20. In other words Ar extends to a system of con-

nections corresponding to small data D0 for P , as intended.

8.2. Restructuring the data Ar. Applying Lemma 6.19 we see that the result-ing Hamiltonian structure H := Sr,Sr,Lr,Ar is A0-admissible. We now furthermold this data for the purposes of computation.

First cap off the ends ei, i 6= 0, of Sr as in Section 6.4. This gives a Hamiltonianstructure

H∧ := S∧r , S

∧r ,L∧

r ,A∧r K=D2

0,

satisfyingarea(A∧

r ) + κ < ~,for each r. Again by Lemma 6.19 H∧ is A0-admissible.

By the now classical gluing of holomorphic curves it follows that

(8.6) [µ4Σ4(γ1, . . . , γ4)] = [ev(H∧, A0)].

It remains to compute the right hand side, to this end we further restructure.Let p1 : [0, 1] → Lag(S2, L0) be the path generated by A(L0, L0), with p1

starting at L0, and where generated has the same meaning as in the previoussection. Suppose we have defined pi−1, set Li−1 := pi−1(1) and define pi to be thepath in Lag(S2, L0) starting at Li−1, generated by A(L0, L0). Set p0 := p1 · . . . ·pd,where · is path concatenation in diagrammatic order. We may assume that L0 istransverse to L4 = p0(1) by adjusting the connection A(L0, L0) if necessary. Thendeform L∧

r in the Hamiltonian data S∧r , S

∧r ,L∧

r ,A∧r as in Figure 3. The resulting

Lagrangian subbundle over ∂Sr will be denoted Lnr , n here stands for ‘new’.


Figure 3. Over the boundary components with black labels Li

the Lagrangian subbundle Lnr is constant with corresponding fiber

Li. Over the i’th red boundary component the Lagrangian sub-bundle corresponds to the path of Lagrangians pi, analogously toDefinition 6.17 further below. Likewise over the blue boundarycomponent the Lagrangian subbundle corresponds to the path ofLagrangians p−1

0 . In the red striped regions we have removed thecurvature of the connection, the blue striped regions we have added it.

We simultaneously deform A∧r to an Ln

r exact Hamiltonian connection Anr which

satisfies the following conditions. Anr is flat in the entire region R+ (which includes

the red shaded finger regions). Along the dotted line (which is contained in the stripchart at the e0 end) An

r is the trivial connection in the distinguished trivialization atthe end, and such that at the e0 end, which is down in the Figure 3, the connectionis unchanged over [0, 1] × (t,∞), for t large. In order to get such a deformation,we introduce curvature in the blue stripped region of Figure 3. We name this newHamiltonian data by

Hn := S∧r , S

∧r ,Ln

r ,Anr.

30 YASHA SAVELYEV

As L±(pi), for each pi, can be arranged to be arbitrarily small, it is clear that wemay choose the deformation from H∧ to Hn to be small near boundary (Definition6.13) and hence be an A0-admissible concordance. More specifically, we may choosea concordance from H∧ to Hn so that for the associated family of connections Ar,t,

Ar,0 = A∧r ,Ar,1 = An

r ,

so that for each r,d

dt|RAr,t

(v, jv)|+ < 0, v ∈ TzSr

for each z ∈ Sr, except for z in the region which is blue striped in Figure 3. However,the area increase in this blue region is bounded from above by L+(p−1

0 ), so that

∀t : | area(Ar,t)− area(A∧r )| ≤ L+(p−1

0 ).

In fact we can arrange thatd

dtarea(Ar,t) = 0,

since the gain of area in the blue striped region is exactly equal to the loss of areain the red striped regions, but this extra precision is not necessary.

Of course:[ev(H∧, A0)] = [ev(Hn, A0)]

since the corresponding Hamiltonian data are A0-admissibly concordant. This fin-ishes our restructuring.

8.3. Computing [ev(Hn, A0)]. If we stretch the neck along the dashed line inFigure 3, the upper half of the resulting building gives us new Hamiltonian data

H0 = S0r , S

0r ,L0

r,A0r.

By the classical theory of continuation maps in Floer homology we clearly havethat

[evn] := [ev(Hn, A0)] ∈ FH(L0, L0)

is non-zero iff[ev0] := [ev(H0, A0] ∈ FH(L0, L4)

is non-zero.Let PL0,L4Lag(M) denote the space of smooth paths in Lag(M) from L0 to L4.

Letf ′ : D2

0 → PL0,L4Lag(S2),

be like f but defined with respect to A0r, so that

(8.7) f ′(t) = g(cr(t))(p0(t)),

if we suppose that the holonomy path of A0r over cr in the trivialization over R+

generates p0 (which can be insured by adjusting A0r or the parametrizations

cr). Here the right hand side of (8.7) means as before: apply an element ofHam(S2) to a Lagrangian to get a new Lagrangian. In this case f ′ takes ∂D2

0 top0 ∈ PL0,L4Lag(M). In particular, f ′ represents a class a ∈ π2(PL0,L4Lag(M), p0).

In what follows we omit specifying the parameter space D20 for r, since it will be

the same everywhere. Let Lp be as in the Definition 6.17.


Lemma 8.2. The A0-admissible Hamiltonian structure H0 = S0r ,S0

r ,L0r,A0

r isA0-admissibly concordant to

Θ′ = D × S2,D,Lf ′(r),Br,

for certain Hamiltonian connections Br (which are not explicitly relevant yet).

Proof. Let R± ⊂ S∧r be as before. Fix a deformation retraction

retr : S0r × I → S0

r ,

of S0r onto R−, smoothly in r. Since A0

r is flat over R+, the pull-back by retr ofthe data H0 then induces an A0-admissible concordance between H0 and

D × S2,D,Lf ′(r),Br = ret∗rA0r,

once we use smooth Riemann mapping theorem to identify each R− ⊂ S0r with its

induced complex structure jr with (D, jst), smoothly in r.

9. Quantum Maslov classes

The class of ev(Θ′, A0) is related to what we christen as quantum Maslov classes.These are relative analogues of the quantum characteristic classes [17]. The namequantum Maslov class is meant to be suggestive, as the classical Maslov numbersare relative analogues of Chern numbers, while quantum characteristic classes aredirectly related (via semi-classical approximation) to Chern classes, [19]. 1

We will not give extensive detail here since we don’t need the full theory, wepresent it because it gives extra perspective. The ordinary relative Seidel morphismappears in Seidel’s [22] in the exact case and further developed in [9] in the monotonecase. Let Lag(M) denote the space whose components are objects of Fuk(M) inthe previous sense, so in particular oriented, spin, Hamiltonian isotopic Lagrangiansubmanifolds of M . We may also denote the component of L by Lag(M,L). Thenthe relative Seidel morphism is a functor

S : ΠLag(M) → DF (M),

where ΠLag(M) is the fundamental groupoid of Lag(M) and DF (M) is the Donaldson-Fukaya category of M , see also [5], [4] which can be understood as an extension.

We sketch how this works. To a path p in Lag(M) from L0 to L1 we have have anassociated Lagrangian subbundle Lp of D ×M over the boundary, as in Definition6.17. Extend this to a Hamiltonian structure

Θp = (D ×M,D,Lp,Ap0)

where Ap0 is as in Lemma 6.18. Assuming Θp is regular, we define S([p]) ∈

DF (L0, L1) by

S([p]) =∑A

[ev(Θp, A)],

where by monotonicity only finitely many A can have non-zero contribution.

1The latter may be renamed “quantum Chern classes”.

32 YASHA SAVELYEV

9.1. Definition of the quantum Maslov classes. Let M be as before, and letP(L0, L1) denote the space of smooth paths in Lag(M) from L0 to L1, constant in[0, ε]∪ [1−ε, 1] for some 0 < ε < 1. There is then an additive group homomorphism:

(9.1) Ψ : H∗(P(L0, L1),Q) → FH(L0, L1)

defined analogously to above and to [17] in non-relative context. Although formallywe will only need the restriction of Ψ to spherical classes.

This works as follows. To a smooth cycle

f : B → P(L0, L1)

for B a smooth closed oriented manifold, we may associate a Hamiltonian structure

D ×M,D,LbB ,

Lb := Lf(b) a Lagrangian subbundle of M × D over ∂D determined by f(b) asbefore. The end of D here is negative. Now let A0 be a Hamiltonian connectionon [0, 1]×M , so that A0(L0) is transverse to L1 where A0(L0) ⊂ 1×M denotesthe A0-transport over [0, 1] of L0 ⊂ 0 ×M .

For each b the space of Hamiltonian connections Lb-exact with respect to A0, (asin Section 6.2) is contractible, c.f. [1]. So we get an induced Hamiltonian structure:

Θf = D ×M,D,Lb,Ab

well defined up to concordance.We may then define Ψ([f ]) by:

Ψ([f ]) =∑A

[ev(Θf , A)],

where again by monotonicity only finitely many A can give non-zero contribution.It is immediate that Ψ is an additive group homomorphism.

Remark 9.1. We should mention that the morphism Ψ extends to a certain functorto DF (M), see [4] for a related discussion.

Given the definition above,

[ev(Θ′, A0)] = Ψ(a)

clearly holds, as A0 is the only class that can contribute to Ψ(a), since by thedimension formula (B.1) only a class A with Maslovvert(A/) = −2 can contribute.

10. Computation of the quantum Maslov class Ψ(a)

10.1. Morse theory for the Hofer length functional. Under certain conditionsthe spaces of perturbation data for certain problems in Gromov-Witten theory ad-mit a Hofer like functional. Although these spaces of perturbations are usuallycontractible, there may be a gauge group in the background that we have to re-spect, so that working equivariantly there is topology. The reader may think of theanalogous situation in Yang-Mills theory [2].

Without elaborating too much, the basic idea of the computation that we willperform consists of cooling the perturbation data as much as possible (in the senseof the functional) to obtain a mini-max (for the functional) data, using which wemay write down our moduli spaces explicitly. This idea was first used in [18].


10.1.1. Hofer length. For p : [0, 1] → Ham(M,ω) a smooth path, define

L+(p) :=

∫ 1

0

maxM

Hpt dt,

L−(p) :=

∫ 1

0

maxM

(−Hpt )dt,

L±(p) := maxL+(p), L−(p),

where Hp : M × [0, 1] → R generates p normalized by the condition that for eacht, Hp

t := Hp|M×t has mean 0, that is∫M

Hpt dvolω = 0. Also define

L+lag : PLag(M) → R,

L+lag(p) :=

∫ 1

0

maxp(t)

Hpt dt,

p(0) = L and where Hp : M × [0, 1] → R is normalized as above and generates a liftp of p to Ham(M) starting at id. By lift we mean that p(t) = p(t)(p(0)). (That isHp generates a path in Ham(M), which moves L0 along p.) Some theory of thislatter functional is developed in [10]. We may however omit the subscript lag fromnotation, as usually there can be no confusion which functional we mean.

Note that Lageq(S2) is naturally diffeomorphic to S2 and moreover it is easyto see that the functional L+|Lageq(S2) is proportional to the Riemannian lengthfunctional Lmet on the path space of S2, with its standard round metric met.

Let now L0, L1 ∈ Lageq(S2) be any transverse pair, and

f ′ : S2 → P(L0, L1) := PL0,L1Lageq(S2),

be the generator of the group H2(P(L0, L1),Z). The idea of the computation isthen this: perturb f ′ to be transverse to the (infinite dimensional) stable manifoldsfor the Riemannian length functional on

P(L0, L1) := PL0,L1Lageq(S2),

push the cycle down by the “infinite time” negative gradient flow for this functional,and use the resulting representative to compute Ψ(a = [f ′]). Although, we will notactually need infinite dimensional topology.

10.1.2. The “energy” minimizing perturbation data. Classical Morse theory [13]tells us that the energy functional

E(p) =

∫[0,1]

〈p(t), p(t)〉met dt

on P(L0, L1) is Morse non-degenerate with a single critical point in each degree.Consequently a (as a homology class) has a representative in the 2-skeleton ofP(L0, L1), for the Morse cell decomposition induced by E. This follows by White-head’s compression lemma which is as follows.

Lemma 10.1 (Whitehead, see [8]). Let (X,A) be a CW pair and let (Y,B) be anypair with B 6= ∅. For each n such that X−A has cells of dimension n, assume thatπn(Y,B, y0) = 0 for all y0 ∈ B. Then every map f : (X,A) → (Y,B) is homotopicrelative to A to a map X → B.

34 YASHA SAVELYEV

Suppose that a has a representative f ′ : S2 → PL0,L1(S2) mapping into the

n-skeleton Bn for the Morse cell decomposition for E, n > 2. Apply the lemmaabove with (X,A) = (S2, pt), Y = Bn and B = Bn−1 as above. Then the quotientBn/Bn−1 is a wedge of n-spheres and since π2(S

n) = 0 for n > 2, f can behomotoped into Bn−1 by the Whitehead lemma. Descend this way until we get arepresentative mapping into B2.

Furthermore since π2(S1) = 0 such a representative cannot entirely lie in the 1-

skeleton. It follows, since we have a single Morse 2-cell that there is a representativef : S2 → PL0,L1(S

2), for a, s.t. the function f∗E is Morse with a maximizer max, ofindex 2, and s.t. γ0 = f(max) is the index 2 geodesic. We call such a representativeminimizing.

Remark 10.2. In principle there maybe more than one such maximizer max, butrecall that we assumed that a is the generator, so by further deformation we mayinsure that there is only one maximizer. The relevant representative f , with a singlemaximizer max as above, can also be constructed by hand.

It follows that max is likewise the unique index 2 maximizer of the functionf∗Lmet by the classical relation between the energy functional and length func-tional. And so max is the index 2 maximizer of f∗L+.

10.1.3. The corresponding minimizing data.

Lemma 10.3. There is a minimizing representative f0 for the class a and a tautHamiltonian structure

Θf0 = D × S2,D,Lf0(b), (Ab = Af0(b)0 ),

where Af0(b)0 is again as in Lemma 6.18, satisfying:

(10.1) ∀b : area(Ab) = L+(f0(b)).

Proof. Note that a geodesic segment p : [0, 1] → S2 for the round metric met onS2 has a unique lift

p : [0, 1] → PU(2) ' SO(3),

p(0) = id with p a segment of a one parameter subgroup, and in this case

L+lag(p) = L+(p).

It then follows that for a piecewise geodesic path p in S2, there is likewise a uniquelift p satisfying

L+lag(p) = L+(p).

Now if f is a minimizing representative of a, we may homotop it to a likewiseminimizing representative f0 so that for all b f0(b) is piecewise geodesic, this followsby the piecewise geodesic approximation theorem Milnor [13, Theorem 16.2] of theloop space.

Let A0 be the trivial Hamiltonian connection on [0, 1]×M . Use the constructionof Lemma 6.18, to get a family of Hamiltonian connections Af0(b)

0 . In this casesince A0 is trivial

area(Af0(b)0 ) = L+(f0(b)).

It remains to verify that Θf0 = D × S2,D,Lf0(b),Ab, is taut. This follows bythe following more general lemma.


Lemma 10.4. Let Lageq(CPn) denote the space of oriented Lagrangian subman-ifolds Hamiltonian isotopic to RPn, then two loops p1, p2 : S1 → Lageq(CPn) aretaut concordant as defined in Section 1.3 iff they are homotopic.

Proof. Let L be a sub-fibration of Cyl×M as in the definition of taut concordanceof loops. Let A be any PU(n) connection on Cyl × CPn which preserves L (thereare no obstructions to constructing this). Then RA is a liePU(n) valued 2-form,such that for all v, w ∈ TzCyl the vector field X = RA(z)(v, w) is tangent to Lz. Inparticular if HX is the Hamiltonian generating X, then since X is an infinitesimalunitary isometry preserving Lz, HX vanishes on Lz. It follows by the definition ofΩA, that it vanishes on L and so we are done.

So given Ab as in the lemma above, since∀b : area(Ab) = L+(f0(b)),

we immediately deduce:

Lemma 10.5. The function area : b 7→ area(Ab) has a unique maximizer, coin-ciding with the maximizer max of f∗

0Lmet and area is Morse at max with index2.

10.1.4. Finding class A0 holomorphic sections for the data. Let us now rename f0by f , Lf0(b) by Lb, and Θf0 by Θ = Θb.

As f(max) is a geodesic for met, its lift f(max) to SO(3) is a rotation aroundan axis intersecting L0 = f(max)(0) in a pair of points, in particular there there isa unique point

xmax ∈⋂t

(Lt = f(max)(t))

maximizing Hmaxt for each t. In our case this follows by elementary geometry but

there is a more general phenomenon of this form c.f. [10].Define

σmax : D → D × S2

to be the constant section z 7→ xmax. Then σmax is a Amax-flat section with bound-ary on Lmax, and is consequently J(Amax)-holomorphic.

Lemma 10.6.[σmax] = A0.

Proof. SetT vertz Lmax := v ∈ TL ⊂ Tz(D × S2) | pr∗v = 0

where pr : D × S2 → D is the projection. Denote byLag(Txmax

S2 ' Lag(R2) ' S1

the space of oriented linear Lagrangian subspaces of TxmaxS2. Let ρ be the path in

Lag(TxmaxS2) defined by

ρ(t) = T vert(ζ(t),xmax)

Lmax, t ∈ [0, 1]

where ζ : R → ∂D is a fixed parametrization as in Definition 6.17.By our conventions for the Hamiltonian vector field:

ω(XH , ·) = −dH(·),

36 YASHA SAVELYEV

ρ is a clockwise oriented path fromTxmaxL0 := T vert

(ζ(0),xmax)Lmax

toTxmax

L1 := T vert(ζ(1),xmax)

Lmax

for the orientation induced by the complex orientation on TxmaxS2.

By the Morse index theorem in Riemannian geometry [13] and by the conditionthat f(max) has Morse index 2, ρ visits initial point ρ(0) exactly twice if we countthe start, as this corresponds to the geodesic f(max) passing through two conjugatepoints in S2. So the concatenation of ρ with the minimal counter-clockwise pathfrom Txmax

L1 back to TxmaxL0 is a degree −1 loop, if S1 ' Lag(R2) is given the

counter-clockwise orientation. ConsequentlyMaslovvert(σ/

max) = −2,

cf. Appendix B, in other words [σmax] = A0.

Proposition 10.7. (σmax,max) is the sole element of M(Θ, A0).

Proof. By Stokes theorem, since ω vanishes on σmax, it is immediate:

(10.2) carea(Θmax, A0) = −∫Dσ∗maxΩmax = L+(f(max)).

Moreover, since Θ = Θb is taut carea(Θb, A0) = L+(f(max)). So by (10.1) andby Lemmas 6.10, 6.11 we have:

L+(f(max)) ≤ area(Ab) = L+(f(b)),

whenever there is an element(σ, b) ∈ M(Θb, A0).

But clearly this is impossible unless b = max, since L+(f(b)) < L+(f(max)) forb 6= max. So to finish the proof of the proposition we just need:

Lemma 10.8. There are no elements σ other than σmax of the moduli spaceM(Θmax, A0).

Proof. We have by (10.2), and by (10.1)

0 = 〈[Ωmax + αΩmax], [σmax]〉,

and so given another element σ we have:

0 = 〈[Ωmax + αΩmax], [σ]〉.

It follows that σ is necessarily Ωmax-horizontal, since

(Ωmax + αΩmax)(v, JΩmax

v) ≥ 0.

Since JΩmaxby assumptions preserves the vertical and Amax-horizontal subspaces

of T (D× S2), and since the inequality is strict for v in the vertical tangent bundleof

S2 → D × S2 → D,

the above inequality is strict whenever v is not horizontal. So σ must be Amax-horizontal. But then σ = σmax since σmax is the only flat section asymptotic toγ0.


10.1.5. Regularity. It will follow that

Ψ(a) = ±[γ0]

if we knew that (σmax,max) be a regular element of

M(Θb, A0).

We won’t answer directly if (σmax,max) is regular, although it likely is. But it isregular after a suitably small Hamiltonian perturbation of the family Ar vanish-ing at Amax. We call this essentially automatic regularity.

Lemma 10.9. There is a family Aregb arbitrarily C∞-close to Ab with Areg

max =Amax and such that

(10.3) M(D × S2,D,Lb,Aregb , A0),

is regular, with (σmax,max) its sole element. In particular

Ψ(a) = ±[γ0].

Proof. The associated real linear Cauchy-Riemann operator

Dσmax: Ω0(σ∗

maxTvertD × S2

max) → Ω0,1(σ∗maxT

vertD × S2max),

has no kernel, by Riemann-Roch [12, Appendix C], as the vertical Maslov numberof [σmax] is −2. And the Fredholm index of (σmax,max) which is -2, is -1 timesthe Morse index of the function area at max, by Lemma 10.5. Given this, ourlemma follows by a direct translation of [21, Theorem 1.20], itself elaborating onthe argument in [18].

To summarize:

Theorem 10.10. For 0 6= a ∈ H2(PL0,L1Lag(S2),Z),

0 6= Ψ(a) ∈ HF (L0, L1).

Proof. We have shown that 0 6= Ψ(a) ∈ HF (L0, L1), for a the generator of thegroup H2(PL0,L1

Lag(S2),Z). Since Ψ is an additive group homomorphism theconclusion follows.

11. Finishing up the proof of Lemma 4.5

Starting with (8.6) we showed that [µ4Σ4(γ1, . . . , γ4)] is non-vanishing in Floer

homology iff[ev(H0, A0)] ∈ HF (L0, L4),

is non-vanishing. We then use Lemma 8.2 to identify [ev(H0, A0)] with [ev(Θ′, A0)],which is also identified with Ψ(a), for a certain spherical 2-class a. Finally, inSection 10 we compute Ψ(a) and show that it is non-zero. This together withLemma 5.1 imply Lemma 4.5.

38 YASHA SAVELYEV

12. Proof of Theorem 1.8

Suppose otherwise, so that

minf,[f ]=a′

maxb∈S2

L+(f(b)) = U < ~,

for a′ = i∗g as in the statement of the theorem. Fix L1 ∈ Lageq(S2) so that L0

intersects L1 transversally, and so that there is a geodesic path p0 ∈ PLageq(L0, L1)with

κ := L±(p0) < ε = (~− U)/2.

Here p0 is the geodesic lift to PU(2) starting at id. Then concatenating f with p0we obtain a smooth family of paths

g : S2 → P(L0, L1)

g(0) = p0,

and g represents the previously appearing class a, that is the generator of thecorresponding π2 group. Let

Θb = D × S2,D,Lb,AbK=S2 ,

be the corresponding Hamiltonian structure, where Ab = Ag(b)0 is as in Lemma

10.3, for A0 the trivial connection, and where Lb := Lg(b). In particular Θb istaut and satisfies:

(12.1) ∀b ∈ S2 : area(Ab) = L+(g(b)) < ~− κ.

By assumption that each f(b) is taut concordant to the constant loop at L0,each Θb is taut concordant to

H = (D × S2,D,L0,A0),

where L0 = Lp0, A0 = Ap0

0 , where the latter is again as in Lemma 6.18. And sofor each b Θ

/b is taut concordant to H/ by Lemma 6.16.

On the other hand by Lemma 10.4 H/ is taut concordant to the trivial Hamil-tonian structure (D2 × S2, D2,Ltr,Atr), Ltr the trivial bundle with fiber L0 andAtr the trivial Hamiltonian connection. So for each b:

(12.2) carea(Θ/b , A0) = carea(H/, A

/0) = ~.

Now by Theorem 10.10

ev(Θb, A0) = Ψ(a) 6= 0.

And so:

M(Θb, A0) 6= ∅,

but this contradicts the conjunction of (12.1), (12.2), and Lemma 6.19.


13. Singular and simplicial connections and curvature bounds

Let A be a G connection on a principal G bundle P → ∆n, and the Finsler normn on lieG be as in Section 1.1.1 of the introduction. A given system U in particularspecifies maps:

u(m1, . . . ,mn, r, n) : Sr → ∆n,

where r ∈ Rn, Sr is the fiber of Sn over r, and where (m1, . . . ,mn) is the composable

chain of morphisms in Π(∆n), mi being the edge morphism from the vertex i − 1to i. Then define

(13.1) areaU (A) = supr

arean(u(m1, . . . ,mn, r, n)∗A),

where arean on the right hand side is as defined in equation (1.1). In the caseG = Ham(M,ω) we take

n : lieHam(M,ω) → Rto be

n(H) = |H|+ = maxM

H.

Let ω be the area 1 Fubini-Study symplectic 2-form on M = CP1. Then thepull-back by the natural map

lieh : liePU(2) → lieHam(CP1, ω) ' C∞0 (CP1)

of the semi-norm: |H|+ = maxM H is the operator norm on PU(2), up to normal-ization. This will be used to get the specific form of Theorem 1.4, from the moregeneral form here.

13.1. Simplicial connections. We now introduce a certain abstraction of singularconnections, which can partly be understood as simplicial resolutions of singularconnections. Let G → P → X be a principal G bundle, where G is a Frechet Liegroup. Denote by X• the simplicial set whose set of n-simplices, X•(n), consistssmooth maps Σ : ∆n → X, with ∆n standard topological n-simplex with verticesordered 0, . . . , n. And denote by Simp(X•) the category with objects ∪nX•(n) andwith hom(Σ0,Σ1) commutative diagrams:

∆n ∆m

X,

Σ0

mor

Σ1

for mor a simplicial face map, that is an injective linear map preserving order ofthe vertices.

Definition 13.1. Define a singular simplicial G-connection A or hereby justsimplicial G-connection on P to be the following data:

• For each Σ : ∆n → X in X•(n) a smooth G-connection AΣ on Σ∗P → ∆n,(a usual Ehresmann G-connection.)

• For a morphism mor : Σ0 → Σ1 in Simp(X•), we ask that mor∗AΣ1= AΣ0

.

Example 13.2. If A if a smooth G-connection on P define a simplicial connectionby AΣ = Σ∗A for every simplex Σ ∈ X•. We call such a simplicial connectioninduced.

40 YASHA SAVELYEV

If we try to “push forward” a simplicial connection to get a “classical” connectionon P over X, then we get a kind of multi-valued singular connection. Multi-valuedbecause each x ∈ X may be in the image of a number of Σ : ∆n → X and Σ itselfmay not be injective, and singular because each Σ is in general singular so thatthe naive push-forward may have blow up singularities. We will call the above thenaive pushforward of a simplicial connection.

Proof of Theorem 1.4 and Corollary 1.6. We will prove this by way of a strongerresult. Let P be a Hamiltonian fibration S2 → P → S4. And let A a simplicialHam(S2) connection on P . Let σ1

0 ∈ S4• be the degenerate 1-simplex at x0, in other

words the constant map: σ10 : [0, 1] → x0. Let κ be the L± length of the holonomy

path of Aσ10

over [0, 1].Let Σ± ∈ X•(4) be a complementary pair as in Section 4.1. The connection A

gives us a simplicial connection as in Example 13.2. By inductive procedure as inPart I, Lemma 5.6, we may find perturbation data D for P so that for this data

∀r : pr1F(L00, . . . , L

n0 ,Σ±, r) 'δ u(m1, . . . ,ms, r, n)

∗AΣ± ,(13.2)A(L0, L0) 'δ Aσ1

0,(13.3)

where Li0 are the objects as before, where 'δ means δ-close in the metrized C∞

topology, and δ is as small as we like. Here we are using notation of Part I asbefore. Set

ur := Σ− u(m1, . . . ,m4, r, 4),

so ur : Sr → S4. Set Sr := u∗rP, set A′

r := pr1F(L00, . . . , L

n0 ,Σ−, r) and set

Θr := Sr,Sr,Lr,A′r.

Definition 13.3. We say that A is perfect, if for every δ, as above, arbitrarilysmall, A(L0, L0) as above can be chosen so that the corresponding Floer chaincomplex CF (L0, L1,A(L0, L0)) is perfect.

Theorem 13.4. Let A be a perfect simplicial Hamiltonian connections on P . IfP is non-trivial as a Hamiltonian bundle then

(areaU (AΣ+) ≥ ~− 5κ) ∨ (areaU (AΣ−) ≥ ~− 5κ),

Proof. Suppose(13.4) areaU (AΣ+) < ~− 5κ,

then by (13.2), (13.2) and by Lemma 6.19 D as defined above can be assumed tobe small provided δ is chosen to be sufficiently small. Take the unital replacementas in Lemma 5.1. Since we know that K(P ) does not admit a section by Theorem3.4, the simplex T of the Lemma 5.1 does not exist. Hence again by this lemma

ev(Θr, A0) = [µ4Σ−

(γ1, . . . , γ4)] 6= 0.

In particularM(Θr, A0) 6= ∅.

So by Lemma 6.19 there exists an r0 so that(13.5) area(A′

r0) ≥ ~− 5κ′.

where κ′ denotes the L± length of the holonomy path in Ham(S2) of A(L0, L0).By (13.3) κ′ → κ as δ → 0. By (13.2), (13.5) passing to the limit as δ → 0 we get:

areaU (AΣ−) ≥ ~− 5κ.


Corollary 13.5. Let A be a PU(2) connection on a non-trivial principal PU(2)bundle P → S4. Then

(areaU (AΣ+) ≥ ~− 5κ) ∨ (areaU (AΣ−) ≥ ~− 5κ).

Proof. A simplicial PU(2) connection A on a principal PU(2) bundle PU(2) →P ′ → S4 is automatically perfect, when understood as a Hamiltonian connectionon the associated bundle S2 → P → S4. So that this is an immediate consequenceof the theorem above.

To prove Corollary 1.6, we just note that for an induced simplicial Hamiltonianconnection A, as defined in Example 13.2, Aσ1

0is trivial. And hence A is automat-

ically perfect. So that this corollary follows by Theorem 13.4.

Appendix A. Homotopy groups of Kan complexes

For convenience let us quickly review Kan complexes just to set notation. Thisnotation is also used in Part I. Let

∆n• (k) := hom∆([k], [n]),

be the standard representable n-simplex, where ∆ is the simplicial category withobjects ordered finite sets [n] = 1, . . . , n and morphisms order preserving setmaps.

Let Λnk ⊂ ∆n

• denote the sub-simplicial set corresponding to the “boundary” of∆n

• with the k’th face removed, 0 ≤ k ≤ n. By k′th face we mean the face oppositeto the k’th vertex. Let X• be an abstract simplicial set. A simplicial map

h : Λnk ⊂ ∆n

• → X•

will be called a horn. A simplicial set S• is said to be a Kan complex if for alln, k ∈ N given a diagram with solid arrows

Λnk S•

∆n• ,

h

ih

there is a dotted arrow making the diagram commute. The map h will be calledthe Kan filling of the horn h. The k’th face of h will be called Kan filled facealong h.

Given a pointed Kan complex (X , x) and n ≥ 1 the n’th simplicial homotopygroup of (X , x): πn(X , x) is defined to be the set of equivalence classes of maps

Σ : ∆n• → X ,

such that Σ takes ∂∆n• to x•, with the latter denoting the image of ∆0

• → X ,induced by the vertex inclusion x → X.

42 YASHA SAVELYEV

More precisely, we have a commutative diagram:

∆n• ∆0

•

X .

Σx

Example A.1. When X = X• is the simplicial set of singular simplices of atopological space X, the maps above are in complete correspondence with maps:

Σ : ∆n → X,

taking the topological boundary of ∆n to x.

For X• general simplicial set, a pair of maps Σ1 : ∆n• → X•,Σ2 : ∆n

• → X•, areequivalent if there is a diagram, called simplicial homotopy:

∆n•

∆n• × I• X•

∆n• .

Σ1i0

i1Σ2

such that ∂∆n• × I• is taken by H to x•. The simplicial homotopy groups of a

Kan complex (X , x) coincide with the classical homotopy groups of the geometricrealization (|X |, x).

Proof of Lemma 3.2. We refer the reader to Part I, Appendix A.2, for more detailson the notions here. We prove a stronger claim.

Lemma A.2. Let p : Y → X be an inner fibration of quasi-categories Y,X , withX a Kan complex. And let K(Y) ⊂ Y denote the maximal Kan subcomplex. Thenp : K(Y) → X is a Kan fibration.

The above is probably well known, but it is simple to just provide the proof forconvenience.

Proof. By definition of an inner fibration, whenever we are given a commutativediagram with solid arrows and with 0 < k < n:

(A.1)

Λnk K(Y) Y

∆n X,

σ

pΣ

there exists a dashed arrow Σ as indicated, making the whole diagram commutative.When n > 2 the edges of Σ are all automatically invertible in Y, as Σ extends σ,and all edges of σ are invertible by definition. For n = 2 the edges of Σ are eitheredges of σ, or are compositions of edges of σ in the quasi-category Y, and henceagain always invertible.

It follows that Σ maps into K(Y) ⊂ Y. Since the starting diagram was arbitrary,we just proved that p : K(Y) → X is an inner fibration. In particular the pre-images


p−1(Σ(∆n)) ⊂ K(Y) are quasi-categories, for all n, where Σ : ∆n → X is any n-simplex, see Part I, Appendix A.2. But K(Y) is a Kan complex, so that also theabove pre-images p−1(Σ(∆n)) are Kan complexes. It readily follows from this thatp : K(Y) → X is a Kan fibration.

The main lemma then follows, since if p : Y → X is a (co)-Cartesian fibration, itis in particular an inner fibration.

Appendix B. On the Maslov number

Let S be obtained from a compact connected Riemann surface S′ with boundary,by removing a finite number of points ei removed from the boundary of S′.

Let V → S be a rank r complex vector bundle, trivialized at the open ends ei,that is so we have distinguished bundle charts Cr×[0, 1]× [0,∞) → V at the ends.

LetΞ → ∂S ⊂ S

be a totally real rank r subbundle of V, which is constant in the coordinates

Cr × [0, 1]× [0,∞),

at the ends. For each end ei and its distinguished chart ei : [0, 1]× [0,∞) → S letbji : [0,∞) → ∂S, j = 0, 1 be the restrictions of ei to i × [0,∞).

We then have a pair of real vector spaces

Ξji = lim

τ 7→∞Ξ|bji (τ).

There is a Maslov number Maslov(V,Ξ, Ξji) associated to this data coinciding

with the boundary Maslov index in the sense of [12, Appendix C3], in the caseΞ0i = Ξ1

i , for the modified pair (V/,Ξ/) obtained from (V,Ξ, Ξji) by naturally

closing off each ei end of V → S.When Ξ0

i is transverse to Ξ1i for each i, Maslov(V,Ξ, Ξj

i) is obtained as theMaslov index for the modified pair (V/,Ξ/) by again closing off the ends ei viagluing (at each end ei) with

(Cr ×D, Ξ, Ξj0),

where D as before is diffeomorphic to D2 with a point e0 on the boundary removed.Here Ξ0

i = Ξ10 and Ξ1

i = Ξ00, while Ξ over the boundary of D is determined by the

“shortest path” from Ξ00 to Ξ1

0, meaning that since these are a pair of transversetotally real subspaces up to a complex isomorphism of Cr (whose choice will notmatter) we may identify them with the subspaces Rr, and iRr after this identifica-tion our shortest path is just eiθRr, θ ∈ [0, π2].

For a real linear Cauchy-Riemann operator on V, with suitable asymptotics, theFredholm index is given by:

r · χ(S) +Maslov(V,Ξ, Ξi).

The proof of this is analogous to [12, Appendix C], we can also reduce it to thatstatement via a gluing argument. (This kind of argument appears for instance in[22])

44 YASHA SAVELYEV

B.1. Dimension formula for moduli space of sections. Suppose we are givena Hamiltonian fiber bundle M2r → S → S, with end structure and S as above.Let L be a Lagrangian sub-bundle of S over the boundary of S, compatible withthe end structure, and such that the Lagrangian submanifolds

Lji = lim

τ 7→∞L|bji (τ),

intersect transversally (identifying the corresponding fibers) or coincide.Given an L-exact Hamiltonian connection A, on S, (see Definition 6.2) which is

additionally assumed to be trivial in the strip coordinate charts at the ends, anda choice of a family jz of compatible almost complex structures on the fibers ofS, set M(A) to be the moduli space of (relative) class A finite vertical L2 energyholomorphic sections of S → S with boundary in L. Define

Maslovvert(A)

to be the Maslov number of the triple (V,Ξ, Ξi) determined by the pullback byσ ∈ M(A) of the vertical tangent bundle of S, L. Then the expected dimension ofM(A) is:

(B.1) r · χ(S) +Maslovvert(A).

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