# Quantitative Heegaard Floer cohomology and the Calabi invariant

Daniel Cristofaro-Gardiner  Vincent Humilière  Cheuk Yu Mak  Sobhan Seyfaddini
Ivan Smith

###### Abstract

We define a new family of spectral invariants associated to certain Lagrangian
links in compact and connected surfaces of any genus. We show that our
invariants recover the Calabi invariant of Hamiltonians in their limit. As
applications, we resolve several open questions from topological surface
dynamics and continuous symplectic topology: we show that the group of
Hamiltonian homeomorphisms of any compact surface with (possibly empty)
boundary is not simple; we extend the Calabi homomorphism to the group of
Hameomorphisms constructed by Oh-Müller; and, we construct an infinite
dimensional family of quasimorphisms on the group of area and orientation
preserving homeomorphisms of the two-sphere. Our invariants are inspired by
recent work of Polterovich and Shelukhin defining and applying spectral
invariants for certain classes of links in the two-sphere.

###### Contents

  1. 1 Introduction
    1. 1.1 Recovering the Calabi invariant
    2. 1.2 The algebraic structure of the group of area-preserving homeomorphisms
    3. 1.3 Quasimorphisms on the sphere
    4. 1.4 Quantitative Heegard Floer cohomology and link spectral invariants
  2. 2 Preliminaries
    1. 2.1 Recollections
    2. 2.2 Hameomorphisms and finite energy homeomorphisms
    3. 2.3 The mass-flow and flux homomorphisms
  3. 3 Non-simplicity and the extension of Calabi
    1. 3.1 The Calabi property
    2. 3.2 Link spectral invariants for Hamiltonian diffeomorphisms and homeomorphisms
    3. 3.3 Infinite twists on positive genus surfaces
    4. 3.4 Calabi on Hameo
  4. 4 Heegaard tori and Clifford tori
    1. 4.1 Set-up and outline
    2. 4.2 Co-ordinates on the symmetric product
    3. 4.3 Relation to the Clifford torus
    4. 4.4 Tautological correspondence
    5. 4.5 Basic disc classes
  5. 5 Curvature and unobstructedness
    1. 5.1 The disc potential
    2. 5.2 Potential in the Clifford-type case
    3. 5.3 Regularity
    4. 5.4 Potential in general
  6. 6 Quantitative Heegaard Floer cohomology
    1. 6.1 The Floer complex
    2. 6.2 A direct system and Hamiltonian invariance
    3. 6.3 The disc potential revisited
    4. 6.4 Proof of Theorem 1.4
  7. 7 Closed-open maps and quasimorphisms
    1. 7.1 Notation review
    2. 7.2 Link spectral invariants are monotone spectral invariants
    3. 7.3 Quasimorphisms on $S^{2}$
    4. 7.4 The commutator and fragmentation lengths

##  1 Introduction

###  1.1 Recovering the Calabi invariant

Let $(\Sigma,\omega)$ denote a compact and connected surface, possibly with
boundary, equipped with an area-form. When the boundary is non-empty, the
group of Hamiltonian diffeomorphisms admits a homomorphism

$\Cal:\Ham(\Sigma,\omega)\rightarrow\R,$

called the Calabi invariant, defined as follows. Let
$\theta\in\Ham(\Sigma,\omega)$. Pick a Hamiltonian
$H:[0,1]\times\Sigma\rightarrow\R$, supported in the interior of $\Sigma$,
such that $\theta=\phi^{1}_{H}$; see Section 2.1 for our conventions in the
definition of $\Ham$. Then,

$\Cal(\theta):=\int_{0}^{1}\int_{\Sigma}H\,\omega\;dt.$

The above integral does not depend on the choice of $H$ and so $\Cal(\theta)$
is well-defined. Moreover, it defines a non-trivial group homomorphism. For
further details on the Calabi homomorphism see [9, 42].

The first goal of the present work is to recover the Calabi invariant from
more modern invariants, called spectral invariants. In fact, we prove a more
general result for closed surfaces. Spectral invariants have by now a long
history of applications in symplectic topology, see for example [63, 56, 43,
19, 44, 60, 37, 17, 2, 16, 14]. For our work here, what is critical is that
the techniques of continuous symplectic topology allow us to define spectral
invariants for area-preserving homeomorphisms, and we will see several
applications below.

To state our result about recovering Calabi, define a Lagrangian link
$\ul{L}\subset\Sigma$ to be a smooth embedding of finitely many pairwise
disjoint circles. We emphasize, because it contrasts the setup for many other
works about Floer theory on surfaces, that the individual components of the
link are not required to be Floer theoretically non-trivial; for example, they
can be small contractible curves. Whenever $\ul{L}$ satisfies a certain
monotonicity assumption, see Definition 1.7, we define a link spectral
invariant $c_{\ul{L}}:C^{\infty}([0,1]\times\Sigma,\omega)\to\R$. The
properties of the invariants $c_{\ul{L}}$ are summarized in Theorem 1.4 below.
We have the following result for suitable sequences of Lagrangian links which
always exist and which we refer to as equidistributed links, see Section 3.1
for the precise definition. A sequence of links being equidistributed in
particular implies that the number of contractible components diverges to
infinity, whilst their diameters in a fixed metric tend to zero; we therefore
think of such links as ‘probing the small-scale geometry’ of the surface.

[Calabi property] Let $\ul{L}^{m}$ be a sequence of equidistributed Lagrangian
links in a closed symplectic surface $(\Sigma,\omega)$. Then, for any $H\in
C^{\infty}([0,1]\times\Sigma)$ we have

$\lim_{m\to\infty}c_{\ul{L}^{m}}(H)=\int_{0}^{1}\int_{\Sigma}H_{t}\;\omega\;dt.$

######  Remark 1.1.

The Calabi property is reminiscent of a property conjectured by Hutchings for
spectral invariants defined using Periodic Floer homology, see [14, Rmk.
1.12], which was verified in [14] for monotone twist maps. We were partly
inspired to think about it because of this conjecture. Hutchings’ conjecture
was in turn inspired by a Volume Property for spectral invariants defined
using Embedded Contact Homology proved in [17] that has had various
applications, see for example [2, 16]. On the other hand, the above Calabi
property is different from a property with the same name appearing in the
works of Entov-Polterovich [19] on Calabi quasimorphisms or the recent paper
of Polterovich-Shelukhin [55]. What these papers refer to as the Calabi
property is equivalent to the Support control property of our Theorem 1.4.

We can think of a result like Theorem 1.1 as asserting that we have “enough”
spectral invariants to recover classical data. We now explain several
applications.

###  1.2 The algebraic structure of the group of area-preserving
homeomorphisms

Our first applications resolve two old questions from topological surface
dynamics that have been key motivating problems in the development of
continuous symplectic topology. The ability to recover Calabi is central for
both proofs.

#### Hamiltonian homeomorphisms

Let $\Homeo_{0}(\Sigma,\omega)$ denote the identity component in the group of
homeomorphisms of $\Sigma$ which preserve the measure induced by $\omega$ and
coincide with the identity near the boundary of $\Sigma$, if the boundary is
non-empty. We say $\varphi\in\Homeo_{0}(\Sigma,\omega)$ is a Hamiltonian
homeomorphism if it can be written as a uniform limit of Hamiltonian
diffeomorphisms. The set of all such homeomorphisms is denoted by
$\overline{\Ham}(\Sigma,\omega)$; this is a normal subgroup of
$\Homeo_{0}(\Sigma,\omega)$. Hamiltonian homeomorphisms have been studied
extensively in the surface dynamics community; see, for example, [40, 34,
35].111We remark that when $\Sigma=S^{2}$, $\overline{\Ham}$ is the group of
area and orientation preserving homeomorphisms, and when $\Sigma=D^{2}$, it is
the group of area preserving homeomorphisms that are the identity near the
boundary.

There exists a homomorphism out of $\Homeo_{0}(\Sigma,\omega)$, called the
mass-flow homomorphism, introduced by Fathi [22], whose kernel is
$\overline{\Ham}(\Sigma,\omega)$. The normal subgroup
$\overline{\Ham}(\Sigma,\omega)$ is proper when $\Sigma$ is different from the
disc or the sphere. In the 1970s, Fathi asked in [22, Section 7] if
$\overline{\Ham}(\Sigma,\omega)$ is a simple group; in higher dimensions, one
can still define mass-flow and Fathi showed [22, Thm. 7.6] that its kernel is
always simple, under a technical assumption on the manifold which always holds
when the manifold is smooth. When $\Sigma$ is a surface with genus $0$,
Fathi’s question was answered in [14, 15]; however, the higher genus case has
remained open. Although the details of mass-flow are not needed for our work,
we recall some facts about it in Section 2.3.

By using our new spectral invariants, we can answer Fathi’s question in full
generality:

$\overline{\Ham}(\Sigma,\omega)$ is not simple.

Theorem 1.2 generalizes the aforementioned results of [14, 15] proving this
result in the genus zero case. Our proof is logically independent of these
works. To prove the theorem, following [14, 15] we construct a normal subgroup
$\FHomeo(\Sigma,\omega)$, called the group of finite energy homeomorphisms,
and we prove that it is proper, see Section 3.3. The group FHomeo is inspired
by Hofer geometry, and one can define Hofer’s metric on it, see [15, Sec.
5.3]. For another proof in the genus $0$ case, see [55].

The group $\FHomeo(\Sigma,\omega)$ contains the commutator subgroup of
$\overline{\Ham}(\Sigma,\omega)$, see Proposition 1, hence we learn from our
main result that $\overline{\Ham}(\Sigma,\omega)$ is not perfect, either.

#### Extending the Calabi invariant

One would like to understand more about the algebraic structure of
$\overline{\Ham}(\Sigma,\omega)$ beyond the simplicity question. Recall that
$\Ham(\Sigma,\omega)$ denotes the subgroup of Hamiltonian diffeomorphisms and
suppose now that the boundary of $\Sigma$ is non-empty.

A question of Fathi from the 1970s [22, Section 7] asks if $\Cal$ admits an
extension to $\overline{\Ham}(\D,\omega)$. An illuminating discussion by Ghys
of this question appears in [26, Section 2]; it follows from results of
Gambaudo-Ghys [25] and Fathi [23] that Calabi is a topological invariant of
Hamiltonian diffeomorphisms, i.e. if $f,g\in\Ham(\Sigma,\omega)$ are conjugate
by some $h\in\Homeo_{0}(\Sigma,\omega)$, then $\Cal(f)=\Cal(g)$. Hence, it
seems natural to try and extend Calabi to $\overline{\Ham}(\Sigma,\omega)$, or
at least to a proper normal subgroup.222Fathi proves in [23] that $\Cal$
extends to Lipschitz area-preserving homeomorphisms. These, however, do not
form a normal subgroup. Our proof of Theorem 1.2 involves constructing an
“infinite twist” Hamiltonian homeomorphism which, heuristically, has infinite
Calabi invariant, so our interest in what follows will be extending the Calabi
homomorphism to a proper normal subgroup rather than the full group.

There is a later conjecture of Fathi about what an appropriate normal subgroup
for the purpose of extending Calabi might be. In the article [48], Oh and
Müller introduced a normal subgroup $\Hameo(\Sigma,\omega),$ called the group
of Hameomorphisms of $\Sigma,$ and whose definition we review in 2.2; the idea
of the definition is that these are elements of
$\overline{\Ham}(\Sigma,\omega)$ that have naturally associated Hamiltonians.
The group $\Hameo(\Sigma,\omega)$ is contained in $\FHomeo(\Sigma,\omega)$,
see Proposition 1, and so our proof of Theorem 1.2 shows that it is proper.
The aforementioned conjecture of Fathi is that the Calabi invariant admits an
extension to $\Hameo(\Sigma,\omega)$ when $\Sigma$ is the disc; see [45, Conj.
6.1]. We prove this for any $\Sigma$ with non-empty boundary.

The Calabi homomorphism admits an extension to a homomorphism from the group
$\Hameo(\Sigma,\omega)$ to the real line.

Theorem 1.2 implies that $\Hameo(\Sigma,\omega)$ is neither simple nor
perfect, when $\partial\Sigma\neq\emptyset$; we do not know whether or not the
kernel of Calabi on $\Hameo$ is simple.

######  Remark 1.2.

  1. 1.

Theorem 1.2 implies that $\FHomeo(\Sigma,\omega)$ is not simple either, when
$\partial\Sigma\neq\emptyset$. This is because by Proposition 1,
$\Hameo(\Sigma,\omega)$ is a normal subgroup of $\FHomeo(\Sigma,\omega)$: we
do not know if $\Hameo(\Sigma,\omega)$ forms a proper subgroup, but if not
then we can conclude that the Calabi invariant extends to
$\FHomeo(\Sigma,\omega)$ and so it cannot be simple. By the same reasoning,
Theorem 1.2 implies Theorem 1.2 in the case where
$\partial\Sigma\neq\emptyset.$

  2. 2.

We also do not know much about the quotient
$\overline{\Ham}(\Sigma,\omega)/\Hameo(\Sigma,\omega),$ although we do know
that it is abelian, by Proposition 1, and that it contains a copy of $\R$; see
Remark 3.7.

###  1.3 Quasimorphisms on the sphere

We now explain one more application of our theory in the case $\Sigma=S^{2}$.
Strictly speaking, this does not use the Calabi property, although it does use
the abundance of our new spectral invariants.

Recall that a homogeneous quasimorphism on a group $G$ is a map
$\mu:G\rightarrow\R$ such that

  1. 1.

$\mu(g^{n})=n\mu(g)$, for all $g\in G$, $n\in\mathbb{Z}$;

  2. 2.

there exists a constant $D(\mu)\geq 0$, called the defect of $\mu$, with the
property that $|\mu(gh)-\mu(g)-\mu(h)|\leq D(\mu).$

Returning now to the algebraic structure of $\Homeo_{0}(S^{2},\omega)$, note
that the vector space of all homogeneous quasimorphisms of a group is an
important algebraic invariant of it; however, it has previously been unknown
whether $\Homeo_{0}(S^{2},\omega)$ has any non-trivial homogeneous
quasimorphisms at all.

The space of homogeneous quasimorphsisms on $\Homeo_{0}(S^{2},\omega)$ is
infinite dimensional.

The same statement was very recently proven for $\Homeo_{0}(\Sigma)$ where
$\Sigma$ is a surface of positive genus, see [7], but in contrast the group
$\Homeo_{0}(S^{2})$ has no non-trivial homogeneous quasimorphisms as we review
in Example 1.3 below. We also note that the space of all homogeneous
quasimorphisms is infinite dimensional for $\Homeo_{0}(\Sigma,\omega)$ when
the genus of $\Sigma$ is at least one, see [20, Thm. 1.2]. The existence of
our quasimorphisms has various implications, as the following illustrates.

######  Example 1.3.

Recall that the commutator length $cl$ of an element $g$ in the commutator
subgroup of a group is the smallest number of commutators required to write
$g$ as a product. The stable commutator length is defined333To use a phrase
from [10], we can think of the commutator length as a kind of algebraic
analogue of the number of handles, and we refer the reader to [10] for further
discussion. by $scl(g):=\text{lim}_{n\to\infty}\frac{cl(g^{n})}{n}.$ It
follows immediately from the existence of a nontrivial homogeneous
quasimorphism that the commutator length and the stable commutator length are
both unbounded. In stark contrast to this, Tsuboi [61] has shown that
$cl(g)=1$ for any $g\in\Homeo_{0}(S^{n})\setminus\\{\id\\}$.444$cl(g)=1$ for
$g\in\Homeo_{0}(S^{1})\setminus\\{\id\\}$ was established earlier by Eisenbud,
Hirsch and Neumann [18].

Moreover, we prove in Proposition 8 that $scl$ is unbounded in any $C^{0}$
neighborhood of the identity. This contrasts [7, Thm. 1.5] on $C^{0}$
continuity of scl in the non-conservative setting; see Section 7.4.

We also explain an application to fragmentation norms in 7.4 below.

In the course of our proof of Theorem 1.3, we answer a question posed by
Entov, Polterovich and Py [20, Question 5.2], which was partly motivated by
the desire to obtain a result like Theorem 1.3, see Remark 1.6; the question
also appears as Problem 23 in the McDuff-Salamon list of open problems [42,
Ch. 14]. The question refers in part to the Hofer metric, defined in Section
2.2.

######  Question 1.4.

Does the group $\Ham(S^{2},\omega)$ admit any homogeneous quasimorphism which
is continuous with respect to the $C^{0}$ topology? If yes, can it be made
Lipschitz with respect the Hofer metric?555The analogue of Question 1.4 for
the $2$ and $4$ (complex) dimensional quadrics was recently settled in the
affirmative by Kawamoto [30].

The space consisting of homogeneous quasimorphsisms on $\Ham(S^{2},\omega)$
which are continuous with respect to the $C^{0}$ topology and Lipschitz with
respect to the Hofer metric is infinite dimensional.

In fact, our quasimorphisms satisfy a simple asymptotic formula — they
converge to $0$ in their limit — and this can be used to recover the Calabi
invariant over $S^{2}$ with more general links, see Proposition 7.

######  Remark 1.5.

In contrast, $\Ham(S^{2},\omega)$ does not admit any non-trivial homomorphisms
to $\R$ since it is simple [3]. As for $\Homeo_{0}(S^{2},\omega)$, it is an
open question whether it admits any non-trivial homomorphisms to $\R$,
although a straightforward modification of the argument in [14, Cor. 2.5]
shows that any such homomorphism could not be $C^{0}$ continuous.

######  Remark 1.6.

As alluded to above, the motivation for the first part of Question 1.4 is
closely connected to our Theorem 1.3: indeed, a result from Entov, Polterovich
and Py [20, Prop. 1.4] implies that any continuous homogeneous quasimorphism
on $\Ham(S^{2},\omega)$ would extend to give such a quasimorphism on
$\Homeo_{0}(S^{2},\omega)$. As for the second part of the question, this is
tuned to applications in Hofer geometry and $C^{0}$ symplectic topology. For
example, it was very recently shown in [15, 55] that $\Ham(S^{2},\omega)$ is
not quasi-isometric to $\R$, thereby settling what is known as the Kapovich-
Polterovich question [42, Prob. 21]; prior to [15, 55], it was shown in [20]
that an affirmative answer to the second question in Question 1.4 would also
settle the Kapovich-Polterovich question.

###  1.4 Quantitative Heegard Floer cohomology and link spectral invariants

We now explain the main tool that we use to prove the aforementioned results.
Let $\Sigma$ be a closed genus $g$ surface equipped with a symplectic form
$\omega$.

Consider a Lagrangian link (or simply a link) if $\ul{L}=\cup_{i=1}^{k}L_{i}$
consisting of $k$ pairwise-disjoint circles on $\Sigma$, with the property
that $\Sigma\setminus\ul{L}$ consists of planar domains $B_{j}^{\circ}$, with
$1\leq j\leq s$, whose closures $B_{j}\subset\Sigma$ are also planar;
throughout the rest of the paper we will only consider links satisfying this
planarity assumption.

Given a link $\ul{L}$, we denote by $k_{j}$ the number of boundary components
of $B_{j}$. Since the Euler characteristic of a planar domain $D$ with $k_{D}$
boundary components is $2-k_{D}$, the Euler characteristic of $\Sigma$ is
$2-2g=\sum_{j=1}^{s}(2-k_{j})=2s-2k$, and hence $s=k-g+1$. Finally, for $1\leq
j\leq s$, let $A_{j}$ denote the $\omega$-area of $B_{j}$.

######  Definition 1.7.

Let $\ul{L}$ be a Lagrangian link satisfying the above planarity assumption.
We call $\ul{L}$ monotone if there exists $\eta\in\R_{\geq 0}$ such that

$2\eta(k_{j}-1)+A_{j}$ (1)

is independent of $j$, for $j\in\\{1,\dots,s\\}$. We will use the terminology
$\eta$-monotone when we need to specify the value of $\eta$. We refer to the
quantity $2\eta(k_{j}-1)+A_{j}$ as the monotonicity constant of
$\ul{L}$.666Our terminology is motivated by Lemma 4.24.

We will write $H_{t}$ for a time-dependent Hamiltonian function
$H:[0,1]\times\Sigma\to\R$; it defines a point of the universal cover
$\widetilde{\Ham}(\Sigma,\omega)$. A Hamiltonian $H_{t}$ is said to be mean-
normalized if $\int_{\Sigma}\,H_{t}\;\omega=0$ for all $t\in[0,1]$. Given
Hamiltonians $H,H^{\prime}$ we define
$H\\#H^{\prime}_{t}(x)=H_{t}(x)+H^{\prime}(t,(\phi^{t}_{H})^{-1}(x))$, which
generates the Hamiltonian flow $\phi^{t}_{H}\circ\phi^{t}_{H^{\prime}}$. We
refer the reader to Section 2.1 for more details on our notations and
conventions.

For every monotone Lagrangian link $\ul{L}=\cup_{i=1}^{k}L_{i}$ there exists a
link spectral invariant

$c_{\ul{L}}:C^{\infty}([0,1]\times\Sigma,\omega)\to\mathbb{R}$

satisfying the following properties.

  * •

(Spectrality) for any $H$, $c_{\ul{L}}(H)$ lies in the spectrum
$\Spec(H:\ul{L})$ (see Definition 6.2 and (54));

  * •

(Hofer Lipschitz) for any $H,H^{\prime}$,

$\int_{0}^{1}\min(H_{t}-H^{\prime}_{t})dt\leq
c_{\ul{L}}(H)-c_{\ul{L}}(H^{\prime})\leq\int_{0}^{1}\max\,(H_{t}-H^{\prime}_{t})dt;$

  * •

(Monotonicity) if $H_{t}\leq H^{\prime}_{t}$ then $c_{\ul{L}}(H)\leq
c_{\ul{L}}(H^{\prime})$;

  * •

(Lagrangian control) if $H_{t}|_{L_{i}}=s_{i}(t)$ for each $i$, then

$c_{\ul{L}}(H)=\frac{1}{k}\,\sum_{i=1}^{k}\int s_{i}(t)dt;$

moreover for any $H$,

$\frac{1}{k}\sum_{i=1}^{k}\int_{0}^{1}\min_{L_{i}}H_{t}\,dt\leq
c_{\ul{L}}(H)\leq\frac{1}{k}\sum_{i=1}^{k}\int_{0}^{1}\max_{L_{i}}H_{t}\,dt;$

  * •

(Support control) if
$\mathrm{supp}(H_{t})\subset\Sigma\backslash\cup_{j}L_{j}$, then
$c_{\ul{L}}(H)=0$;

  * •

(Subadditivity) $c_{\ul{L}}(H\\#H^{\prime})\leq
c_{\ul{L}}(H)+c_{\ul{L}}(H^{\prime})$;

  * •

(Homotopy invariance) if $H,H^{\prime}$ are mean-normalized and determine the
same point of the universal cover $\widetilde{\Ham}(\Sigma,\omega)$, then
$c_{\ul{L}}(H)=c_{\ul{L}}(H^{\prime})$;

  * •

(Shift) $c_{\ul{L}}(H+s(t))=c_{\ul{L}}(H)+\int_{0}^{1}s(t)\,dt$.

We prove this theorem in Section 6.4. The spectral invariant $c_{\ul{L}}$ is
defined in Equation (54).

######  Remark 1.8.

The idea of looking for spectral invariants suitable for our applications
through Lagrangian links was inspired by the recent work of Polterovich and
Shelukhin [55]: they prove a similar result for certain classes of links in
$S^{2}$, consisting of parallel circles, in [55, Thm. F] and demonstrate many
applications.

The above theorem yields spectral invariants for Hamiltonians. We will explain
how to use this result to define spectral invariants for Hamiltonian
diffeomorphisms in 3.2. To prove our results we will also need spectral
invariants for Hamiltonian homeomorphisms. We will do this in 3.2 as well.

In Section 7.3, we consider the case $\Sigma=S^{2}$ and introduce maps
$\mu_{\ul{L}}:\Ham(S^{2},\omega)\rightarrow\R$ obtained from homogenization of
the link spectral invariant $c_{\ul{L}}$; see Equation (63). The
$\mu_{\ul{L}}$ are homogeneous quasimorphisms which inherit some of the
properties listed above. It is with these quasimorphisms that we prove Theorem
1.3 and 1.3.

#### Context for Theorem 1.4

We briefly discuss the ideas entering into the proof of Theorem 1.4. Following
an insight from [39], although some of the individual components $L_{j}$ are
Floer-theoretically trivial in $\Sigma$, the link $\ul{L}$ defines a
Lagrangian submanifold $\Sym(\ul{L})$ of the symmetric product
$X=\Sym^{k}(\Sigma)$ which may be non-trivial. A Hamiltonian function
$H:[0,1]\times\Sigma\to\R$ determines canonically a function
$\Sym(H):[0,1]\times\Sym^{k}(\Sigma)\to\R$. Although this is only Lipschitz
continuous across the diagonal, the fact that $\Sym(\ul{L})$ lies away from
the diagonal makes it possible to work with (modified versions of) these
Hamiltonians unproblematically.

The spectral invariant $c_{\ul{L}}$ is constructed using Lagrangian Floer
cohomology of $\Sym(\ul{L})$ in $X$, which can be viewed as a ‘quantitative’
version of the Heegaard Floer cohomology for links from [50], cf. Remarks 4.1
and 4.2. This quantitative version counts essentially the same holomorphic
discs as in Heegaard Floer theory, but we keep track of holonomy contributions
(working with local systems), and of intersection numbers of holomorphic discs
with the diagonal. The parameter $\eta\in\R_{\geq 0}$ of Theorem 1.4 plays the
role of a bulk deformation; when the assumption (1) of Definition 1.7 is
satisfied, our variant of Lagrangian Floer cohomology is both Hamiltonian-
invariant and non-zero. To prove the non-vanishing of Floer cohomology, we
show that for certain links $\ul{L}\subset\PP^{1}$ the symmetric product
Lagrangian $\Sym(\ul{L})$ is smoothly isotopic to a Clifford-type torus
supported in a small ball (Corollary 4.3), and use that isotopy to control the
holomorphic discs with boundary on $\Sym(\ul{L})$ and to compute its disc
potential (in the sense of [13, 11], see Proposition 4, 5). A combination of
the tautological correspondence, relating discs in the symmetric product
$\Sym^{k}(\Sigma)$ with holomorphic maps of branched covers of the disc to
$\Sigma$, together with embeddings of the planar domains in
$\Sigma\backslash\ul{L}$ into $\PP^{1}$, allows us to reduce the general
computation of the disc potential to this special case (Theorem 5.4). Once
Floer cohomology of $\Sym(\ul{L})$ is defined and non-trivial, the
construction and properties of the spectral invariant closely follow the usual
arguments [24, 37] with only minor modifications. We remark that, in contrast
to [39], this paper does not use orbifold Floer cohomology and does not
require virtual perturbation techniques.

######  Remark 1.9.

When $g=0$ or $\eta=0$, the arguments can be simplified by working with
spherically monotone symplectic forms on $X$, with respect to which
$\Sym(\ul{L})$ is a monotone Lagrangian. (See Remarks 4.29 and 6.8 as well as
Section 7.2). In this case, the spectral invariant we define coincides with
the classical monotone Lagrangian spectral invariant associated to
$\Sym(\ul{L})$ in $X$ with an appropriate symplectic form (see Lemma 7.3).

The above allows us to prove Corollary 7.2 establishing an inequality between
our link spectral invariants and the Hamiltonian Floer spectral invariants of
$\Sym(H)$. With the help of this inequality, we prove that our link spectral
invariants yield quasimorphisms in the $g=0$ case.

### Organization of the paper

In Section 2, we set our notation, introduce our groups of homeomorphisms on
surfaces and recall Fathi’s mass flow homomorphism. In Section 3, we use the
properties of spectral invariants stated in Theorem 1.4 to prove the Calabi
property (Theorem 1.1), non-simplicity of the group of Hamiltonian
homeomorphisms (Theorem 1.2) and the extension of the Calabi homomorphism to
Hameomoprhisms (Theorem 1.2). In Section 4 we study pseudo-holomorphic discs
with boundary on $\Sym(\ul L)$, which allows us to compute the disc potential
function of $\Sym(\ul L)$ in Section 5. This is used in Section 6 to show that
the relevant Floer cohomology is well-defined and non-vanishing. We also
define our spectral invariants and prove Theorem 1.4 in Section 6.4. Finally,
we prove our results on quasimorphisms in Section 7.3, and our results on
commutator and fragmentation lengths in Section 7.4.

### Acknowledgments

C.M. thanks the organizers of the ‘Symplectic Zoominar’ for the opportunity to
speak in the seminar, where this collaboration was initiated. We thank
Frédéric Le Roux for helpful conversations about Section 7.4 and Yusuke
Kawamoto for helpful discussions about Remark 7.6 and Section 7.3.

D.C.G. is partially supported by NSF grant DMS-1711976 and an Institute for
Advanced Study von Neumann fellowship. V.H. is partially supported by the
grant “Microlocal” ANR-15-CE40-0007 from Agence Nationale de la Recherche.
C.M. is supported by ERC Starting grant number 850713. S.S. is supported by
ERC Starting grant number 851701. I.S. is partially supported by Fellowship
EP/N01815X/1 from the Engineering and Physical Sciences Research Council, U.K.

##  2 Preliminaries

In this section we introduce parts of our notation and review some necessary
background.

###  2.1 Recollections

Let $(M,\omega)$ be a symplectic manifold. We denote by
$C^{\infty}([0,1]\times M)$ the set of time-varying Hamiltonians that vanish
near the boundary when $M$ has non-empty boundary. Our convention is such that
the (time-varying) Hamiltonian vector field associated to $H$ is defined by
$\omega(X_{H_{t}},\cdot)=dH_{t}.$ The homotopy class of a Hamiltonian path
$\\{\phi^{t}_{H}:0\leq t\leq 1\\}$ determines an element of the universal
cover $\widetilde{\Ham}(M,\omega)$. In the case of a surface $\Sigma\neq
S^{2}$, the fundamental group of $\Ham$ is trivial and so
$\widetilde{\Ham}=\Ham$; see [53, Sec. 7.2]. The fundamental group of
$\Ham(S^{2},\omega)$ is $\Z/2\Z$ and so $\widetilde{\Ham}(S^{2},\omega)$ is a
two-fold covering of $\Ham(S^{2},\omega)$.

###  2.2 Hameomorphisms and finite energy homeomorphisms

Denote by $C^{0}([0,1]\times M)$ the set of continuous time-dependent
functions on $M$ that vanish near the boundary if $\partial M\neq\emptyset$.
The energy, or the Hofer norm, of $H\in C^{0}([0,1]\times M)$ is defined by
the quantity

$\|H\|_{(1,\infty)}=\int_{0}^{1}\left(\max_{x\in M}H_{t}-\min_{x\in
M}H_{t}\right)dt.$

The Hofer distance between $\varphi,\psi\in\Ham(M,\omega)$ is defined by

$d_{H}(\varphi,\psi):=\inf\\{\|H\|_{(1,\infty)}:\varphi\psi^{-1}=\phi^{1}_{H}\\}.$
(2)

This is a bi-invariant distance on $\Ham(M,\omega)$; see [29, 32, 53].

######  Definition 2.1.

An element $\phi\in\overline{\Ham}(M,\omega)$ is a finite energy homeomorphism
if there exists a sequence of smooth Hamiltonians $H_{i}\in
C^{\infty}([0,1]\times M)$ such that

$\phi^{1}_{H_{i}}\xrightarrow{C^{0}}\phi,\text{ with
}\|H_{i}\|_{(1,\infty)}\leq C$

for some constant $C$. An element $\phi\in\overline{\Ham}(M,\omega)$ is called
a Hameomorphism if there exists a continuous $H\in C^{0}([0,1]\times M)$ such
that

$\phi^{1}_{H_{i}}\xrightarrow{C^{0}}\phi,\text{ and
}\|H-H_{i}\|_{(1,\infty)}\to 0.$

The set of all finite energy homeomorphisms is denoted by $\FHomeo(M,\omega)$
and the set of all Hameomorphisms is denoted by $\Hameo(M,\omega)$.777Oh and
Müller use the terminology _Hamiltonian homeomorphisms_ for the elements of
$\Hameo_{c}(M,\omega)$. We have chosen to avoid this terminology because in
the surface dynamics literature it is commonly used for elements of
$\overline{\Ham}(M,\omega)$.

There is an inclusion $\Hameo(M,\omega)\subset\FHomeo(M,\omega)$.

######  Proposition 1.

The groups $\Hameo(M,\omega)$ and $\FHomeo(M,\omega)$ satisfy the following
properties.

  1. (i)

They are both normal subgroups of $\Homeo_{0}(M,\omega)$;

  2. (ii)

$\Hameo(M,\omega)$ is a normal subgroup of $\FHomeo(M,\omega)$;

  3. (iii)

If $M$ is a compact surface, they both contain the commutator subgroup of
$\Homeo_{0}(M,\omega)$.

######  Proof 2.2.

The fact that $\Hameo(M,\omega)$ is a normal subgroup of
$\Homeo_{0}(M,\omega)$ is proven in [48]; the same statement for
$\FHomeo(M,\omega)$ is proven in [14, Prop. 2.1], in the case where $M$ is the
disc; the same argument generalizes, in a straightforward way, to any $M$.
This proves the first item.

The second item follows from the first and the inclusion
$\Hameo(M,\omega)\subset\FHomeo(M,\omega)$.

The third item follows from a general argument, involving fragmentation
techniques [21, 28, 22], which proves that any normal subgroup of
$\Homeo_{0}(M,\omega)$ contains the commutator subgroup
$[\Homeo_{0}(M,\omega),\Homeo_{0}(M,\omega)]$. A proof of this in the case
where $M=D^{2}$ is presented in [14, Prop. 2.2]; the argument therein
generalizes, in a straightforward way, to any $M$.

We end this section with the observation that $\phi\in\Homeo_{0}(M,\omega)$ is
a finite energy homeomorphism (resp. Hameomorphism) if it can be written as
the $C^{0}$ limit of a sequence $\phi_{i}\in\Ham(M,\omega)$ which is bounded
(resp. Cauchy) in Hofer’s distance.

###  2.3 The mass-flow and flux homomorphisms

Let $M$ denote a manifold equipped with a volume form $\omega$ and denote by
$\Homeo_{0}(M,\omega)$ the identity component in the group of volume-
preserving homeomorphisms of $M$ that are the identity near $\partial M$. In
[22], Fathi constructs the mass-flow homomorphism

$\mathcal{F}:\Homeo_{0}(M,\omega)\rightarrow H_{1}(M)/\Gamma,$

mentioned above, where $H_{1}(M)$ denotes the first homology group of $M$ with
coefficients in $\R$ and $\Gamma$ is a discrete subgroup of $H_{1}(M)$ whose
definition we will not need here. Clearly, $\Homeo_{0}(M,\omega)$ is not
simple when the mass-flow homomorphism is non-trivial. This is indeed the case
when $M$ is a closed surface other than the sphere. As we explained in 1.2,
Fathi proved that $\mathrm{ker}(\mathcal{F})$ is simple if the dimension of
$M$ is at least three.

For the convenience of the reader, we recall here a (symplectic) description
of the mass-flow homomorphism in the case of surfaces; we will be very brief
as the precise definition of the mass-flow homomorphism is not needed for our
purposes in this article.

Denote by $\Diff_{0}(\Sigma,\omega)$ the identity component in the group of
area-preserving diffeomorphisms $\Sigma$ that are the identity near the
boundary if $\partial\Sigma\neq\emptyset$. There is a well-known homomorphism,
called flux,

$\mathrm{Flux}:\Diff_{0}(\Sigma,\omega)\rightarrow H^{1}(\Sigma)/\Gamma,$

where $H^{1}(\Sigma)$ denotes the first cohomology group of $\Sigma$ with
coefficients in $\R$ and $\Gamma\subset H^{1}(\Sigma)$ is a discrete subgroup;
see [42] for the precise definition. The kernel of this homomorphism is
$\Ham(\Sigma,\omega)$. It can be shown that, in the case of surfaces, the flux
homomorphism extends continuously with respect to the $C^{0}$ topology to
yield a homomorphism

$\mathrm{Flux}:\Homeo_{0}(\Sigma,\omega)\rightarrow H^{1}(\Sigma)/\Gamma,$

which coincides with the mass-flow homomorphism
$\mathcal{F}:\Homeo_{0}(,\omega)\rightarrow H_{1}(M)/\Gamma$, after applying
Poincaré duality. As we said above, its kernel, whose non-simplicity we
establish in this paper, is exactly the group of Hamiltonian homeomorphisms
$\overline{\Ham}(\Sigma,\omega)$.

In dimensions greater than $2$, the mass-flow homorphism can be described
similarly in terms of the Poincaré dual of the volume flux homomorphism.

##  3 Non-simplicity and the extension of Calabi

Here we assume Theorem 1.4 and establish our applications to non-simplicity of
surface transformation groups and the extension of the Calabi invariant.
Theorem 1.4 will be proven in the subsequent sections.

###  3.1 The Calabi property

We begin by defining equidistributed sequences of Lagrangian links and prove
Theorem 1.1.

Throughout this section, we fix a Riemannian metric $d$ on the surface
$\Sigma$ and let $\omega$ be the associated area form. Define the diameter of
a Lagrangian link $\ul{L}=\cup_{i=1}^{k}L_{i}$ to be the maximum of the
diameters of the contractible components of $\ul{L}$; we will denote it by
$\mathrm{diam}(\ul{L})$.

We call a sequence of Lagrangian links $\ul{L}^{m}$ equidistributed if

  1. (i)

$\mathrm{diam}(\ul{L}^{m})\to 0$;

  2. (ii)

the number of non-contractible components of $\ul{L}^{m}$ is bounded above by
a number $N$ independent of $m$;

  3. (iii)

the contractible components of $\ul{L}^{m}$ are not nested: more precisely,
each such circle bounds a unique disc of diameter no more than
$\mathrm{diam}(\ul{L}^{m})$ and we require these discs to be disjoint;

  4. (iv)

each $\ul{L}^{m}$ is monotone, in the sense of Definition 1.7, for some $\eta$
which may depend on $m$.

Figure 1: A typical example of a link $\ul{L}^{m}$ for $m$ large in an
equidistributed sequence. Here, $\Sigma$ has genus 2, there are 4 non-
contractible components in $\ul{L}^{m}$ (in blue). The disc components in
$\Sigma\setminus\ul{L}^{m}$ are colored in grey.

Note that any disc associated to a contractible component $\ul L^{m}$ as in
(iii) must be a connected component of $F\setminus\ul{L}^{m}$: indeed, if it
contained a component of $\ul L^{m}$ then this component would have to be
contractible and then the disc associated with it would violate the uniqueness
property in (iii). It also follows from (iv) that all these discs have equal
area. We denote this common area by $\alpha_{m}$. Note that the other
components of $\Sigma\setminus\ul L^{m}$ all have area smaller than or equal
to $\alpha_{m}$.

It is straightforward to check that equidistributed sequences of Lagrangian
links exist; see Figure 1.

######  Example 3.1.

Let $\eta_{m}$ be a sequence of real numbers such that

$\eta_{m}<\frac{1}{2m(m-1)}$ (3)

for all $m$. Then, there is an equidistributed sequence of $\eta_{m}$-monotone
links $\ul L^{m}$ on $S^{2}$.

Indeed, for each $m$, one can take $\ul L^{m}$ to be the boundaries of a
collection of $m$ pairwise disjoint discs of equal area
$\lambda=\frac{1}{m+1}+2\eta_{m}\frac{m-1}{m+1}$. The complement of these
discs then has area $1-m\lambda$, which is positive by (3).

######  Proof 3.2 (Proof of Theorem 1.1).

We will suppose throughout the proof that $\int_{\Sigma}\omega=1$. Denote by
$L_{1},\ldots,L_{k_{m}}$ the contractible components in $\ul{L}^{m}$. These
bound closed and pairwise disjoint discs $B_{1},\ldots,B_{k_{m}}$ associated
via (iii) above.

Now fix $\eps>0$. Then, since $\mathrm{diam}(\ul L^{m})\to 0$, for
sufficiently large $m$ we can find a smooth Hamiltonian $G_{m}$ such that

$G_{m}|_{B_{i}}=s_{i}(t),\quad\text{max}|H-G_{m}|\leq\eps,$

where each $s_{i}:[0,1]\to\R.$ We have that

$\left|\int_{0}^{1}\int_{\Sigma}H\;\omega\,dt\,-c_{\ul{L}^{m}}(H)\right|\leq$
$\left|\int_{0}^{1}\int_{\Sigma}H-{G_{m}}\;\omega\,dt\,\right|+\left|\int_{0}^{1}\int_{\Sigma}{G_{m}}\;\omega\,dt\,-c_{\ul{L}^{m}}({G_{m}})\right|+\left|c_{\ul{L}^{m}}({G_{m}})-c_{\ul{L}^{m}}(H)\right|$

and so we must bound the three terms from the previous line.

The term $\left|\int_{0}^{1}\int_{\Sigma}H-{G_{m}}\;\omega\,dt\,\right|$ is
bounded by $\eps$ because $\max|H-{G_{m}}|\leq\eps$ and
$\mathrm{Area}(\Sigma)=1$. Similarly, we have
$\left|c_{\ul{L}^{m}}({G_{m}})-c_{\ul{L}^{m}}(H)\right|\leq\varepsilon$ by the
Hofer Lipschitz property from Theorem 1.4.

To bound the final term, use the Lagrangian control property of Theorem 1.4 to
get

$c_{\ul{L}^{m}}({G_{m}})=\frac{1}{k_{m}+\ell_{m}}\sum_{i=1}^{k_{m}}\int_{0}^{1}s_{i}(t)\,dt+E_{m},$

where $\ell_{m}$ is the number of non-contractible components of $\ul L^{m}$
and $E_{m}$ satisfies

$\frac{\ell_{m}}{k_{m}+\ell_{m}}(\min{H}-\eps)\leq\frac{\ell_{m}}{k_{m}+\ell_{m}}\min{G_{m}}\leq
E_{m}\leq\frac{\ell_{m}}{k_{m}+\ell_{m}}\max{G_{m}}\leq\frac{\ell_{m}}{k_{m}+\ell_{m}}(\max
H+\eps).$

In particular, since $\ell_{m}$ is bounded, $E_{m}$ converges to $0$ as
$k_{m}$ goes to $\infty$.

Now, noting that
$\int_{0}^{1}s_{i}(t)dt=\frac{1}{\alpha_{m}}\int_{0}^{1}\int_{B_{i}}{G_{m}}\,\omega\,dt$,
because $\mathrm{Area}(B_{i})=\alpha_{m}$, we can rewrite the above as

$\displaystyle
c_{\ul{L}^{m}}({G_{m}})=\frac{1}{\alpha_{m}(k_{m}+\ell_{m})}\sum_{i=1}^{k_{m}}\int_{0}^{1}\int_{B_{i}}{G_{m}}\;\omega\,dt+E_{m}=\frac{1}{\alpha_{m}(k_{m}+\ell_{m})}\int_{0}^{1}\int_{\Sigma\setminus
C_{m}}{G_{m}}\;\omega\;dt+E_{m},$

where $C_{m}$ denotes the complement
$C_{m}:=\Sigma\setminus\cup_{i=1}^{k_{m}}B_{i}$. We claim that

$\text{lim}_{m\to\infty}\frac{1}{\alpha_{m}(k_{m}+\ell_{m})}=1,\quad\text{lim}_{m\to\infty}\text{area}(C_{m})=0,\quad\text{lim}_{m\to\infty}k_{m}=\infty;$
(4)

from this, it follows by the third limit that $E_{m}$ converges to zero in
view of the above, and then from the first two limits that

$\left|c_{\ul{L}}({G_{m}})-\int_{0}^{1}\int_{\Sigma}{G_{m}}\;\omega\,dt\right|\leq\eps$

for $m$ large enough, as desired.

It remains to show (4).

We claim the inequality

$\frac{1}{k_{m}}\geq\alpha_{m}\geq\frac{1}{k_{m}+2N+1}.$ (5)

The first inequality here is immediate. To see the second, consider the
surface $C^{\prime}_{m}$ given by removing the noncontractible components of
$\ul{L}^{m}$ from $C_{m}$. Then, a coarse bound is that $C^{\prime}_{m}$ has
at most $2N+1$ components, and so $C_{m}$ satisfies

$\mathrm{area}(C_{m})\leq(2N+1)\alpha_{m}.$

Using that $\mathrm{area}(C_{m})+\sum\mathrm{area}(B_{j})=1$, we can now
deduce (5).

To finish the proof of (4), since $\mathrm{diam}(\ul L^{m})\to 0$, we have
$\alpha_{m}\to 0$ which, in combination with the inequality immediately above,
gives the second limit in (4); it also gives in combination with (5), the
third limit. The first limit in (4) now follows from (5), since $\ell_{m}$ is
bounded.

###  3.2 Link spectral invariants for Hamiltonian diffeomorphisms and
homeomorphisms

Theorem 1.4 yields link spectral invariants for Hamiltonians. To prove our
results we will also need to define these invariants for Hamiltonian
diffeomorphisms and homeomorphisms.

We begin by defining our invariants for Hamiltonian diffeomorphisms. Suppose
that $\Sigma$ is a closed surface and let $\ul{L}$ be a monotone Lagrangian
link in $\Sigma$. Given $\tilde{\varphi}$, an element in the universal cover
$\widetilde{\Ham}(\Sigma,\omega)$, we pick a mean normalized Hamiltonian $H$
whose flow represents $\tilde{\varphi}$. Then, we define

$c_{\ul{L}}(\tilde{\varphi}):=c_{\ul{L}}(H).$ (6)

This is well-defined by the homotopy invariance property from Theorem 1.4.
When $\Sigma\neq S^{2}$, this yields a well-defined map

$c_{\ul{L}}:\Ham(\Sigma,\omega)\rightarrow\R,$ (7)

because $\Ham(\Sigma,\omega)$ is simply connected.

For clarity of exposition, we will suppose that $\Sigma$ has positive genus
throughout the rest of Section 3; we will see below that this suffices to
prove Theorems 1.2 and 1.2.

The spectral invariant $c_{\ul{L}}:\Ham(\Sigma,\omega)\rightarrow\R$ inherits
appropriately reformulated versions of the properties listed in Theorem1.4. We
list the following properties which will be used below. For
$\phi,\psi\in\Ham(\Sigma,\omega)$ we have

  1. 1.

(Hofer Lipschitz) $|c_{\ul{L}}(\varphi)-c_{\ul{L}}(\psi)|\leq
d_{H}(\varphi,\psi)$, where $d_{H}$ is the Hofer distance defined in (2).

  2. 2.

(Triangle inequality) $c_{\ul{L}}(\phi\psi)\leq
c_{\ul{L}}(\phi)+c_{\ul{L}}(\psi)$.

We now turn to defining invariants of homeomorphisms. An individual
$c_{\ul{L}}$ is not in general $C^{0}$-continuous, as the following example
shows.

######  Example 3.3.

Let $D$ be a disc that does not meet $\ul{L}$ and let $\varphi$ be supported
in $D$. Then, by the Shift and Support control properties from Theorem 1.4, we
have that

$c_{\ul{L}}=-\Cal(\varphi).$

Now it is known that $\Cal$ is not $C^{0}$-continuous. For example, identify
$D$ with a disc of radius one centered at the origin in $\R^{2}$, equipped
with an area form, and take a sequence of Hamiltonians $H_{i}$ that are
compactly supported in discs $D_{i}$ centered at the origin with radius $1/i$,
such that $\Cal(\phi^{1}_{H_{i}})=1$. Then the maps $\phi^{1}_{H_{i}}$ are
converging in $C^{0}$ to the identity, which has Calabi invariant $0$.

On the other hand, if we consider a difference of spectral invariants
$c_{\ul{L}}-c_{\ul{L^{\prime}}}$ and $D$ is disjoint from $\ul{L}$ and
$\ul{L^{\prime}}$, then $c_{\ul{L}}-c_{\ul{L^{\prime}}}$ vanishes on any
$\varphi$ supported in $D$. In fact, we will see in Proposition 2 below that
this difference is continuous on $\Ham(\Sigma,\omega)$.

We now state the result that allows us to define invariants for
homeomorphisms. The notation $d_{C^{0}}$ in the proposition stands for the
$C^{0}$ distance which is defined to be

$\displaystyle
d_{C^{0}}(\varphi,\psi)=\sup_{x\in\Sigma}d(\varphi(x),\psi(x)),$

where $d$ is a Riemannian distance on $\Sigma$.

######  Proposition 2.

Let $\ul{L},\ul{L}^{\prime}$ be monotone Lagrangian links. The mapping
$\Ham(\Sigma,\omega)\rightarrow\R$ defined via

$\varphi\mapsto c_{\ul{L}}(\varphi)-c_{\ul{L^{\prime}}}(\varphi)$

is uniformly continuous with respect to $d_{C^{0}}$. Consequently, it extends
continuously to $\overline{\Ham}(\Sigma,\omega)$.

To treat surfaces with boundary, we will need a variant of Proposition 2. Let
$\Sigma_{0}$ be a compact surface with boundary contained in a closed surface
$\Sigma$. Then, by the above discussion, any monotone Lagrangian link $\ul{L}$
in $\Sigma$, yields a spectral invariant

$c_{\ul{L}}:\Ham(\Sigma_{0},\omega)\rightarrow\R$

obtained from restricting $c_{\ul{L}}$ to
$\Ham(\Sigma_{0},\omega)\subset\Ham(\Sigma,\omega)$.

######  Proposition 3.

Let $\ul{L}$ be a monotone Lagrangian link. The mapping
$\Ham(\Sigma_{0},\omega)\rightarrow\R$ defined via

$\varphi\mapsto c_{\ul{L}}(\varphi)+\Cal(\varphi)$ (8)

is uniformly continuous with respect to $d_{C^{0}}$. Consequently, it extends
continuously to $\overline{\Ham}(\Sigma,\omega)$.

Note that $c_{\ul{L}}(\varphi)+\Cal(\varphi)$ corresponds to the value of
$c_{\ul{L}}(H)$ where $H$ is any Hamiltonian generating $\varphi$ whose
support is included in the interior of $\Sigma_{0}$.

The proofs of the above results follow from standard arguments from $C^{0}$
symplectic topology; see [59, 14, 15, 55]. We will now prove these results.

######  Proof 3.4 (Proof of Proposition 2).

Define $\zeta:\Ham(\Sigma,\omega)\rightarrow\R$ by

$\zeta(\varphi)=c_{\ul{L}}(\varphi)-c_{\ul{L^{\prime}}}(\varphi).$

We need to prove that $\zeta$ is uniformly continuous with respect to the
$C^{0}$ distance.

Let $\eps>0$ and fix a closed disc
$B\subset\Sigma\setminus(\ul{L}\cup\ul{L^{\prime}})$. By888The lemma is stated
for $\Sigma=S^{2}$, but the argument works just as well for the case of
general $\Sigma$. [15, Lemma 3.11], there exists a real number $\delta>0$ such
that for any $g\in\Ham(\Sigma,\omega)$ satisfying $d_{C^{0}}(g,\id)<\delta$,
there exists $h\in\Ham(\Sigma,\omega)$ with support in $B$ and

$d_{H}(g,h)\leq\eps.$

Let $\phi_{1},\phi_{2}\in\Ham(\Sigma,\omega)$ be such that
$d_{C^{0}}(\phi_{1},\phi_{2})<\delta$. We will prove that
$|\zeta(\phi_{1})-\zeta(\phi_{2})|\leq 2\eps$ and this will conclude our
proof.

Since
$d_{C^{0}}(\phi_{1}\phi_{2}^{-1},\id)=d_{C^{0}}(\phi_{1},\phi_{2})\leq\delta$,
we may pick $h\in\Ham(\Sigma,\omega)$ supported in $B$ and such that

$d_{H}(\phi_{1}\phi_{2}^{-1},h)\leq\eps.$ (9)

We now claim that

$c_{\ul{L}}(h)=-c_{\ul{L}}(h^{-1})=c_{\ul{L^{\prime}}}(h)=-c_{\ul{L^{\prime}}}(h^{-1}).$
(10)

Indeed, this follows from the Lagrangian control property of Theorem 1.4,
since we can find a mean normalized Hamiltonian $H$ for $h$ such that $H_{t}$
is constant in the complement of $B$, and so $h^{-1}$ has a mean normalized
Hamiltonian equal to $-H$ in the complement of $B$.

Now observe that

$\displaystyle
c_{\ul{L}}(\phi_{1})=c_{\ul{L}}(\phi_{1}\phi_{2}^{-1}\phi_{2})\leq
c_{\ul{L}}(\phi_{1}\phi_{2}^{-1}h^{-1})+c_{\ul{L}}(h\phi_{2})$
$\displaystyle\leq\eps+c_{\ul{L}}(h\phi_{2})\leq
c_{\ul{L}}(h)+c_{\ul{L}}(\phi_{2})+\eps.$ (11)

Here, the first inequality holds by the Triangle inequality property from
above; the second holds by the Hofer Lipschitz property combined with (9); and
the third holds by again applying the Triangle inequality.

Similarly,

$\displaystyle
c_{\ul{L^{\prime}}}(\phi_{2})=c_{\ul{L^{\prime}}}((\phi_{1}\phi_{2}^{-1})^{-1}\phi_{1})\leq
c_{\ul{L^{\prime}}}(h^{-1}\phi_{1})+\eps\leq
c_{\ul{L^{\prime}}}(h^{-1})+c_{\ul{L^{\prime}}}(\phi_{1})+\eps.$

The above inequalities together with (10) give

$\displaystyle\zeta(\phi_{1})$
$\displaystyle=c_{\ul{L}}(\phi_{1})-c_{\ul{L^{\prime}}}(\phi_{1})$
$\displaystyle\leq
c_{\ul{L}}(h)+c_{\ul{L}}(\phi_{2})+\eps+c_{\ul{L^{\prime}}}(h^{-1})-c_{\ul{L^{\prime}}}(\phi_{2})+\eps$
$\displaystyle=\zeta(\phi_{2})+2\eps.$

Switching the roles of $\phi_{1}$ and $\phi_{2}$, we obtain
$|\zeta(\phi_{1})-\zeta(\phi_{2})|\leq 2\eps$, which shows that $\zeta$ is
uniformly continuous.

######  Proof 3.5 (Proof of Proposition 3).

As in the previous proof, we start by letting $\eps>0$ and fix a closed
$B\subset\Sigma_{0}\setminus(L\cup L^{\prime})$. We then follow step by step
the same argument until we arrive at Inequality 3.4:

$\displaystyle c_{\ul{L}}(\phi_{1})\leq
c_{\ul{L}}(h)+c_{\ul{L}}(\phi_{2})+\eps.$

Since the Calabi homomorphism is 1-Lipschitz with respect to Hofer’s distance,
inequality (9) yields

$\Cal(\phi_{1})\leq\Cal(\phi_{2})+\Cal(h)+\eps.$

Now, by the Shift property from Theorem 1.4, $c_{\ul{L}}(h)=-\Cal(h)$, as can
be seen by choosing a Hamiltonian for $h$ that vanishes outside $B$ and then
mean normalizing. Thus we obtain from the two previous inequalities:

$c_{\ul{L}}(\phi_{1})+\Cal(\phi_{1})\leq
c_{\ul{L}}(\phi_{2})+\Cal(\phi_{2})+2\eps.$

We conclude by switching the roles of $\phi_{1}$ and $\phi_{2}$ as in the
proof of Proposition 2.

###  3.3 Infinite twists on positive genus surfaces

We can now prove Theorem 1.2.

######  Proof 3.6 (Proof of Theorem 1.2).

We showed in Proposition 1 that $\FHomeo$ is a normal subgroup of
$\overline{\Ham}$. It remains to show that it is proper. To do this, we adapt
the strategy from [14, Thm. 1.7], namely we construct an example of a
Hamiltonian homeomorphism that does not have finite energy.

We first consider the case where $\Sigma$ is closed. Let $\ul{L}^{m}$ be an
equidistributed sequence of Lagrangian links. Define
$\zeta_{m}:\Ham(\Sigma,\omega)\rightarrow\R$ by

$\zeta_{m}(\varphi)=c_{\ul{L}^{m}}(\varphi)-c_{\ul{L}^{1}}(\varphi).$

By Proposition 2, $\zeta_{m}$ admits a continuous extension to
$\overline{\Ham}(\Sigma,\omega)$.

We now claim that if $\phi\in\FHomeo$, then $\zeta_{m}(\phi)$ remains bounded
as $m$ varies. To see this, let $\phi_{i}=\varphi^{1}_{H_{i}}$ be a sequence
of diffeomorphisms converging to $\phi$ such that the $H_{i}$ are mean
normalized and have Hofer norm bounded by $C$. Then by the Hofer Lipschitz
property from Theorem 1.4, applied with $H^{\prime}=0$, we have that the
$\zeta_{m}(\phi_{i})$ are also bounded by $C$. Hence, by continuity, the
$\zeta_{m}(\phi)$ are bounded as well.

Next, let $D\subset\Sigma\setminus\cup_{i=1}^{k_{1}}L_{i}^{1}$ be a smoothly
embedded closed disc, which we identify with the disc of radius $R$ in
$\R^{2}$ centered at the origin with its standard area form. We now define an
“infinite twist” homeomorphism $\psi$ supported in $D$ as follows. Let
$(\theta,r)$ denote polar coordinates. Let $f:(0,R]\to\R$ be a smooth function
which vanishes near $R$, is decreasing, and satisfies

$\int^{1}_{0}r^{3}f(r)\,dr=\infty.$ (12)

We now define $\psi$ by $\psi(0)=0$ and

$\psi(r,\theta)=(r,\theta+2\pi f(r))$ (13)

for $r>0$. The heuristic behind the condition (12) is that it forces $\psi$ to
have “infinite Calabi invariant”. Indeed, if $f$ was defined on the closed
interval $[0,R]$, then $\psi$ would be a Hamiltonian diffeomorphism with
Calabi invariant $\int^{1}_{0}r^{3}f(r)\,dr$.

We now claim that $\psi$ is a Hamiltonian homeomorphism with the property that
$\zeta_{m}(\psi)$ diverges as $m$ varies. By [14, Lem. 1.14]999[14, Lem. 1.14]
uses the condition $\int^{1}_{0}\int^{1}_{r}sf(s)\,ds\,r\,dr=\infty$, but this
is equivalent to (12) since
$\int^{1}_{0}\int^{1}_{r}sf(s)\,ds\,r\,dr=\frac{1}{2}\int^{1}_{0}r^{3}f(r)\,dr$
by integration by parts., there are Hamiltonians $F_{i}$, compactly supported
in the interior of $D$, with the following properties:

  1. 1.

The sequence $\psi^{1}_{F_{i}}$ converges in $C^{0}$ to $\psi$.

  2. 2.

$F_{i}\leq F_{i+1}.$

  3. 3.

$\text{lim}_{i\to\infty}\int^{1}_{0}\int_{\Sigma}F_{i}\omega=\infty.$

By the first property above, $\psi$ is a Hamiltonian homeomorphism. We now
apply several properties from Theorem 1.4. By the Shift property,
$\zeta_{m}(\psi^{1}_{F_{i}})=c_{\ul{L}^{m}}(F_{i})-c_{\ul{L}^{1}}(F_{i})$, and
by the Support control property from the same theorem,
$c_{\ul{L}^{1}}(F_{i})=0.$ It then follows by continuity and the Monotonicity
property that

$\zeta_{m}(\psi)\geq c_{\ul{L}^{m}}(F_{i}),$

hence by the Calabi property from Theorem 1.1,

$\text{lim}_{m\to\infty}\zeta_{m}(\psi)\geq\int^{1}_{0}\int_{\Sigma}F_{i}\hskip
5.69054pt\omega$

for any $i$. Hence by the third property above, the $\zeta_{m}(\psi)$ diverge.

In the case when $\Sigma$ is not closed, we reduce to the above by embedding
$\Sigma$ into a closed surface $\Sigma^{\prime}$. Now define an infinite twist
exactly as above, except in addition the infinite twist is supported in
$\Sigma$: by the above, this map is not in
$\FHomeo(\Sigma^{\prime},\omega^{\prime})$, hence can not be in
$\FHomeo(\Sigma,\omega).$

######  Remark 3.7.

The infinite twist $\psi$, introduced above in (13), is the time-1 map of the
1-parameter subgroup $\psi^{t}$ of $\Homeo_{0}(\Sigma,\omega)$ defined by
$\psi^{t}(0)=0$ and

$\psi^{t}(r,\theta):=(r,\theta+2\pi tf(r)).$

It follows immediately from the above proof that $\psi^{t}$ is not a finite-
energy homeomorphism for $t\neq 0$. This yields an injective group
homomorphism from the real line $\R$ into the quotient
$\overline{\Ham}(\Sigma,\omega)/\FHomeo(\Sigma,\omega)$.

Since $\Hameo(\Sigma,\omega)\subset\FHomeo(\Sigma,\omega)$, we see that
$\psi^{t}$ yields an injective group homomorphism from $\R$ into the quotient
$\overline{\Ham}(\Sigma,\omega)/\Hameo(\Sigma,\omega)$, as well.

One can show that the above injections are not surjections; see [55]. However,
we have not been able to determine whether or not the quotients are isomorphic
to $\R$ as abelian groups.

###  3.4 Calabi on Hameo

We will now provide a proof of Theorem 1.2. The proof closely follows the
argument from [14, Section 7.4], except that we use the Lagrangian spectral
invariants defined here instead of the PFH spectral invariants studied in
[14].

######  Proof 3.8.

Let $\phi\in\Hameo(\Sigma,\omega)$, and take an $H\in
C^{0}([0,1]\times\Sigma)$ such that

$\phi^{1}_{H_{i}}\xrightarrow{C^{0}}\phi,\text{ and
}\|H-H_{i}\|_{(1,\infty)}\to 0,$

where the $H_{i}$ are smooth Hamiltonians as in Definition 2.1. For future
use, we record $H$ in the notation by writing $\phi=\phi_{H}$.

We now define

$\Cal(\phi):=\int_{0}^{1}\int_{\Sigma}H\,\omega\,dt.$ (14)

We claim this is well-defined. To show this, it suffices to show that if
$\phi=\text{Id}$, then

$\int_{0}^{1}\int_{\Sigma}H\,\omega\,dt=0,$ (15)

since Cal is a homomorphism on $\Ham(\Sigma,\omega).$ In other words, we will
show that if $\phi^{1}_{H_{i}}\xrightarrow{C^{0}}\id$ and
$\|H-H_{i}\|_{(1,\infty)}\to 0$, then (15) holds.

As in Proposition 3, embed $\Sigma$ into a closed surface $\Sigma^{\prime}$,
choose a sequence of equidistributed Lagrangian links $\ul{L}^{m}$ in
$\Sigma^{\prime}$, and consider $\xi_{m}:\Ham(\Sigma,\omega)\rightarrow\R$ by

$\xi_{m}(\varphi)=c_{\ul{L}^{m}}(\varphi)+\Cal(\varphi).$

By Proposition 3, $\xi_{m}$ extends continuously to
$\overline{\Ham}(\Sigma,\omega)$. This in particular implies that

$\lim_{j\to\infty}\xi_{m}(\phi^{1}_{H_{j}})=0.$ (16)

For any fixed $m,i$, we can write

$\displaystyle\left|\int_{0}^{1}\int_{\Sigma}H\,\omega\,dt\right|$
$\displaystyle\leq\left|\int_{0}^{1}\int_{\Sigma}H\,\omega\,dt\;-\Cal(\phi^{1}_{H_{i}})\right|$
$\displaystyle\qquad\qquad+\left|\Cal(\phi^{1}_{H_{i}})-\xi_{m}(\phi^{1}_{H_{i}})\right|+\left|\xi_{m}(\phi^{1}_{H_{i}})\right|.$

The right hand side of the above inequality is a sum of three terms. We know
that

$\left|\int_{0}^{1}\int_{\Sigma}H\,\omega-\Cal(\phi^{1}_{H_{i}})\,dt\right|\leq||H-H_{i}||_{(1,\infty)},$

since $H_{i}$ are smooth and compactly supported Hamiltonians and so
$\Cal(\phi^{1}_{H_{i}})=\int_{0}^{1}\int_{\Sigma}H_{i}\,\omega\,dt.$ We claim
that the third term has the same bound. Indeed, by the Hofer Lipschitz
property from Theorem 1.4, we have
$|\xi_{m}(\phi^{1}_{H_{j}})-\xi_{m}(\phi^{1}_{H_{i}})|\leq||H_{j}-H_{i}||_{(1,\infty)}$
for all $i,j$, and then fixing $i$ and taking the limit as $j\to\infty$ gives

$|\xi_{m}(\phi^{1}_{H_{i}})|\leq||H-H_{i}||_{(1,\infty)}$

by (16). Hence, whatever $m$, the first and third terms of the above
inequality can be made arbitrarily small by choosing $i$ sufficiently large.
As for the second term, for fixed $i$, this can be made arbitrarily small by
choosing $m$ sufficiently large, by the Calabi property proved in Theorem 1.1.

Hence, Cal is well-defined. It remains to show that it is a homomorphism. The
fact that $\Cal$ is a homomorphism if well-defined was in fact previously
shown in [45] so we will be brief. Let $\psi_{1}$ and $\psi_{2}$ be elements
of $\Hameo(\Sigma,\omega)$, and choose corresponding $H$, $G$. By
reparametrizing, we can assume that $H$ and $G$ vanish near $0$ and $1$, and
we can then form the concatenation

$K(t,x)=\begin{cases}2H(2t,x),&\quadif t\in[0,\frac{1}{2}]\\\
2G(2t-1,x),&\quadif t\in[\frac{1}{2},1]\end{cases}.$

One now checks that $\phi_{K}=\phi_{G}\circ\phi_{H}$, and it now follows
immediately from this formula for $K$ and (14) that
$\Cal(\phi_{G}\circ\phi_{H})=\Cal(\phi_{H})+\Cal(\phi_{G}).$ The proof that
$\Cal((\phi_{H})^{-1})=-\Cal(\phi_{H})$ is similar.

##  4 Heegaard tori and Clifford tori

The proof of Theorem 1.4 occupies the next three sections of the paper. Recall
from the introduction that this result will be obtained by studying a Floer
cohomology for symmetric product Lagrangians $\Sym(\ul{L})$ in the symmetric
products of the surface. This section is mainly devoted to the proof of a
monotonicity result (Lemma 4.24), which will later on guarantee that we have a
well-defined Lagrangian Floer cohomology. Section 5 computes the potential
function of $\Sym(\ul{L})$ and Section 6 defines the Floer cohomology and
spectral invariants.

###  4.1 Set-up and outline

We recall the set-up. Fix a closed genus $g$ surface $\Sigma$, and equip
$\Sigma$ with a symplectic form $\omega$. We can choose a complex structure
$J_{\Sigma}$ on $\Sigma$ such that $\omega$ is a Kähler form. Consider a
Lagrangian link $\ul{L}=\cup_{i=1}^{k}L_{i}$ consisting of $k$ pairwise-
disjoint circles on $\Sigma$, with the property that $\Sigma\setminus\ul{L}$
consists of planar domains $B_{j}^{\circ}$, with $1\leq j\leq s$, whose
closures $B_{j}\subset\Sigma$ are also planar. Let $B_{j}$ have $k_{j}$
boundary components. Since the Euler characteristic of a planar domain $D$
with $k_{D}$ boundary components is $2-k_{D}$, the Euler characteristic of
$\Sigma$ is $2-2g=\sum_{j=1}^{s}(2-k_{j})=2s-2k$, and hence $s=k-g+1$. We
assume throughout that $s\geq 2$. Finally, for $1\leq j\leq s$, let $A_{j}$
denote the $\omega$-area of $B_{j}$.

Let $(M,\omega_{M})=(\Sigma^{k},\omega^{\oplus k})$. Let $X:=\Sym^{k}(\Sigma)$
be the $k$-fold symmetric product. It has a complex structure $J_{X}$ induced
from $J_{\Sigma}$ making $X$ a complex manifold and the quotient map $\pi:M\to
X$ holomorphic. We equip $X$ with the singular Kähler current $\omega_{X}$
which naturally descends from $(M,\omega_{M})$ under $\pi$. Let $\Sym(\ul{L})$
be the Lagrangian submanifold in $X$ given by the image of
$L_{1}\times\dots\times L_{k}$ under $\pi$. The spectral invariant
$c_{\ul{L}}$ of Theorem 1.4 will be constructed using a variant of Lagrangian
Floer cohomology of $\Sym(\ul{L})$ in $X$, ‘bulk deformed’ by $\eta$ times the
diagonal divisor.

######  Remark 4.1.

In Heegaard Floer theory for links in 3-manifolds, one begins with a surface
$\Sigma$ of genus $g$, two sets of attaching circles
$\alpha_{1},\ldots,\alpha_{k}$ and $\beta_{1},\ldots,\beta_{k}$ and two sets
of base-points $z_{1},\ldots,z_{l}$ and $w_{1},\ldots,w_{l}$, where $k=g+l-1$,
see [50, Definition 3.1]. This data encodes a link in a $3$-manifold; one can
take $g=0$ for links in $S^{3}$. Link Floer homology is obtained from a
version of Lagrangian Floer cohomology of product-like tori associated to
$\alpha$ and $\beta$ in $\Sym^{k}(\Sigma)$. For link invariants the crucial
topological information is contained in the filtrations associated to the
intersection numbers with divisors $D_{p}=p+\Sym^{k-1}(\Sigma)$, for
$p\in\\{z_{i},w_{j}\\}$ one of the base-points, which play no role in this
paper. Our ‘quantitative version’ instead keeps track of holonomies of local
systems and of intersection number with the diagonal divisor. We also work
with ‘anchored’ or ‘capped’ Floer generators, so that the action functional
becomes well-defined.

######  Remark 4.2.

It is crucial for our purposes that our Floer cohomology is invariant under
Hamiltonian isotopies (at least those inherited from isotopies of the link
$\ul{L}$), whereas in Heegaard Floer homology the important invariance
properties are those which give different presentations of a fixed link
(handleslide moves and stabilisations, which one shows respect the topological
information held by the filtrations determined by the $D_{p}$). The following
illustrative example may be helpful. Consider two circles on $\PP^{1}$ whose
complementary domains have closures (disjoint) discs $B_{1}$ of area $A_{1}$,
$B_{2}$ of area $A_{2}$ and an annulus $B_{3}$ of area $A_{3}$. The Maslov
index two discs on $\Sym(\ul{L})$ are given by $B_{1}$, $B_{2}$ and a double
covering of $B_{3}$. The fact that such branched covers arise makes it natural
to keep track of branch points, and hence intersections with the diagonal
divisor; this is the role of our bulk parameter $\eta$. If $A_{1}\neq A_{2}$
then the link is displaceable, even though its Heegaard Floer cohomology would
be defined (over $\Z/2$) and non-vanishing. It is well-known that Floer
cohomology over $\C$ is Hamiltonian invariant only under monotonicity
hypotheses, which is where the hypothesis of Theorem 1.4 arises. Hamiltonian
invariance for the bulk-deformed version relies on restricting to values
$\eta\geq 0$. Our analysis of the curvature in the Floer complex of
$\Sym(\ul{L})$ would apply equally well over the Novikov field, cf. Definition
5.5.

The unobstructedness of $\Sym(\ul{L})$ follows broadly as in its Heegaard
Floer counterpart. (More precisely, in the link setting, a ‘weak
admissibility’ condition is imposed on Heegaard diagrams to rule out bubbling
which would obstruct the Floer complex over $\Z/2$, whereas we compute the
curvature in the Floer complex directly.) To compute Floer cohomology, we
first consider the special case in which $\Sigma=\PP^{1}$ and the $B_{j}$ are
discs for $j=1,\dots,k$. We show the corresponding $\Sym(\ul{L})$ is isotopic
to a Clifford-type torus in $X=\Sym^{k}(\PP^{1})=\PP^{k}$, and use that
isotopy to compare the holomorphic discs they bound. In the general case, the
fact that the regions $B_{j}\subset\Sigma$ are planar domains enables us to
reduce aspects of the holomorphic curve theory to the case $\Sigma=\PP^{1}$.
Our proof incorporates local systems because non-vanishing of Floer cohomology
is detected, as in [11, 13], by considering the Floer boundary operator under
variation of the local system. We obtain a spectral invariant $c_{\cE}$ for
any local system $\cE\to\Sym(\ul{L})$ with respect to which Floer cohomology
is non-trivial. In fact Floer cohomology is non-zero for the trivial local
system on $\Sym(\ul{L})$, and (after rescaling by the number of components) it
is the spectral invariant $c_{\cE}$ for the trivial local system which is the
$c_{\ul{L}}$ which appears in Theorem 1.4.

For unobstructedness of the Floer cohomology of $\Sym(\ul{L})$, we will need
control over the Maslov indices of holomorphic discs with boundary on that
torus. To that end, we next show that when $\Sigma=\PP^{1}$ and the circles
$L_{j}$ bound pairwise disjoint discs $B_{j}$, with $1\leq j\leq k=s-1$, the
torus $\Sym(\ul{L})$ is isotopic to a Clifford-type torus in projective space.

###  4.2 Co-ordinates on the symmetric product

The symmetric product $\Sym^{k}(\mathbb{P}^{1})$ is naturally a complex
manifold, biholomorphic to $\PP^{k}$. To fix notation, we recall that
isomorphism. Let $x_{0,i},x_{1,i}$ denote homogeneous co-ordinates on the
$i$-th factor of $(\PP^{1})^{k}$. Define
$Q_{0}(x),\dots,Q_{k}(x)\in\C[x_{0,1},x_{1,1},\dots,x_{0,k},x_{1,k}]$ by the
identity

$\displaystyle\prod_{j=1}^{k}(x_{0,j}X+x_{1,j}Y)=\sum_{j=0}^{k}Q_{j}(x)X^{k-j}Y^{j}.$

Let $Y_{0},\dots,Y_{k}$ be the homogenous co-ordinates of $\PP^{k}$. We define
$\pi:(\PP^{1})^{k}\to\PP^{k}$ by

$[Y_{0}:\dots:Y_{k}]=\pi([x_{0,1}:x_{1,1}],\dots,[x_{0,k}:x_{1,k}])=[Q_{0}(x):\dots:Q_{k}(x)].$

It is an $S_{k}$-invariant holomorphic map which descends to a biholomorphism
$\Sym^{k}(\mathbb{P}^{1})\simeq\mathbb{P}^{k}$.

Let $a_{1},\dots,a_{k+1}$ be $k+1$ pairwise distinct points in $\PP^{1}$. We
identify $\PP^{1}$ as $\C\cup\\{\infty\\}$ and assume that $a_{k+1}=\infty$.
For each $i=1,\dots,k+1$, we define

$\displaystyle\widetilde{D}_{i}\ :=\
\left\\{\prod_{j=1}^{k}(x_{1,j}-a_{i}x_{0,j})=0\right\\}\subset(\PP^{1})^{k}.$

Note that $\widetilde{D}_{i}$ is $S_{k}$-invariant and it descends to

$\displaystyle
D_{i}:=\pi(\widetilde{D}_{i})=\left\\{\sum_{j=0}^{k}(-a_{i})^{k-j}Y_{j}=0\right\\}\subset\PP^{k}.$

When $i=k+1$, the divisors $\widetilde{D}_{k+1}$ and $D_{k+1}$ are understood
as $\\{\prod_{j=1}^{k}x_{0,j}=0\\}$ and $\\{Y_{0}=0\\}$ respectively.

######  Remark 4.3.

The divisor $D_{i}$ is precisely the image of
$\Sym^{k-1}(\Sigma)\to\Sym^{k}(\Sigma)=X$ under the map $D\mapsto a_{i}+D$
(i.e. $(p_{1},\dots,p_{k-1})\mapsto(p_{1},\dots,p_{k-1},a_{i})$ but written in
a form that regards $D=p_{1}+\dots+p_{k-1}=(p_{1},\dots,p_{k-1})$ as a divisor
in $\Sigma$). In particular, the $D_{i}$ are pairwise homologous, and
$\cup_{i=1}^{k+1}D_{i}$ is an anticanonical divisor of $X$.

Note that $(\PP^{1})^{k}\setminus\tilde{D}_{k+1}=\C^{k}$ and
$\pi|_{\C^{k}}:\C^{k}\to\C^{k}$ is a $S_{k}$-invariant holomorphic map which
descends to a biholomorphism $\Sym^{k}(\C)\simeq\C^{k}$. For $i=1,\dots,k$, we
define $x_{i}:=\frac{x_{1,i}}{x_{0,i}}$ and $y_{i}:=\frac{Y_{i}}{Y_{0}}$,
which give coordinates on the complements of $\tilde{D}_{k+1}$ and $D_{k+1}$
respectively. Since $q_{j}:=\frac{Q_{j}}{Q_{0}}$ is precisely the $j^{th}$
elementary symmetric polynomial of $\\{x_{i}\\}_{i=1}^{k}$, the map
$\pi|_{\C^{k}}$ can be written as

$\displaystyle(y_{1},\dots,y_{k})$
$\displaystyle=\pi(x_{1},\dots,x_{k})=(q_{1}(x),\dots,q_{k}(x))$
$\displaystyle q_{j}(x)$ $\displaystyle=\frac{1}{j!(k-j)!}\sum_{\sigma\in
S_{k}}x_{\sigma(1)}\dots x_{\sigma(j)}.$

In affine coordinates, for $i=1,\dots,k$, we have

$\displaystyle\widetilde{D}_{i}\setminus\widetilde{D}_{k+1}$ $\displaystyle=\
\left\\{\prod_{j=1}^{k}(x_{j}-a_{i})=0\right\\}\text{ and }D_{i}\setminus
D_{k+1}=\left\\{\sum_{j=0}^{k}(-a_{i})^{k-j}y_{j}=0\right\\}.$

Since the $\\{a_{i}\\}$ are pairwise distinct, the Vandermonde matrix

$A=\begin{pmatrix}(-a_{1})^{k-1}&(-a_{1})^{k-2}&\dots&1\\\
(-a_{2})^{k-1}&(-a_{2})^{k-2}&\dots&1\\\ \vdots&&&\vdots\\\
(-a_{k})^{k-1}&(-a_{k})^{k-2}&\dots&1\\\ \end{pmatrix}$

is non-degenerate. We define $g_{i}=\sum_{j=0}^{k}(-a_{i})^{k-j}y_{j}$ so that
$D_{i}\setminus D_{k+1}=\\{g_{i}=0\\}$; the non-degeneracy of $A$ implies that
$\\{g_{i}\\}_{i=1}^{k}$ is an invertible linear change of coordinates of
$\\{y_{i}\\}_{i=1}^{k}$.

###  4.3 Relation to the Clifford torus

For $\eps>0$ small, we define the Clifford torus in $X$ as

$\displaystyle L_{\eps}:=\\{(g_{1},\dots,g_{k})\in\C^{k}:|g_{i}|=\eps\text{
for each }i\\}.$

The main result of this section, Corollary 4.3 below, asserts that when $\eps$
is small, $L_{\eps}$ is $C^{1}$ close to $\Sym(\ul{L})$ for an appropriate
$\ul{L}$.

For a small neighborhood $G$ of $(g_{1},\dots,g_{k})=0$,
$\pi|_{\pi^{-1}(G)}:\pi^{-1}(G)\to G$ is a trivial covering map with $k!$
sheets. For example,

$\pi^{-1}(\\{(g_{1},\dots,g_{k})=0\\})\ =\ \bigcup_{\sigma\in
S_{k}}\\{x_{i}=a_{\sigma(i)}\text{ for }1\leq i\leq k\\}.$

Therefore, when $\eps>0$ is small, $\pi^{-1}(L_{\eps})$ is a collection of
$k!$ pairwise-disjoint totally real $k$-tori in $(\PP^{1})^{k}$. More
explicitly, we have

$\displaystyle\pi^{-1}(L_{\eps})=\left\\{\left|\prod_{j=1}^{k}(x_{j}-a_{i})\right|=\eps\text{
for each }i=1,\dots,k\right\\}.$

Let $\widetilde{G}$ be the connected component of $\pi^{-1}(G)$ containing the
point with co-ordinates $x_{i}=a_{i}$ for each $i$. For $\eps>0$ sufficiently
small, there exists $\delta>\eps$ such that

$\displaystyle\left\\{|x_{i}-a_{i}|<\delta\text{ for each }i,1\leq i\leq
k\right\\}\subset\widetilde{G}$
$\displaystyle\widetilde{L}_{\eps}:=\pi^{-1}(L_{\eps})\cap\widetilde{G}=\left\\{\left|\prod_{j=1}^{k}(x_{j}-a_{i})\right|=\eps\text{
and }|x_{i}-a_{i}|<\delta\text{ for each }i,\ 1\leq i\leq k\right\\}$
$\displaystyle\text{If }x\in\widetilde{L}_{\eps}\text{ and }i\neq j,\text{
then }|x_{i}-a_{j}|>3\delta.$

######  Lemma 4.4.

For $\kappa>0$, there exists a small $\eps>0$ and a family of diffeomorphisms
$(\Phi^{t})_{t\in[0,1]}$ of $(\PP^{1})^{k}$ supported inside $\widetilde{G}$
with the following properties:

  * •

$\Phi^{0}$ is the identity;

  * •

the $C^{1}$-norm of $\Phi^{t}$ is less than $\kappa$ for all $t\in[0,1]$;

  * •

$\Phi^{1}(\\{|x_{i}-a_{i}||\prod_{j\neq i}(a_{j}-a_{i})|=\eps\text{ for all
}i\\})=\widetilde{L}_{\eps}$;

  * •

$\Phi^{t}(\widetilde{D}_{i}\cap\widetilde{G})=\widetilde{D}_{i}\cap\widetilde{G}$
for all $t\in[0,1]$ and all $i=1,\dots,k$.

######  Proof 4.5.

For simplicity of notation, we will give the proof in the case in which
$a_{i}\in\R$ for each $i$.

Let $u_{i}+\sqrt{-1}v_{i}=x_{i}-a_{i}$. Then
$\widetilde{D}_{i}\cap\widetilde{G}=\\{(u,v)\in\widetilde{G}:u_{i}=v_{i}=0\\}$.
The system of equations

$\displaystyle(x_{i}-a_{i})\left((1-t)\prod_{j\neq
i}(a_{j}-a_{i})+t\prod_{j\neq i}(x_{j}-a_{i})\right)\ =\
\alpha_{i}+\sqrt{-1}\beta_{i}\qquad 1\leq i\leq k$ (17)

for $t\in[0,1]$ and $\alpha_{i},\beta_{i}\in\R$ becomes, in the $u_{i},v_{i}$
co-ordinates,

$\displaystyle(u_{i}+\sqrt{-1}v_{i})\left((1-t)\prod_{j\neq
i}(a_{j}-a_{i})+t\prod_{j\neq
i}(u_{j}+\sqrt{-1}v_{j}+a_{j}-a_{i})\right)=\alpha_{i}+\sqrt{-1}\beta_{i}.$
(18)

Taking real and imaginary parts, we obtain

$\displaystyle u_{i}\prod_{j\neq i}(a_{j}-a_{i})+tH_{u_{i}}(u,v)=\alpha_{i},$
$\displaystyle v_{i}\prod_{j\neq i}(a_{j}-a_{i})+tH_{v_{i}}(u,v)=\beta_{i},$

where $H_{u_{i}}(u,v)$ and $H_{v_{i}}(u,v)$ are polynomials in $u_{j},v_{j}$
in which each term has degree at least two.

Let $\rho:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a cut-off function
such that $\rho(s)=1$ for $s<\frac{\eps^{2}}{2}$, $\rho(s)=0$ for $s>\eps^{2}$
and $|\rho^{\prime}(s)|<\frac{C}{\eps^{2}}$ for some constant $C$ independent
of $\eps$ and for all $s$. We denote $\sum_{i}u_{i}^{2}+v_{i}^{2}$ by
$|(u,v)|^{2}$. Let

$\displaystyle F_{u_{i}}^{t}(u,v,\alpha,\beta):=u_{i}\prod_{j\neq
i}(a_{j}-a_{i})+t\rho(|(u,v)|^{2})H_{u_{i}}(u,v)-\alpha_{i},$ $\displaystyle
F_{v_{i}}^{t}(u,v,\alpha,\beta):=v_{i}\prod_{j\neq
i}(a_{j}-a_{i})+t\rho(|(u,v)|^{2})H_{v_{i}}(u,v)-\beta_{i}.$

The $2k\times 2k$ square matrix

$D_{u,v}F^{t}:=\begin{pmatrix}\frac{\partial F_{u_{1}}^{t}}{\partial
u_{1}}&\frac{\partial F_{u_{1}}^{t}}{\partial v_{1}}&\dots&\frac{\partial
F_{u_{1}}^{t}}{\partial v_{k}}\\\ \frac{\partial F_{v_{1}}^{t}}{\partial
u_{1}}&\frac{\partial F_{v_{1}}^{t}}{\partial v_{1}}&\dots&\frac{\partial
F_{v_{1}}^{t}}{\partial v_{k}}\\\ \vdots&&&\vdots\\\ \frac{\partial
F_{v_{k}}^{t}}{\partial u_{1}}&\frac{\partial F_{v_{k}}^{t}}{\partial
v_{1}}&\dots&\frac{\partial F_{v_{k}}^{t}}{\partial v_{k}}\\\ \end{pmatrix}$

can be written as a sum $A+t\rho B_{1}+t\rho^{\prime}B_{2}$, where $A$ is the
diagonal matrix with entries $\prod_{j\neq i}(a_{j}-a_{i})$ at both the
$(2i-1,2i-1)^{th}$ and $(2i,2i)^{th}$ positions, for each $i=1,\dots,k$, and
where the entries of $B_{\ell}$ are polynomials, with each non-zero term
having degree at least $1$ when $\ell=1$ and degree at least $3$ when
$\ell=2$.

Since the support of $\rho$ is $[0,\eps^{2}]$, when $\eps>0$ is small we have

$\displaystyle\|D_{u,v}F^{t}-A\|=t\,O(\eps)$

for all $t\in[0,1]$ and for all points $(u,v)$. By the implicit function
theorem, there exists a unique $g^{t}(\alpha,\beta)$ such that

$\displaystyle F_{u_{i}}^{t}(g^{t}(\alpha,\beta),\alpha,\beta)=0$ (19)
$\displaystyle F_{v_{i}}^{t}(g^{t}(\alpha,\beta),\alpha,\beta)=0$ (20)

for all $i=1,\dots,k$. Define a smooth isotopy starting at the identity by

$\Phi^{t}(u,v):=g^{t}\circ(g^{0})^{-1}(u,v).$

We can control the $C^{1}$-norm of the isotopy as follows. We have
$D\Phi^{t}=D(g^{t})\circ D((g^{0})^{-1})$. Since $g^{t}$ solves the equations
(19) and (20) for each $i$, by differentiating with respect to $\alpha$ and
$\beta$, we have

$Dg^{t}=-(D_{u,v}F^{t})^{-1}\begin{pmatrix}\frac{\partial
F_{u_{1}}^{t}}{\partial\alpha_{1}}&\frac{\partial
F_{u_{1}}^{t}}{\partial\beta_{1}}&\dots&\frac{\partial
F_{u_{1}}^{t}}{\partial\beta_{k}}\\\ \frac{\partial
F_{v_{1}}^{t}}{\partial\alpha_{1}}&\frac{\partial
F_{v_{1}}^{t}}{\partial\beta_{1}}&\dots&\frac{\partial
F_{v_{1}}^{t}}{\partial\beta_{k}}\\\ \vdots&&&\vdots\\\ \frac{\partial
F_{v_{k}}^{t}}{\partial\alpha_{1}}&\frac{\partial
F_{v_{k}}^{t}}{\partial\beta_{1}}&\dots&\frac{\partial
F_{v_{k}}^{t}}{\partial\beta_{k}}\\\ \end{pmatrix}=(D_{u,v}F^{t})^{-1}.$

As a result, we have

$\displaystyle\|Dg^{t}-A^{-1}\|=\frac{t\,O(\eps)}{\|A\|^{2}}.$

Moreover, when $t=0$, we have exactly $Dg^{0}=A^{-1}$. Therefore, we have

$\displaystyle\|D\Phi^{t}-\id\|=\frac{t\,O(\eps)}{\|A\|}$

so the $C^{1}$-norm of $\Phi^{t}$ is smaller than the prescribed $\kappa$
whenever $\eps$ is sufficiently small.

We now check the remaining conditions. Equation (18) implies that

$(g^{t})^{-1}(\\{u_{i}=v_{i}=0\\})\subset\\{\alpha_{i}=\beta_{i}=0\\},$

so

$\Phi^{t}(\widetilde{D}_{i}\cap\widetilde{G})=\widetilde{D}_{i}\cap\widetilde{G}$

for every $t,i$. It is also clear from the construction, cf. (17), that there
exists $0<\eps^{\prime}\ll\eps$ such that

$\displaystyle\Phi^{1}\left(\left\\{|x_{i}-a_{i}|\left|\prod_{j\neq
i}(a_{j}-a_{i})\right|=\eps^{\prime}\text{ for all
}i\right\\}\right)=\widetilde{L}_{\eps^{\prime}}.$

Therefore, replacing $\eps$ by $\eps^{\prime}$, the final claim of the
statement holds.

If $\eps>0$ is sufficiently small, and if $\ul{L}_{a_{1},\dots,a_{k},\eps}$ is
the union of circles

$\ul{L}_{a_{1},\dots,a_{k},\eps}=\bigcup_{i=1}^{k}\left\\{|x_{i}-a_{i}|\left|\prod_{j\neq
i}(a_{j}-a_{i})\right|=\eps\right\\}\subset\C,$

there is a $C^{1}$-small isotopy $\Phi^{t}_{G}$ supported in
$G\subset\PP^{k}$, with

$\Phi^{1}_{G}(\Sym(\ul{L}_{a_{1},\dots,a_{k},\eps}))=L_{\eps}\qquad\textrm{and}\qquad\Phi^{t}_{G}(D_{i})=D_{i}\
\textrm{for all}\ i,t.$

######  Proof 4.6.

The submanifold

$\displaystyle\widetilde{L}_{a_{1},\dots,a_{k},\eps}:=\left\\{|x_{i}-a_{i}|\left|\prod_{j\neq
i}(a_{j}-a_{i})\right|=\eps\text{ for all }i\right\\}\cap\tilde{G}$

is a product of circles, and $\pi(\widetilde{L}_{a_{1},\dots,a_{k},\eps})$ is
precisely $\Sym(\ul{L}_{a_{1},\dots,a_{k},\eps})$. Since the isotopy
$\Phi^{t}$ constructed in Lemma 4.4 is supported in $\tilde{G}$ and
$\pi|_{\tilde{G}}:\tilde{G}\to G$ is a diffeomorphism, we can descend
$\Phi^{t}$ to a family of diffeomorphisms $\Phi^{t}_{G}$ supported in $G$ such
that $\Phi^{1}_{G}(\Sym(\ul{L}_{a_{1},\dots,a_{k},\eps}))=L_{\eps}$ and
$\Phi^{t}_{G}(D_{i})=D_{i}$ for all $i$.

We will use Corollary 4.5 to obtain control over holomorphic discs on
$\Sym(\ul{L}_{a_{1},\dots,a_{k},\eps})$. We will discuss how to extend that
control from $\ul{L}_{a_{1},..,a_{k},\eps}$ to a more general $\ul{L}$
associated to a collection of disjoint discs in Corollary 4.5 and Proposition
5.

###  4.4 Tautological correspondence

We return to the general setting, in which the Riemann surface $\Sigma$ has
genus $g$ and $\ul{L}\subset\Sigma$ comprises $k$ pairwise disjoint circles.
Let $S$ denote the unit disc. We can understand holomorphic discs in
$\Sym^{k}(\Sigma)$ with boundary on $\Sym(\ul{L})$ via the ‘tautological
correspondence’ between a holomorphic map

$u:(S,\partial S)\to(X,\Sym(\ul{L}))$ (21)

and a pair of holomorphic maps $(v,\pi_{\WH{S}})$, where

$v:(\WH{S},\partial\WH{S})\to(\Sigma,\ul{L})$ (22)

and $\pi_{\WH{S}}:(\WH{S},\partial\WH{S})\to(S,\partial S)$ is a $k:1$
branched covering with all the branch points lying inside the interior of $S$.
The correspondence arises as follows (see also [38, Section 13], [39, Section
3.1] and the references therein). Let $\Delta\subset X$ be the ‘big diagonal’
comprising all unordered $k$-tuples of points in $\Sigma$ at least two of
which co-incide. We denote by $J_{X}$ the standard complex structure on $X$
induced by $J_{\Sigma}$. Given a continuous map $u:(S,\partial
S)\to(X,\Sym(\ul{L}))$ that is $J_{X}$-holomorphic near $\Delta$, we have a
pull-back diagram

$\textstyle{\WT{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\scriptstyle{\pi_{\WT{S}}}$$\textstyle{\Sigma^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X}$
(23)

By construction, $V$ is $S_{k}$-equivariant, and there is a unique conformal
structure $J_{\WT{S}}$ on $\WT{S}$ such that $\pi_{\WT{S}}$ is holomorphic.
Moreover, $V$ is $J_{\WT{S}}$ holomorphic if and only if $u$ is $J_{X}$
holomorphic.

Let $\pi_{1}:\Sigma^{k}\to\Sigma$ be the projection to the first factor. The
map $\pi_{1}\circ V$ is invariant under the subgroup $S_{k-1}\subset S_{k}$
which stabilises that first factor, so $\pi_{1}\circ V$ factors through a
$(k-1)!$-fold branched covering $\WT{S}\to\WH{S}$. We denote the induced map
$\WH{S}\to\Sigma$ by $v$. We also have an induced $k$-fold branched covering
$\pi_{\WH{S}}:\WH{S}\to S$, which is holomorphic with respect to the induced
complex structure $J_{\WH{S}}$ on $\WH{S}$. Note that $\partial\WH{S}$ has $k$
connected components and different connected components are mapped to
different connected components of $\ul{L}$ under $v$.

On the other hand, given a $k$-fold branched covering $\pi_{\WH{S}}$ and a
continuous map $v$ as in (22) such that different connected components of
$\partial\WH{S}$ are mapped to different connected components of $\ul{L}$, we
define a map as in (21) by $u(z)=v(\pi_{\WH{S}}^{-1}(z))$. The map $v$ is
$J_{\WH{S}}$ holomorphic if and only if $u$ is $J_{X}$ holomorphic.

######  Remark 4.7.

Note that if $z$ is a branch point of $\pi_{\WH{S}}$, then $u(z)\in\Delta$. In
general, $u(z)\in\Delta$ does not guarantee that $z$ is a branch point of
$\pi_{\WH{S}}$.

######  Remark 4.8.

Fix two collections of pairwise-disjoint circles $\ul{L}$ and $\ul{K}$ (there
may however be intersections between circles from $\ul{L}$ and ones from
$\ul{K}$). A continuous map

$\displaystyle
u:(\R\times[0,1],\R\times\\{0\\},\R\times\\{1\\})\to(X,\Sym(\ul{L}),\Sym(\ul{K}))$

that is $J_{X}$-holomorphic near $\Delta$ analogously gives rise to a
tautologically corresponding pair, comprising a $k$-fold branched covering
$\pi_{\WH{S}}:\WH{S}\to\R\times[0,1]$ together with a map
$v:(\WH{S},\partial_{0}\WH{S},\partial_{1}\WH{S})\to(\Sigma,\ul{L},\ul{K})$
where $\partial_{i}\WH{S}=\pi_{\WH{S}}^{-1}(\R\times\\{i\\})$.

###  4.5 Basic disc classes

The identification of the Heegaard torus
$\Sym(\ul{L}_{a_{1},\dots,a_{k},\eps})$ with a Clifford-type torus in
Corollary 4.3 yields a helpful basis of $H_{2}(X,\Sym(\ul{L}))$.

Suppose that $\Sigma=\PP^{1}$ and the $B_{i}$ are discs for $i=1,\dots,k=s-1$.
Suppose also that $a_{i}\in B_{i}^{\circ}$ for $i=1,\dots,k+1$. Then
$H_{2}(X,\Sym(\ul{L}))$ is freely generated by $k+1$ primitive classes
$\\{[u_{i}]\\}_{i=1}^{k+1}$ such that $[u_{i}]\cdot D_{j}=\delta_{ij}$.
Moreover, each of these primitive classes has Maslov index $\mu(u_{i})=2$.

######  Proof 4.9.

First we consider the special case that
$\ul{L}=\ul{L}_{a_{1},\dots,a_{k},\eps}$ for small $\eps$. Since
$\Sym(\ul{L})$ is smoothly isotopic to $L_{\eps}$, there is an isomorphism of
relative homology groups

$H_{2}(X,\Sym(\ul{L}))\cong H_{2}(X,L_{\eps}).$ (24)

Since $\Phi^{t}_{G}$ is $C^{1}$-small we can take the isotopy to be through
totally real tori, in which case the isomorphism (24) preserves the Maslov
class. Furthermore, the isotopy $\Phi^{t}_{G}$ is supported away from the
anticanonical divisor $\cup_{i=1}^{k+1}D_{i}$, so the isomorphism (24) does
not change the intersection number with $D_{i}$. Since $L_{\eps}$ is a
Clifford torus, it is known that $H_{2}(X,L_{\eps})$ admits a basis
$\\{[u_{i}]\\}_{i=1}^{k+1}$ such that $[u_{i}]\cdot D_{j}=\delta_{ij}$.
Moreover, it is also known that $\mu(u_{i})=2$ for all $i$. Hence, the same is
true for $\Sym(\ul{L})$.

For general $\ul{L}$ in $\PP^{1}$ such that $B_{i}$ are discs for
$i=1,\dots,k=s-1$, we can find a smooth family of $(\ul{L}_{t})_{t\in[0,1]}$
in $\PP^{1}$ such that $\ul{L}_{0}=\ul{L}$ and
$\ul{L}_{1}=\ul{L}_{a_{1},\dots,a_{k},\eps}$ for some small $\epsilon>0$.
Moreover, we can assume that $\ul{L}_{t}$ is disjoint from
$\\{a_{1},\dots,a_{k}\\}$ for all $t$. Therefore, we get a smooth family of
Lagrangian tori $\Sym(\ul{L}_{t})$ that is disjoint from $D_{i}$ for all $i$
and all $t$. The result follows.

In the course of the proof of the next lemma, we explain how to construct the
disc classes $[u_{i}]$ in Corollary 4.5 from the tautologically corresponding
pairs of maps $(v_{i},\pi_{\WH{S}_{i}})$, and use this to compute the
intersection numbers $[u_{i}]\cdot\Delta$.

######  Lemma 4.10.

Suppose $\Sigma=\PP^{1}$ and the $B_{i}$ are discs for $i=1,\dots,k=s-1$.
Suppose also that $a_{i}\in B_{i}^{\circ}$ for $i=1,\dots,k+1$ and $[u_{i}]$
is as in Corollary 4.5. We have $[u_{i}]\cdot\Delta=0$ for $1\leq i\leq k$ and
$[u_{k+1}]\cdot\Delta=2(k-1).$

######  Proof 4.11.

Let $\WH{S}^{\prime}=\sqcup_{j=1}^{k}S_{j}$ where $S_{j}=S$ is a unit disc for
each $j$. For $i=1,\dots,k$, let $v^{\prime}_{i}:\WH{S}^{\prime}\to S^{2}$ be
a map such that $v^{\prime}_{i}|_{S_{j}}$ is a constant map to a point in
$L_{j}$ if $j\neq i$ and $v^{\prime}_{i}|_{S_{i}}$ is a biholomorphism to
$B_{i}$. Let $\pi_{\WH{S}^{\prime}}:\WH{S}^{\prime}\to S$ be the trivial
covering map. Since $[v_{i}^{\prime}]\cdot a_{j}=\delta_{ij}$ for
$j=1,\dots,k+1$ (see Remark 4.3), the map $u^{\prime}_{i}:S\to X$ obtained
from the tautological correspondence from
$(v^{\prime}_{i},\pi_{\WH{S}^{\prime}})$ satisfies $[u^{\prime}_{i}]\cdot
D_{j}=\delta_{ij}$ for $j=1,\dots,k+1$. Therefore, $u_{i}^{\prime}$ represents
the class $[u_{i}]$. Since $\pi_{\WH{S}^{\prime}}$ is a trivial covering and
$\\{v_{i}^{\prime}|_{S_{j}}\\}_{j=1}^{k}$ have pairwise disjoint images, the
image of $u^{\prime}_{i}$ is disjoint from $\Delta$ so we have
$[u^{\prime}_{i}]\cdot\Delta=0$ for $i=1,\dots,k$.

On the other hand, if $\WH{S}^{\prime\prime}=\PP^{1}$, the Riemann-Hurwitz
formula shows that a simple $k$-fold branched covering
$\pi_{\WH{S}^{\prime\prime}}:\WH{S}^{\prime\prime}\to\PP^{1}$ has $2(k-1)$
branch points. Let $v^{\prime\prime}:\WH{S}^{\prime\prime}\to\PP^{1}$ be a
biholomorphism, and $u^{\prime\prime}:\PP^{1}\to X$ be the $J_{X}$-holomorphic
map tautologically corresponding to
$(v^{\prime\prime},\pi_{\WH{S}^{\prime\prime}})$. We know that
$u^{\prime\prime}$ represents the class $\sum_{i=1}^{k+1}[u_{i}]$, because
$[v^{\prime\prime}]\cdot a_{j}=[u^{\prime\prime}]\cdot D_{j}=1$ for every $j$.
Since $v^{\prime\prime}$ is a biholomorphism, $u^{\prime\prime}(z)\in\Delta$
if and only if $z$ is a branch point of $\pi_{\WH{S}^{\prime\prime}}$. The
assumption that $\pi_{\WH{S}^{\prime\prime}}$ is a simple branched covering
guarantees that the intersection multiplicity between $u$ and $\Delta$ at
every branch point of $\pi_{\WH{S}^{\prime\prime}}$ is $1$ (this fact can be
checked by a local calculation). Therefore, we know that
$[u^{\prime\prime}]\cdot\Delta=2(k-1)$ because $\pi_{\WH{S}^{\prime\prime}}$
is a simple branched covering with $2(k-1)$ branch points. As a result,
$[u_{k+1}]\cdot\Delta=([u^{\prime\prime}]-\sum_{i=1}^{k}[u_{i}^{\prime}])\cdot\Delta=2(k-1)$.

In the situation of Lemma 4.10, $k_{i}=1$ for $i\leq k$ and $k_{k+1}=k$, so
one can write the conclusion as saying that

$[u_{i}]\cdot\Delta=2(k_{i}-1)\quad\text{for }i=1,\ldots,s=k+1.$ (25)

We next establish the analogue of Corollary 4.5, and in particular establish
(25), for general $\Sigma$ and $\ul{L}$. Recall that $B_{1},\ldots,B_{s}$
enumerate the closures of the planar regions comprising
$\Sigma\backslash\ul{L}$. Pick a point $a_{i}\in B_{i}^{\circ}\subset B_{i}$
for each $i$. Let $D_{i}$ be the divisor of $\Sym^{k}(\Sigma)$ which is the
image of the map (cf. Remark 4.3)

$\displaystyle\Sym^{k-1}(\Sigma)$ $\displaystyle\to\Sym^{k}(\Sigma)$
$\displaystyle D$ $\displaystyle\mapsto D+a_{i}.$

Let $\Delta$ be the diagonal.

For each $i$, we can construct a continuous map $u_{i}:S\to(X,\Sym(\ul{L}))$
using a pair of maps $v_{i}$ and $\pi_{\WH{S}_{i}}$ as in the proof of Lemma
4.10. More precisely, let

$\WH{S}_{i}=B_{i}\sqcup\sqcup_{j=1}^{k-k_{i}}S_{j}\text{ where }S_{j}=S\text{
for all }j.$

Let $\pi_{\WH{S}_{i}}:\WH{S}_{i}\to S$ be a $k$-fold branched covering such
that $\pi_{\WH{S}_{i}}|_{S_{j}}$ is a biholomorphism and
$\pi_{\WH{S}_{i}}|_{B_{i}}$ is a $k_{i}$-fold simple branched covering to $S$.
Let $v_{i}:\WH{S}_{i}\to\Sigma$ be such that $v_{i}|_{B_{i}}$ is the identity
map to $B_{i}$ and the $v_{i}|_{S_{j}}$ are constant maps to the various
connected components of $\ul{L}$ that are not boundaries of $B_{i}$. We define
$u_{i}:=v_{i}\circ\pi_{\WH{S}_{i}}^{-1}$. It is clear that $[u_{i}]\cdot
D_{j}=\delta_{ij}$.

######  Lemma 4.12.

The image of $\pi_{2}(X,\Sym(\ul{L}))\to H_{2}(X,\Sym(\ul{L}))$ is freely
generated by $\\{[u_{i}]\\}_{i=1}^{s}$. The image of $\pi_{2}(X)\to
H_{2}(X,\Sym(\ul{L}))$ is freely generated by $\sum_{i=1}^{s}[u_{i}]$.

######  Proof 4.13.

By considering intersection numbers with the $D_{i}$, it is clear that
$\\{[u_{i}]\\}_{i=1}^{s}$ is a linearly independent set of primitive elements
in $H_{2}(X,\Sym(\ul{L}))$. We have the following commutative diagram with
exact rows.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{2}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{2}(X,\Sym(\ul{L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(\Sym(\ul{L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{2}(\Sym(\ul{L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{2}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{2}(X,\Sym(\ul{L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(\Sym(\ul{L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(X)}$

Let $I:=\operatorname{im}(\pi_{2}(X)\to\pi_{2}(X,\Sym(\ul{L})))$ and
$K:=\ker(\pi_{1}(\Sym(\ul{L}))\to\pi_{1}(X))$ so we have a short exact
sequence

$\displaystyle 0\to I\to\pi_{2}(X,\Sym(\ul{L}))\to K\to 0.$

The image of $\pi_{2}(X)\to H_{2}(X)$ is isomorphic to $\mathbb{Z}$ (see [4,
Theorem 9.2]). Therefore, the rank of the image of $I\to
H_{2}(X,\Sym(\ul{L}))$ is at most $1$.

On the other hand, the map
$\pi_{1}(\Sym(\ul{L}))\to\pi_{1}(X)=H_{1}(X)=H_{1}(\Sigma)$ (see e.g. [49,
Lemma 2.6]) can be identified with the following map (induced by the inclusion
$\ul{L}\to\Sigma$)

$\displaystyle H_{1}(\ul{L})\to H_{1}(\Sigma).$

The latter one sits inside the relative long exact sequence for the pairs
$(\Sigma,\ul{L})$

$\displaystyle
H_{2}(\Sigma)\xrightarrow{f_{1}}H_{2}(\Sigma,\ul{L})\xrightarrow{f_{2}}H_{1}(\ul{L})\xrightarrow{f_{3}}H_{1}(\Sigma)$

where $H_{2}(\Sigma)=\mathbb{Z}$, $H_{2}(\Sigma,\ul{L})=\mathbb{Z}^{s}$ and
$f_{1}$ is injective. Therefore, we have
$K=\ker(f_{3})=\operatorname{im}(f_{2})=\operatorname{coker}(f_{1})=\mathbb{Z}^{s-1}$.

Since $K$ is free, we have $\pi_{2}(X,\Sym(\ul{L}))\simeq I\oplus K$.
Therefore, image of $\pi_{2}(X,\Sym(\ul{L}))\to H_{2}(X,\Sym(\ul{L}))$ is
isomorphic to the image of a linear map $\mathbb{Z}^{s}\to
H_{2}(X,\Sym(\ul{L}))$. It implies that the rank of the image of
$\pi_{2}(X,\Sym(\ul{L}))\to H_{2}(X,\Sym(\ul{L}))$ is at most $s$ and if the
rank is $s$, then the map is injective and hence the image has no torsion.
Since $\\{[u_{i}]\\}_{i=1}^{s}$ is a linearly independent set of primitive
elements, we conclude that it freely generates the image of
$\pi_{2}(X,\Sym(\ul{L}))\to H_{2}(X,\Sym(\ul{L}))$. Moreover, we know that the
image of $I\to H_{2}(X,\Sym(\ul{L}))$ is isomorphic to $\mathbb{Z}$. To
conclude the proof, it suffices to find a continuous map $u:\PP^{1}\to X$
representing the class $\sum_{i=1}^{s}[u_{i}]$.

We can construct $u$ using tautological correspondence. Let $\WH{S}=\Sigma$
and $v:\WH{S}\to\Sigma$ be the identity map. Let
$\pi_{\WH{S}}:\WH{S}\to\PP^{1}$ be a topological $k$-fold simple branched
covering. The map $u=v\circ\pi_{\WH{S}}^{-1}$ satisfies $[u]\cdot D_{i}=1$ for
all $i=1,\dots,s$, so we have $[u]=\sum_{i=1}^{s}[u_{i}]$.

######  Remark 4.14.

$\pi_{2}(\Sym^{k}(\Sigma))$ may have rank $>1$ (see [6, Theorem 5.4]). The
hypothesis on the link $\ul{L}$ (that the $B_{j}$ are planar) implies that the
number of components $k\geq g+1$. If we restrict to links with $k\geq 2g-1$
components, then $\Sym^{k}(\Sigma)$ is a projective bundle over
$\mathrm{Jac}(\Sigma)$, and $\pi_{2}(\Sym^{k}(\Sigma))=\Z$ (see [1, Ch VII,
Proprosition 2.1]); this gives a simpler proof of Lemma 4.12 for such cases.

In [52, Section 7], Perutz explains how, given an open neighbourhood
$V\supset\Delta$ of the diagonal, one can modify $\omega_{X}$ inside $V$, and
in particular away from $\Sym(\ul{L})$ if $V$ is sufficiently small, to get a
smooth Kähler form $\omega_{V}$ such that

$[\omega_{V}]=(1/k!)\,\pi_{*}[\omega_{M}]=:[\omega_{X}].$ (26)

The space of Kähler forms one obtains in this way (as $V$ varies) is
connected. We will refer to such forms as being of ‘Perutz-type’.

######  Definition 4.15 (Topological energy).

Let $\omega_{V}$ be a Perutz-type Kähler form smoothing the current
$\omega_{X}$. Then we set

$\omega_{X}(u):=\omega_{V}(u)$ (27)

for any $u\in H_{2}(X,\Sym(\ul{L}))$ in the span of the
$\\{[u_{i}]\\}_{i=1}^{s}$.

The following definition is a variant of that from [49].

######  Definition 4.16.

Let $V$ be an open neighborhood of $\Delta\cup\cup_{i=1}^{s}D_{i}$. The space
$\mathcal{J}(V)$ of nearly symmetric almost complex structures on $X$ consists
of those $J$ such that

  * •

$J=J_{X}$ in $V$

  * •

$J$ tames $\omega_{X}$ outside $V$.

If $V$ is only an open neighborhood of $\Delta$, then we use
$\mathcal{J}_{\Delta}(V)$ to denote the space satisfying the two conditions
above.

######  Remark 4.17.

Note that $\omega_{V}$ tames $J$ for any $J\in\mathcal{J}_{\Delta}(V)$ and any
choice of Perutz-type Kähler form $\omega_{V}$ as above.

When we consider $J$-holomorphic maps with boundary on $\Sym(\ul{L})$ for some
$J\in\mathcal{J}(V)$ or $J\in\mathcal{J}_{\Delta}(V)$, we always assume that
the open neighborhood $V$ is disjoint from $\Sym(\ul{L})$. When the particular
choice of $V$ is not important, we will write $\cJ$ and $\cJ_{\Delta}$ for
$\cJ(V)$ and $\cJ_{\Delta}(V)$, respectively. Since $\Sym(\ul{L})$ is totally
real with respect to any $J\in\cJ_{\Delta}$, a smooth disc $(S,\partial
S)\to(X,\Sym(\ul{L}))$ has a well-defined Maslov index with respect to any
such $J$.

######  Lemma 4.18.

If $u:(S,\partial S)\to(X,\Sym(\ul{L}))$ has class
$[u]=\sum_{i}c_{i}[u_{i}]\in H_{2}(X,\Sym(\ul{L}))$, its Maslov index is
$2\sum_{i}c_{i}=2\sum_{i}[u]\cdot D_{i}$ with respect to $J\in\cJ_{\Delta}$.

######  Proof 4.19.

It suffices to prove that $\mu(u_{i})=2$ for all $i$. Since $\cJ_{\Delta}$ is
connected, it suffices to consider $J_{X}$.

Let $(v_{i},\pi_{\WH{S}_{i}})$ tautologically correspond to $u_{i}$. Recall
that $\WH{S}_{i}=B_{i}\sqcup_{j=1}^{k-k_{i}}S_{j}$, $v_{i}|_{B_{i}}$ is the
identity map and the $v_{i}|_{S_{j}}$ are constant maps. It follows that
$u_{i}$ factors through the following holomorphic embedding (i.e. $\im(u_{i})$
lies inside the image of the following map)

$\displaystyle\Sym^{k_{i}}(B_{i})\times\prod_{L_{j}\nsubseteq\partial
B_{i}}D^{*}L_{j}$ $\displaystyle\to\Sym^{k}(\Sigma)=X$ (28)
$\displaystyle([x_{1},\dots,x_{k_{i}}],p_{1},\dots,p_{k-k_{i}})$
$\displaystyle\mapsto[x_{1},\dots,x_{k_{i}},p_{1},\dots,p_{k-k_{i}}]$ (29)

where $D^{*}L_{j}$ is a neighborhood of $L_{j}\subset\Sigma$ such that
$\\{B_{i}\\}\cup\\{D^{*}L_{j}\\}_{L_{j}\nsubseteq\partial B_{i}}$ are pairwise
disjoint. With respect to the product decomposition of the LHS of (28), we can
write $u_{i}=(\bar{u}_{i},c_{1},\dots,c_{k-k_{i}})$ where
$\bar{u}_{i}:(S,\partial S)\to(\Sym^{k_{i}}(\Sigma),\Sym(\partial B_{i}))$ and
$c_{i}$ are constant maps. It follows that $u_{i}^{*}(TX,T\Sym(\ul{L}))$ has
$k-k_{i}$ trivial factors, which contribute $0$ to the Maslov index.
Therefore, it suffices to prove that $\mu(\bar{u}_{i})=2$. Notice that
$\bar{u}_{i}$ tautologically corresponds to the pair
$(v_{i}|_{B_{i}},\pi_{\WH{S}_{i}}|_{B_{i}})$. Since $B_{i}$ is a planar
domain, we may choose an embedding $B_{i}\hookrightarrow\PP^{1}$ to obtain a
map (of the same Maslov index)
$\bar{u}_{i}:S\to(\Sym^{k_{i}}(B_{i}),\Sym(\partial
B_{i}))\subset(\Sym^{k_{i}}(\PP^{1}),\Sym(\partial B_{i}))$. By Corollary 4.5,
the Maslov index of $\bar{u}_{i}$ is $2$.

######  Lemma 4.20.

For $u_{i}$ as in Lemma 4.12, we have $[u_{i}]\cdot\Delta=2(k_{i}-1)$ for
$i=1,\dots,s$.

######  Proof 4.21.

We use the notation of the proof of Lemma 4.18. Since $v_{i}|_{S_{j}}$ are
constant maps, we have $[u_{i}]\cdot\Delta=[\bar{u}_{i}]\cdot\bar{\Delta}$
where $\bar{\Delta}$ is the diagonal in $\Sym^{k_{i}}(B_{i})$. By regarding
$\bar{u}_{i}$ as a map from $S$ to
$\Sym^{k_{i}}(B_{i})\subset\Sym^{k_{i}}(\PP^{1})$, we can apply Lemma 4.10 and
(25) to conclude the result.

If $u:\PP^{1}\to X$ is a non-constant $J$-holomorphic map for some
$J\in\mathcal{J}_{\Delta}$, then $\mu(u)\geq 4$ and
$[u]\cdot\Delta\geq\sum_{i=1}^{s}2(k_{i}-1)$.

######  Proof 4.22.

Suppose $J\in\mathcal{J}$. By Lemma 4.12, $[u]$ is a multiple of
$\sum_{i=1}^{s}[u_{i}]$. By positivity of intersection with $D_{i}$, $u$ is a
positive multiple of $\sum_{i=1}^{s}[u_{i}]$. Hence the result follows from
Lemma 4.18 because $B_{i}$ being all planar implies that $s\geq 2$.

Now suppose $J\in\mathcal{J}_{\Delta}$. If the image of $u$ is contained in
$\Delta$, then $u$ is actually $J_{X}$-holomorphic and we reduce to the
previous case. If the image of $u$ is not contained in $\Delta$, we have
positivity of intersection between $u$ and $\Delta$, so $[u]$ is still a
positive multiple of $\sum_{i=1}^{s}[u_{i}]$. The result follows from Lemma
4.20.

######  Remark 4.23.

Let $x$ be Poincaré dual to the divisor
$D_{p}=\\{p\\}\times\Sym^{k-1}(\Sigma)$ and $\theta$ be the pullback of the
theta-divisor from the Jacobian under the Abel-Jacobi map. The first Chern
class of $X$ is $-\theta-(g-k-1)x$ (see [1, Ch VII, Section 5]). When $s\geq
2$ and hence $k+1-g\geq 2$, we have $\langle
c_{1}(X),[u]\rangle=[u]\cdot(-\theta-(g-k-1)x)=-(g-k-1)[u]\cdot x\geq 2$. This
gives a more direct proof that $\mu(u)\geq 4$ for sphere components $u$.

######  Lemma 4.24 (Monotonicity).

Suppose that there is an $\eta\geq 0$ such that $A_{j}+2(k_{j}-1)\eta$ is
independent of $j$ and denote this common value by $\lambda$. Then for all
$u\in\pi_{2}(X,\Sym(\ul{L}))$, we have

$\displaystyle\omega_{X}(u)+\eta[u]\cdot\Delta=\frac{\lambda}{2}\mu(u).$ (30)

As a result, $\Sym(\ul{L})$ does not bound any non-constant $J$-holomorphic
disc of non-positive Maslov index for any $J\in\mathcal{J}_{\Delta}$.

######  Proof 4.25.

It is easy to check that $\omega_{X}(u_{i})=A_{i}$ (see Definition 4.15 and
(26)). Therefore, (30) is a direct consequence of applying Lemmas 4.12, 4.18
and 4.20 to all the $[u_{i}]$. The last sentence follows from the positivity
of $\omega_{X}(u)$ and non-negativity of $\eta[u]\cdot\Delta$ for a non-
constant $J$-holomorphic disc $u$ such that $J\in\mathcal{J}_{\Delta}$.

If $\ul{L}$ is not $\eta$-monotone, we still have the following.

######  Lemma 4.26.

The Lagrangian $\Sym(\ul{L})$ does not bound any non-constant $J$-holomorphic
disc of non-positive Maslov index for any $J\in\mathcal{J}$.

######  Proof 4.27.

For $J\in\mathcal{J}$, we have positivity of intersection between $D_{i}$ and
a $J$-holomorphic disc $u$ with boundary on $\Sym(\ul{L})$. Therefore, Lemma
4.18 guarantees that $u$ has non-positive Maslov index if and only if $[u]=0$.
In this case, $u$ is a constant map.

######  Remark 4.28.

Suppose $\Sigma=\PP^{1}$ and $\ul{L}\subset\PP^{1}$ is an $\eta$-monotone
link. Suppose moreover that the total $\omega$-area of $\Sigma$ is $1$. If the
link has a unique component, necessarily it is an equator, which is
$0$-monotone for $\eta=0$. If $k>1$, there is at least one planar domain
$B_{j}$ with $k_{j}\geq 2$, from which one sees that the monotonicity constant
$\lambda:=A_{j}+2(k_{j}-1)\eta>2\eta$. On the other hand, considering
$\sum_{j=1}^{s}A_{j}+2(k_{j}-1)\eta$ shows that $s(2\eta-\lambda)=4\eta-1$, so
$\eta$-monotone links can only exist for $\eta\in[0,\frac{1}{4})$. Moreover,
links consisting of $k\geq 2$ parallel circles on the sphere can take any
value of $\eta\in[0,\frac{1}{4})$. Hence, we see that the set of all values of
$(k,\eta)$ for which there exists a $k$-component $\eta$-monotone link
$\ul{L}$ with $k\geq 2$ is exactly $\left\\{(k,\eta):k\in\N_{\geq
2},\eta\in[0,\frac{1}{4})\right\\}.$

######  Remark 4.29.

If $\ul{L}$ is a $0$-monotone link (i.e. the areas of the $B_{i}$ are the same
for all $i$), then $\Sym(\ul{L})$ is a monotone Lagrangian submanifold with
respect to a Perutz-type Kähler form $\omega_{V}$ as in Definition 4.15 for
any $V\supset\Delta$ disjoint from $\Sym(\ul{L})$.

When $g=0$ and $\ul{L}$ is an $\eta$-monotone link for $\eta>0$, we can
‘inflate’ the symplectic form on $X$ near the diagonal to make $\Sym(\ul{L})$
a monotone Lagrangian submanifold, as follows.

Let $V\supset\Delta$ be disjoint from $\Sym(\ul{L})$ and $\omega_{V}$ be a
Perutz-type Kähler form. Since $\Sym^{k}(\PP^{1})=\PP^{k}$, we know that
$\Delta$ is a very ample divisor and its complement is affine. In particular,
we can find a neighborhood $V^{\prime}$ of $\Delta$ such that its closure
$\bar{V}^{\prime}\subset V$ and $\bar{V}^{\prime}$ admits a concave contact
boundary (see e.g. [57, Section 4b] for the existence of $V^{\prime}$).

Let $X_{-}:=X\setminus V^{\prime}$ be equipped with the restricted symplectic
form $\omega_{V}|_{X_{-}}$. For $R>1$, let
$X_{0,R}:=([1,R]\times\partial\bar{V}^{\prime},d(r\theta))$ where $r$ is the
coordinate on $[1,R]$ and $\theta$ is the contact form on
$\partial\bar{V}^{\prime}$ induced by $\omega_{V}$. Let
$X_{+,R}:=(\bar{V}^{\prime},R\omega_{V}|_{\bar{V}^{\prime}})$. We can form a
symplectic manifold by gluing their boundaries

$\displaystyle(X(R),\omega_{X(R)}):=X_{-}\cup_{\\{1\\}\times\partial\bar{V}^{\prime}}X_{0,R}\cup_{\\{R\\}\times\partial\bar{V}^{\prime}}X_{+,R}.$

We can also find a diffeomorphism $F:X\to X(R)$ such that $F$ is the identity
map over $X_{-}$ and near $\Delta$. The symplectic form $F^{*}\omega_{X(R)}$
lies in the cohomology class $[\omega_{V}]+f(R)\mathrm{PD}[\Delta]$ for a
strictly increasing function $f$ such that $f(1)=0$ and
$\lim_{R\to\infty}f(R)=\infty$. Therefore, it is clear that $\Sym(\ul{L})$ is
a monotone Lagrangian in $(X,F^{*}\omega_{X(R_{\eta})})$ for the $R_{\eta}$
such that $f(R_{\eta})=\eta$.

We denote $F^{*}\omega_{X(R_{\eta})}$ by $\omega_{V,\eta}$. The dependence on
the choices made in the construction will not be important in the paper.

######  Remark 4.30.

When $\Sigma=\PP^{1}$, the symplectic forms $\omega_{V,\eta}$ have cohomology
class varying with $\eta$, cf. Remark 7.5, but they can be rescaled to be
cohomologous and hence isotopic, even as one varies $\eta$. They are therefore
related by a global smooth isotopy, by Moser’s theorem, so if
$\omega(\Sigma)=1$ then $(X,\omega_{V,\eta})$ is symplectomorphic to the
Fubini-Study form normalized so that the symplectic area of $[\PP^{1}]$ is
$(k+1)\lambda$, where $\lambda=A_{j}+2(k_{j}-1)\eta$.

However, this isotopy will not respect the diagonal, and the resulting isotopy
of $\Sym(\ul{L})\subset\PP^{k}$ is not through Lagrangian submanifolds
associated to links. For the purposes of studying links and the geometry of
$\Sigma$, it therefore makes sense to keep track of $\eta$ even in this case.

######  Remark 4.31.

Suppose $k\geq 2g-1$, so $\Sym^{k}(\Sigma_{g})$ is a projective bundle
$\PP(V)$ over the Jacobian. We follow the notation of Remark 4.23. If $\omega$
is an integral Kähler form on $\Sigma$ of area 1, the current $\omega_{X}$
defines the cohomology class $x$. (This is ample, and indeed the tautological
class $\mathcal{O}_{\PP(V)}(1)$.) The diagonal divisor $\Delta$ has class
$2[(k+g-1)x-\theta]$. Cones of divisors of $\Sym^{k}(\Sigma)$ were studied in
[31, 51]; the diagonal is on the boundary of the pseudo-effective cone. It
follows that $[\omega_{X}]+\eta\cdot\mathrm{PD}[\Delta]$ will not lie in the
ample cone for sufficiently large $\eta\gg 0$.

##  5 Curvature and unobstructedness

Our next goal is to define a version of Floer cohomology for the torus
$\Sym(\ul{L})$, and to determine when it is non-zero. As in many examples of
this nature, the non-triviality of the Floer cohomology will be determined by
the disc potential function associated to $\Sym(\ul{L})$ (see Definition 5.3
and Lemma 6.12). We are going to compute the disc potential function in this
section.

###  5.1 The disc potential

We recall the spaces of almost complex structures $\cJ(V)$ and
$\cJ_{\Delta}(V)$ from Definition 4.16. For a fixed $\ul{L}\subset\Sigma$ and
hence $\Sym(\ul{L})\subset X$, we continue to use $\cJ$ (resp. $\cJ_{\Delta}$)
to denote $\cJ(V)$ (resp. $\cJ_{\Delta}(V)$) for an open neighborhood $V$ for
$\Delta\cup\cup_{i=1}^{s}D_{i}$ (resp. $\Delta$) that is disjoint from
$\Sym(\ul{L})$. For $J\in\cJ$, consider the moduli space
$\cM_{A}(\Sym(\ul{L});J)$ of Maslov index $2$ $J$-holomorphic discs
$u:(S,\partial S)\to(X,\Sym(\ul{L}))$ with $1$ boundary marked point and in
the relative homology class $A\in H_{2}(X,\Sym(\ul{L}))$. The evaluation map
at the boundary marked point defines a map
$ev:\cM_{A}(\Sym(\ul{L});J)\to\Sym(\ul{L})$.

######  Lemma 5.1.

If $J\in\cJ$ is generic, $\cM_{A}(\Sym(\ul{L});J)$ is a compact manifold of
dimension $k$. The same is true for generic $J\in\cJ_{\Delta}$ if $\ul{L}$ is
$\eta$-monotone.

######  Proof 5.2.

By Lemma 4.24 and 4.26, $\Sym(\ul{L})$ does not bound non-constant
$J$-holomorphic discs with non-positive Maslov index, so the Gromov
compactification of $\cM_{A}(\Sym(\ul{L});J)$ is the space itself. The
condition $\mu(A)=2$ implies that $A$ is primitive because $\Sym(\ul{L})$
cannot bound discs of Maslov index $1$. Therefore discs in class $A$ are
necessarily somewhere injective, by [33]. The existence of a somewhere
injective point implies that elements in $\cM_{A}(\Sym(\ul{L});J)$ are regular
for a generic $J\in\cJ$ and a generic $J\in\cJ_{\Delta}$ (see [46, Theorem
10.4.1, Corollary 10.4.8]). In this case, $\cM_{A}(\Sym(\ul{L});J)$ is a
manifold of dimension the same as the virtual dimension which equals to
$(k-3)+\mu(A)+1=k$.

A choice of orientation and spin structure on $\Sym(\ul{L})$ defines an
orientation of $\cM_{A}(\Sym(\ul{L});J)$, with respect to which the evaluation
map has a well-defined degree. (Equivalently, the fiber product between
$\cM_{A}(\Sym(\ul{L});J)$ and a generic point in $\Sym(\ul{L})$ under the
evaluation map therefore defines a compact oriented zero dimensional
manifold.) In a minor abuse of notation, we denote the algebraic count of
points of this $0$-manifold by $\\#\cM_{A}(\Sym(\ul{L});J)$.

######  Definition 5.3.

For $\eta$-monotone $\ul{L}$ and generic $J\in\mathcal{J}_{\Delta}$, the disc
potential function

$W:=W_{\Sym(\ul{L})}(x,J):H^{1}(\Sym(\ul{L});\C^{*})\longrightarrow\C$

is defined by

$\displaystyle W_{\Sym(\ul{L})}(x,J)=\sum_{A\in
H_{2}(X,\Sym(\ul{L}))}(\\#\cM_{A}(\Sym(\ul{L});J))x^{\partial A}.$ (31)

The notation $x^{\partial A}$ can be written in a more explicit way as
follows. Let $\\{q_{1},\dots,q_{k}\\}$ be a basis of $H_{1}(\Sym(\ul{L}),\Z)$.
We have $\partial A=\sum_{i=1}^{k}c_{i}q_{i}$ for some $c_{i}\in\Z$. In this
case, we have $x^{\partial A}=\prod_{i=1}^{k}x_{i}^{c_{i}}$.

######  Remark 5.4.

When elements in $\cM_{A}(\Sym(\ul{L});J)$ are regular off a set of real
codimension $2$, the degree of the evaluation map is still well-defined (see
[41, Chapter 6.5 and 6.6]). In this case, $\\#\cM_{A}(\Sym(\ul{L});J)$ is
well-defined and the potential function is defined in the same way (31).

The potential function depends on the choice of orientation and spin structure
on $\Sym(\ul{L})$, but these choices will not play a significant role in the
sequel (we will be interested in the existence of critical points of the disc
potential; a different choice of orientation or spin structure will change the
value of the critical point, not the existence of critical points).
Concretely, we will fix an orientation by orienting and ordering the
constituent circles $L_{i}\subset\ul{L}$, and will take the unique
translation-invariant spin structure (this follows the usual convention for
Lagrangian toric fibres from [12, 13]).

We compute $W_{\Sym(\ul{L})}(x,J)$ in the subsequent sections.

######  Definition 5.5.

Let $\Lambda$ be the Novikov field with real exponent. That is

$\displaystyle\Lambda:=\left\\{\sum_{i=0}^{\infty}c_{i}T^{b_{i}}|c_{i}\in\C,b_{i}\in\R,b_{0}<b_{1}<\dots,\lim_{i\to\infty}b_{i}=\infty\right\\}.$

The non-Archimedean valuation $val:\Lambda^{*}:=\Lambda\setminus\\{0\\}\to\R$
is defined to be
$val(\sum_{i=0}^{\infty}c_{i}T^{b_{i}})=\inf\\{b_{i}|c_{i}\neq 0\\}$. For not
necessarily $\eta$-monotone $\ul{L}$ and generic $J\in\mathcal{J}$, we define
the $\eta$-disc potential function as a function
$H^{1}(\Sym(\ul{L});U_{\Lambda})\to\Lambda^{*}$, where
$U_{\Lambda}=val^{-1}(0)$ is the unitary subgroup of $\Lambda$.

In that case, the $\eta$-disc potential is given by

$\displaystyle W_{\Sym(\ul{L})}^{\eta}(x,J)=\sum_{A\in
H_{2}(X,\Sym(\ul{L}))}(\\#\cM_{A}(\Sym(\ul{L});J))\,T^{\omega_{X}(A)+\eta
A\cdot\Delta}\;x^{\partial A}.$

When $\ul{L}$ is $\eta$-monotone, then
$W_{\Sym(\ul{L})}=W_{\Sym(\ul{L})}^{\eta}|_{T=1}$.

###  5.2 Potential in the Clifford-type case

We return to the running example in which
$\ul{L}=\ul{L}_{a_{1},\dots,a_{k},\eps}=\cup_{i=1}^{k}\\{|x-a_{i}||\prod_{j\neq
i}(a_{j}-a_{i})|=\eps\\}$ from above. The general case will be explained in
the next section.

We orient the circles as boundaries of complex discs in $\C$, and take the
product orientation on $\Sym(\ul{L})$. The fundamental classes of the circles
$L_{i}\subset\ul{L}$ also give us preferred basis co-ordinates $x_{i}$ on
$H^{1}(\Sym(\ul{L});\C^{*})$.

######  Proposition 4.

Let $\ul{L}=\ul{L}_{a_{1},\dots,a_{k},\eps}$. For sufficiently small $\eps>0$,
sufficiently small open sets $V\supset\Delta\cup\cup_{i=1}^{s}D_{i}$ that is
disjoint from $\Sym(\ul{L})$ and generic $J\in\mathcal{J}(V)$, we have
$W_{\Sym(\ul{L})}^{\eta}(x,J)=\sum_{i=1}^{k}T^{A_{i}}x_{i}+\frac{T^{A_{k+1}+2(k-1)\eta}}{x_{1}\dots
x_{k}}$.

######  Proof 5.6.

We return to the setting and notation of Lemma 4.4, Corollary 4.3 and their
proofs. Recall that this introduced a small $\eps>0$ and an open neighbourhood
$G$ of $L_{\eps}$. Moreover, since $(\Phi^{t}_{G})^{*}$ is $C^{1}$-small when
$\eps$ is small, we can assume that $(\Phi^{t}_{G})^{*}\omega_{X}$ tames the
standard complex structure $J_{X}$ for all $t$. We fix once and for all such
an $\eps$, and recall that $\Phi^{t}_{G}$ is supported away from $\Delta$. The
idea of the proof is to show that under the identification
$H_{2}(X,\Sym(\ul{L}))=H_{2}(X,L_{\eps})$ induced by $\Phi^{t}_{G}$, we have
$\\#\cM_{A}(\Sym(\ul{L});J)=\\#\cM_{A}(L_{\eps};J^{\prime})$ for appropriate
almost complex structures $J$ and $J^{\prime}$.

The $4$-tuple $(X,J_{X},(\Phi_{G}^{t})^{*}\omega_{X},\Phi_{G}^{-t}(L_{\eps}))$
is isomorphic to $(X,(\Phi_{G}^{t})_{*}J_{X},\omega_{X},L_{\eps})$, so we can
take the perspective that $\Phi_{G}^{t}$ induces a one-parameter family of
$\omega_{X}$-tamed almost complex structures $(\Phi_{G}^{t})_{*}J_{X}$, and
that we work with a fixed Lagrangian and a fixed symplectic form (more
properly, a fixed symplectic current which is singular along $\Delta$). Note
that $(\Phi_{G}^{t})_{*}J_{X}=J_{X}$ near $\Delta$ and $\cup_{i=1}^{k+1}D_{i}$
is preserved under $\Phi^{t}_{G}$. Therefore, we may fix an open neighborhood
$V$ of $\Delta\cup\cup_{i=1}^{k+1}D_{i}$ such that
$(\Phi_{G}^{t})_{*}J_{X}=J_{X}$ in $V$. It follows that
$(\Phi_{G}^{t})_{*}J_{X}\in\mathcal{J}(V)$ for all $t$.

For any $J\in\cJ(V)$ a $J$-holomorphic disc $u$ with boundary on $L_{\eps}$
has Maslov index

$\displaystyle\mu(u)=2\sum_{j=0}^{k}[u]\cdot[D_{j}].$ (32)

By positivity of intersections, $L_{\eps}$ cannot bound non-constant
$J$-holomorphic discs with non-positive Maslov index for any $J\in\cJ(V)$.

It remains to relate the $(\Phi_{G}^{t})_{*}J_{X}$-holomorphic discs with
Maslov index $2$ for $t=0,1$. When $t=0$, we have
$(\Phi_{G}^{t})_{*}J_{X}=J_{X}$ and the Maslov $2$ discs with boundary on
$L_{\eps}$ are well-known to be regular [12, 13].

At this point, we do not know that the Maslov two discs for
$(\Phi^{1}_{G})_{*}J_{X}$ are regular. We instead choose a generic
$C^{2}$-small perturbation $J^{\prime}_{t}$ of the path
$((\Phi_{G}^{t})_{*}J_{X})_{t\in[0,1]}$ relative to the end-point $t=0$ (but
not necessarily fixing the end-point at $t=1$). In particular,
$J^{\prime}_{1}$ is a generic perturbation of $(\Phi_{G}^{1})_{*}J_{X}$.

The parametrized moduli space of Maslov $2$ $J^{\prime}_{t}$-holomorphic discs
$u$ with boundary on $L_{\eps}$ for some $t$ could in general fail to be
regular: there can be finitely many interior times $t$ where bifurcation
occurs. A necessary condition for bifurcation to occur at time $t_{0}$ is that
there are at least two non-constant $J^{\prime}_{t_{0}}$-holomorphic discs
$v,v^{\prime}$ with $\mu(v)+\mu(v^{\prime})\leq 2$. At least one of
$v,v^{\prime}$ then has Maslov index strictly less than $2$, and hence (by
orientability of $\Sym(\ul{L})$) index less than or equal to $0$, which
contradicts (32). Therefore, there is no bifurcation and the parametrized
moduli space for a generic path $J^{\prime}_{t}$ is a smooth compact cobordism
between the moduli spaces for $t=0,1$.

Since $J^{\prime}_{1}$ is a generic perturbation of $(\Phi_{G}^{1})_{*}J_{X}$,
it implies that for generic $J\in\mathcal{J}(V)$, the algebraic counts of
Maslov $2$ $J$-holomorphic discs with boundary on $\Sym(\ul{L})$ are the same
as those of the Clifford-type torus $L_{\eps}$. The result now follows from
the fact that $\\#\cM_{[u_{i}]}(L_{\eps};J_{X})=1$ for all $i=1,\dots,k+1$ and
$\\#\cM_{A}(L_{\eps};J_{X})=0$ for $A\neq[u_{i}]$ (see [12]).

Now we consider a slightly more general class of $\ul{L}$ in $\Sigma=\PP^{1}$.
We still assume that $B_{j}$ are topological discs with smooth boundary for
$j=1,\dots,k$ but we do not require that
$\ul{L}=\ul{L}_{a_{1},\dots,a_{k},\eps}$.

######  Proposition 5.

For sufficiently small open sets $V\supset\Delta\cup\cup_{i=1}^{s}D_{i}$ and
generic $J\in\mathcal{J}(V)$, we have
$W_{\Sym(\ul{L})}^{\eta}(x,J)=\sum_{i=1}^{k}T^{A_{i}}x_{i}+\frac{T^{A_{k+1}+2(k-1)\eta}}{x_{1}\dots
x_{k}}$.

Moreover, if $\ul{L}$ is $\eta$-monotone, then for generic
$J\in\mathcal{J}_{\Delta}$ we have
$W_{\Sym(\ul{L})}(x,J)=\sum_{i=1}^{k}x_{i}+\frac{1}{x_{1}\dots x_{k}}$.

######  Proof 5.7.

Similar to the proof of Corollary 4.5, we can find a smooth family of
$(\ul{L}_{t})_{t\in[0,1]}$ such that $\ul{L}_{0}=\ul{L}$ and
$\ul{L}_{1}=\ul{L}_{a_{1},\dots,a_{k},\eps}$ for some $a_{i}$ and small
$\eps$. We can assume that $\ul{L}_{t}$ is disjoint from
$\\{a_{1},\dots,a_{k}\\}$ for all $t\in[0,1]$. We can assume that $V$ is
disjoint from $\Sym(\ul{L}_{t})$ for all $t$. By Lemma 4.26,
$\Sym(\ul{L}_{t})$ does not bound non-constant $J$-holomorphic disc with non-
positive Maslov index for all $J\in\mathcal{J}(V)$. As in the proof of
Proposition 4, we can form a smooth compact cobordism between the moduli
spaces of Maslov $2$ holomorphic discs for $t=0,1$. This proves the first
statement.

For the second statement, we want to show that the potential function can be
computed for generic $J\in\mathcal{J}_{\Delta}$ that are not necessarily in
$\mathcal{J}(V)$. It follows from applying a further cobordism argument, using
Lemma 4.24 instead of 4.26, to a family of almost complex structures in
$\mathcal{J}_{\Delta}$.

###  5.3 Regularity

We can upgrade Proposition 5 to a statement for the canonical complex
structure $J_{X}$ if $J_{\Sigma}$ is chosen appropriately relative to
$\ul{L}$101010A similar claim is made in [49, Proposition 3.9]. This (as well
as its generalization to the cases $\Sigma\neq\PP^{1}$) will be explained in
Section 5.4. The key result we prove in this subsection is that elements in
$\cM_{[u_{i}]}(\Sym(\ul{L});J_{X})$ are regular off a set of codimension at
least $2$ (see Corollary 5.3) when the complex structure on $B_{i}$ is
appropriate in the same sense. We do not assume
$\ul{L}=\ul{L}_{a_{1},\dots,a_{k},\eps}$ in this section.

The tautological correspondence of Section 4.4 shows that $S_{k}$-equivariant
maps of any regularity
$V:(\WT{S},\partial\WT{S})\to(\Sigma^{k},\pi^{-1}(\Sym(\ul{L})))$ can be
identified with maps (of the same regularity) from
$v:(\WH{S},\partial\WH{S})\to(\Sigma,\ul{L})$, see [39, Section 3.1] for more
details. There is a similar dictionary for maps valued in vector fields or
endomorphisms. In particular, we have

$\displaystyle
H^{j}_{\bar{\partial}}((\WT{S},\partial\WT{S}),(V^{*}T\Sigma^{k},V|_{\partial\WT{S}}^{*}T(\pi^{-1}(\Sym(\ul{L})))))^{S_{k}}=H^{j}_{\bar{\partial}}((\WH{S},\partial\WH{S}),(v^{*}T\Sigma,v|_{\partial\WH{S}}^{*}T\ul{L}))$
(33)

for all $j$, where $H^{j}_{\bar{\partial}}$ denotes Dolbeault cohomology.

######  Proposition 6.

Let $u:(S,\partial S)\to(X,\Sym(\ul{L}))$ be a $J_{X}$-holomorphic map and
$(v,\pi_{\WH{S}}):\WH{S}\to\Sigma\times S$ the map tautologically
corresponding to $u$. Suppose that $v$ is regular and that $\pi_{\WH{S}}$ is a
simple branched covering with $[u]\cdot\Delta$ simple branch points. Then $u$
is regular.

Before the proof, we formulate a lemma comparing virtual dimensions of the
maps $u$ and $v$. Let $\WH{S}$ be a Riemann surface (so its conformal
structure is fixed). Let $\vdim(v,\WH{S})$ be the virtual dimension of the
space of maps $v:\WH{S}\to\Sigma$ with boundary on $\ul{L}$. Let $\vdim(u)$ be
the virtual dimension of the moduli space of discs $u$, where we divide out by
the action of the $3$-dimensional automorphism group $\PP SL(2,\R)$ of $S$.

######  Lemma 5.8.

Let $u$ and $(v,\pi_{\WH{S}})$ be as in Proposition 6, then

$\displaystyle\vdim(u)+3=\vdim(v,\WH{S})+2[u]\cdot\Delta.$ (34)

######  Proof 5.9.

We can write $[u]$ as a sum $\sum_{i=1}^{s}c_{i}[u_{i}]$ where $c_{i}\geq 0$
for all $i$. The LHS of (34) is

$\displaystyle\vdim(u)+3=k+\mu(u)=k+2\sum_{i=1}^{s}c_{i}.$ (35)

On the other hand, we have $[v]=\sum_{i=1}^{s}c_{i}[v_{i}]$ and

$\displaystyle\vdim(v,\WH{S})=\chi(\WH{S})+\sum_{i=1}^{s}c_{i}\mu(v_{i})=\chi(\WH{S})+2\sum_{i=1}^{s}(2-k_{i})c_{i}.$
(36)

where $\mu(v_{i})$ is the Maslov index of the class $[v_{i}]\in
H_{2}(\Sigma,\ul{L})$ and it is given by $2(2-k_{i})$ because the inclusion
$B_{i}\hookrightarrow\Sigma$ represents $[v_{i}]$ and the Maslov index of a
planar domain with $k_{i}$ boundary components is $2(2-k_{i})$. By Lemma 4.20,
we have $[u]\cdot\Delta=\sum_{i=1}^{s}2(k_{i}-1)c_{i}$. Since we assume that
$\pi_{\WH{S}}$ is a simple branched covering with $[u]\cdot\Delta$ many branch
points, the Riemann-Hurwitz formula yields

$\displaystyle\chi(\WH{S})$ $\displaystyle=k-\sum_{i=1}^{s}2(k_{i}-1)c_{i}.$
(37)

Combining (35), (36) and (37), we get (34).

######  Proof 5.10 (Proof of Proposition 6).

Recall from (23) the pull-back diagram

$\textstyle{\WT{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\scriptstyle{\pi_{\WT{S}}}$$\textstyle{\Sigma^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X.}$

We have a short exact sequence of sheaves over $\WT{S}$

$\displaystyle 0\to V^{*}T\Sigma^{k}\to\pi_{\WT{S}}^{*}(u^{*}TX)\to Z\to 0$

where under the identification $V^{*}(\pi^{*}TX)=\pi_{\WT{S}}^{*}(u^{*}TX)$,
the second arrow is induced by $\pi_{*}:T\Sigma^{k}\to\pi_{*}TX$ and $Z$ is
defined to be the cokernel. Since $\pi$ is a ramified covering, $Z$ is
supported on the critical points of $\pi_{\WT{S}}$; at each critical point,
the stalk has complex rank equal to the ramification index minus $1$, see [27,
Ch IV, Prop 2.2]. Consider the induced long exact sequence in cohomology
(where for simplicity we omit the boundary condition from the notation)

$\displaystyle 0\to H^{0}_{\bar{\partial}}(\WT{S},V^{*}T\Sigma^{k})\to
H^{0}_{\bar{\partial}}(\WT{S},\pi_{\WT{S}}^{*}(u^{*}TX))\to
H^{0}_{\bar{\partial}}(\WT{S},Z)\to
H^{1}_{\bar{\partial}}(\WT{S},V^{*}T\Sigma^{k})\to\dots$

Taking $S_{k}$-invariants is an exact functor over $\mathbb{C}$, so we have

$\displaystyle 0\to H^{0}_{\bar{\partial}}(\WT{S},V^{*}T\Sigma^{k})^{S_{k}}\to
H^{0}_{\bar{\partial}}(\WT{S},\pi_{\WT{S}}^{*}(u^{*}TX))^{S_{k}}\to
H^{0}_{\bar{\partial}}(\WT{S},Z)^{S_{k}}\to
H^{1}_{\bar{\partial}}(\WT{S},V^{*}T\Sigma^{k})^{S_{k}}$

By (33) and the assumption that $v$ is regular, this reduces to

$\displaystyle 0\to H^{0}_{\bar{\partial}}(\WH{S},v^{*}T\Sigma)\to
H^{0}_{\bar{\partial}}(\WT{S},\pi_{\WT{S}}^{*}(u^{*}TX))^{S_{k}}\to
H^{0}_{\bar{\partial}}(\WT{S},Z)^{S_{k}}\to 0$

Since $\pi_{\WT{S}}$ is a branched covering, we have

$\displaystyle
H^{0}_{\bar{\partial}}(\WT{S},\pi_{\WT{S}}^{*}(u^{*}TX))^{S_{k}}=H^{0}_{\bar{\partial}}(S,u^{*}TX).$

Since $\pi_{\WH{S}}$ is simply branched, the complex rank of
$H^{0}_{\bar{\partial}}(\WT{S},Z)^{S_{k}}$ is precisely the number of critical
points, so it has real dimension $2[u]\cdot\Delta$. Therefore, we have

$\displaystyle\dim_{\R}H^{0}_{\bar{\partial}}(S,u^{*}TX)$ $\displaystyle=$
$\displaystyle\dim_{\R}H^{0}_{\bar{\partial}}(\WH{S},v^{*}T\Sigma)+\dim_{\R}H^{0}_{\bar{\partial}}(\WT{S},Z)^{S_{k}}$
$\displaystyle=$ $\displaystyle\vdim(v,\WH{S})+2[u]\cdot\Delta$
$\displaystyle=$ $\displaystyle\vdim(u)+3,$

where the last equality comes from Lemma 5.8. Given that we have not divided
out by the automorphism group of $S$, this exactly says that $u$ is regular.

Suppose that the non-simple $k_{i}$-fold branched coverings from
$(B_{i},J_{\Sigma}|_{B_{i}})$ to $S$ form a set of real codimension two among
all $k_{i}$-fold branched coverings. Then $\cM_{[u_{i}]}(\Sym(\ul{L}),J_{X})$
is regular off a set of real codimension $2$ and
$\\#\cM_{[u_{i}]}(\Sym(\ul{L}),J_{X})=1$.

######  Proof 5.11.

If $u$ is a holomorphic map which gives rise to an element in
$\cM_{[u_{i}]}(\Sym(\ul{L}),J_{X})$ and $(v,\pi_{\hat{S}})$ is tautologically
corresponding to $u$, then $[v]=[v_{i}]$. By the open mapping theorem,
$\im(v)\cap B_{j}^{\circ}$ is either a point or the entire $B_{j}^{\circ}$ for
each $j$. Therefore, the Lagrangian boundary condition of $v$ together with
$[v]=[v_{i}]$ implies that there is a connected component $\WH{S}_{0}$ of
$\hat{S}$ such that $v|_{\WH{S}_{0}}$ is a degree $1$ map to $B_{i}$.
Moreover, the other connected components of $\WH{S}$ are biholomorphic to $S$
and $v$ restricts to a constant map on these components. Clearly, $v$ is
regular.

By Proposition 6, to show that $\cM_{[u_{i}]}(\Sym(\ul{L}),J_{X})$ is regular
off a set of real codimension $2$, it suffices to show that among all the
$k$-fold branched coverings $\WH{S}\to S$, the ones that are not simply
branched with $[u]\cdot\Delta$ many critical points form a subset of real
codimension at least $2$. The Riemann-Hurwitz formula shows that all $k$-fold
branched coverings $\WH{S}\to S$ have $[u]\cdot\Delta$ many critical points
when counted with multiplicity. Therefore, we just need to show that the locus
of non-simple branched coverings forms a subset of real codimension at least
$2$. This immediately follows from our assumption because
$\WH{S}=\WH{S}_{0}\sqcup\sqcup_{j=1}^{k-k_{i}}S=B_{i}\sqcup\sqcup_{j=1}^{k-k_{i}}S$.

Therefore, $\\#\cM_{[u_{i}]}(\Sym(\ul{L}),J_{X})$ is well-defined (see Remark
5.4). Moreover, it can be computed using the algebraic count of the
tautologically corresponding pair $(v,\pi_{\WH{S}})$, which can in turn be
computed by embedding $B_{i}$ into $\PP^{1}$ so we have
$\\#\cM_{[u_{i}]}(\Sym(\ul{L}),J_{X})=1$ by Proposition 5.

Note that there exists a complex structure on $B_{i}$ such that the hypothesis
of Corollary 5.3 is satisfied. It is because branched coverings of $S$
correspond to a choice of (branch) points in $S$ with monodromy data. In
particular, $2(k_{i}-1)$ branch points in $S$ (counted with multiplicity)
together with a monodromy representation into $S_{k_{i}}$ (the symmetric group
on $k_{i}$ elements) determines a complex structure on $B_{i}$. Non-simple
branched coverings arise when branch points co-incide, which is a codimension
two phenomenon. Therefore, by a dimension count, the hypothesis of Corollary
5.3 holds for the generic complex structure on $B_{i}$.

We are going to make use of Corollary 5.3 to calculate the potential function
in the next subsection.

###  5.4 Potential in general

We return to the case in which $\Sigma$ is a closed surface with arbitrary
genus, and is equipped with a symplectic form $\omega$. In contrast to the
previous sections, we do not fix the conformal structure on $\Sigma$ at this
point. Let $\ul{L}\subset\Sigma$ be a $k$-component $\eta$-monotone link whose
complement comprises $s$ domains with planar closures $B_{i}$. Recall
$s=k-g+1$.

Let $\Sigma$ and $\ul{L}\subset(\Sigma,\omega)$ be as above. There is a
complex structure $J_{\Sigma}$ on $\Sigma$, for which $\omega$ is a Kähler
form, and moreover, for the induced complex structure $J_{X}$ on $X$, the
Maslov $2$ $J_{X}$-holomorphic discs with boundary on $\Sym(\ul{L})$ are
regular (off a set of real codimension two) and the disc potential is given by

$\displaystyle W_{\Sym(\ul{L})}(x,J_{X})=\sum_{i=1}^{s}x^{\partial B_{i}},$
(38)

where the term $x^{\partial B_{i}}$ should be understood via the isomorphism
$H_{1}(\ul{L})\simeq H_{1}(\Sym(\ul{L}))$111111 Let $c_{i}$ be a point in
$L_{i}$ and $\gamma_{i}(t):[0,1]\to L_{i}$ be a loop representing the
fundamental class of $L_{i}$. Then the isomorphism is given by sending
$[\gamma_{i}(t)]$ to $[\Gamma_{i}(t)]$ where
$\Gamma_{i}(t)=[c_{1},\dots,c_{i-1},\gamma_{i}(t),c_{i+1},\dots,c_{s}]$ and
extending linearly. The isomorphism is independent of the choice of $c_{i}$
and $\gamma_{i}$..

Furthermore, $x=(1,\dots,1)$ is a critical point of
$W_{\Sym(\ul{L})}(x,J_{X})$.

######  Proof 5.12.

We can find a Hamiltonian diffeomorphism $\varphi$ of $(\Sigma,\omega)$
supported near the connected components of $\ul{L}$ such that
$\varphi(\ul{L})$ consists of real analytic curves. Therefore, it suffices to
consider the case that $\ul{L}$ consists of real analytic curves.

For each $i=1,\dots,s$, we pick a complex structure on $B_{i}$ such that the
set of non-simple $k_{i}$-fold branched coverings to the unit disc is of real
codimension at least two among all $k_{i}$-fold branched coverings. We can
glue the complex structures on $B_{i}$ for all $i$ because their common
boundaries are real analytic. It gives a complex structure on $\Sigma$.

Choose a Kähler form $\omega’$ on $\Sigma$, and let $g’$ be the Kähler metric
on $\Sigma$ induced by $\omega^{\prime}$. For any smooth funciton
$f:\Sigma\to\R_{>0}$, the metric $fg^{\prime}$ is also a Kähler metric on
$\Sigma$. We can pick $f$ such that the Kähler form $\omega^{\prime\prime}$
induced by $fg^{\prime}$ has area $A_{i}$ over $B_{i}$. By applying Weinstein
neighborhood theorem to $\ul{L}$ in $(\Sigma,\omega)$ and
$(\Sigma,\omega^{\prime\prime})$, we can find a diffeomorphsim
$G:\Sigma\to\Sigma$ such that $G(\ul{L})=\ul{L}$,
$G^{*}\omega^{\prime\prime}=\omega$ near $\ul{L}$ and
$G^{*}\omega^{\prime\prime}(B_{i})=A_{i}$ for all $i$. By Moser argument,
$G^{*}\omega^{\prime\prime}$ is isotopic to $\omega$ relative to $\ul{L}$.
Therefore, we can find a symplectomorphism
$F:(\Sigma,\omega)\to(\Sigma,\omega^{\prime\prime})$ such that
$F(\ul{L})=\ul{L}$. The pull-back of the complex structure on $\Sigma$ along
$F$ is the $J_{\Sigma}$ we need. It has the property that
$J_{\Sigma}|_{B_{i}}$ satisfies the hypothesis of Corollary 5.3 for all $i$.

Notice that if $A\in H_{2}(X,\Sym(\ul{L}))$, $\mu(A)=2$ and $A\neq[u_{i}]$,
then Lemma 4.18 implies that $A=\sum c_{i}[u_{i}]$ with some $c_{i}$ being
negative. Therefore, by positivity of intersection, we have
$\cM_{A}(\Sym(\ul{L}),J_{X})=\emptyset$. On the other hand, by our choice of
$J_{\Sigma}$, we have $\\#\cM_{[u_{i}]}(\Sym(\ul{L}),J_{X})=1$ by Corollary
5.3. All this together give us (38).

For each $i$, there are precisely two terms involving $x_{i}$, one with
exponent $1$ and the other with exponent $-1$. From this, it is
straightforward to check that $x=(1,\dots,1)$ is a critical point of
$W_{\Sym(\ul{L})}(x,J_{X})$.

In the next section, we will only consider the Floer cohomology between
$\Sym(\ul{L})$ and its Hamiltonian translates. We will always assume that
$J_{\Sigma}$ is chosen as in Theorem 5.4 so that the potential function of
$\Sym(\ul{L})$ is given by (38).

######  Remark 5.13.

A cobordism argument, as in Proposition 4 and 5, implies that for generic
$J\in\mathcal{J}_{\Delta}$, the potential function of $\Sym(\ul{L})$ (as well
as its Hamiltonian translates) will also be given by the RHS of (38).

######  Remark 5.14.

In the more general setting of Definition 5.5, when $A_{i}+2(k_{i}-1)\eta$ is
not independent of $i$, the $\eta$-disc potential function is

$\displaystyle
W_{\Sym(\ul{L})}^{\eta}(x,J_{X})=\sum_{i=1}^{s}T^{A_{i}+2(k_{i}-1)\eta}x^{\partial
B_{i}}.$

######  Lemma 5.15.

Suppose that $\Sigma=\PP^{1}$ and $\ul{L}$ is $\eta$-monotone. Then the
critical points of $W_{\Sym(\ul{L})}(x,J_{X})$ are non-degenerate.

######  Proof 5.16.

First we consider the case that $\eta=0$. When $B_{1},\dots,B_{k}$ are discs,
$W_{\Sym(\ul{L})}(x,J_{X})$ is given by
$\sum_{i=1}^{k}x_{i}+\frac{1}{x_{1}\dots x_{k}}$ (cf. Proposition 5 and
Theorem 5.4). One sees that all the critical points of
$W_{\Sym(\ul{L})}(x,J_{X})$ are non-degenerate. It remains to see how changing
the configuration of circles affects the disc potential.

Consider two components $L_{1},L_{2}$ of $\ul{L}$ which are boundary
components of a planar domain $B_{1}$. There is a ‘handleslide’ move,
depending on a choice of arc connecting $L_{1}$ and $L_{2}$ (and lying in the
complement of other other circles $L_{i}$), which replaces $(L_{1},L_{2})$ by
the pair $(L_{1}^{\prime},L_{2})$ where $L_{1}^{\prime}$ is obtained as the
connect sum of $L_{1}$ and $L_{2}$ along the arc. Let $B_{1}^{\prime}$ denote
the planar component after the handleslide which contains $L_{1}^{\prime}$ but
not $L_{2}$, and $B_{2}^{\prime}$ the component containing both
$L_{1}^{\prime}$ and $L_{2}$. By smoothly isotoping $L_{1}^{\prime}$
appropriately (in the complement of $L_{i}$ for $i\geq 2$), we can assume that
the area of $B_{1}$, $B_{1}^{\prime}$ and $B_{2}^{\prime}$ are the same.

Let $\ul{L}^{\prime}=L_{1}^{\prime}\cup\cup_{i=2}^{k}L_{i}$. By the assumption
on the area, $\ul{L}^{\prime}$ is also a $0$-monotone link. The disc potential
functions $W_{\Sym(\ul{L})}$ and $W_{\Sym(\ul{L}^{\prime})}$ differ in the two
terms which previously contained the monomial $x_{1}$, which is replaced by a
monomial $x_{1}^{\prime}$ and which arises from the two terms in the potential
given by the regions $B_{i}^{\prime}$. More precisely, for $\eps\in\\{-1,1\\}$
depending on the orientation of $L_{2}$, the modified potential
$W_{\Sym(\ul{L}^{\prime})}$ is obtained from $W_{\Sym(\ul{L})}$ by setting
$x_{1}^{\prime}=x_{1}x_{2}^{\eps}$ and $x_{i}^{\prime}=x_{i}$ for $i\neq 1$.
Direct computation shows that such a change of coordinates preserves non-
degeneracy of the critical points. Finally, any two planar unlinks can be
related by a sequence of such handleslide moves.

For general $\eta$, we can find a smooth family of $\ul{L}_{t}$ such that
$\ul{L}_{0}$ is $\eta$-monotone and $\ul{L}_{1}$ is $0$-monotone. There is a
cobordism between the Maslov $2$ holomorphic discs that $\Sym(\ul{L}_{0})$ and
$\Sym(\ul{L}_{1})$ bound. We can hence deduce the result for the $\eta>0$ case
from the $\eta=0$ case.

##  6 Quantitative Heegaard Floer cohomology

In this section, we assume that $\ul{L}$ is $\eta$-monotone. We now introduce
the version of Lagrangian Floer cohomology that will underlie our link