# The pong algebra and the wrapped Fukaya category

Peter Ozsváth  Department of Mathematics, Princeton University
Princeton, New Jersey 08544<EMAIL_ADDRESS>and  Zoltán Szabó
Department of Mathematics, Princeton University
Princeton, New Jersey 08544<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to identify the pong algebra defined in our earlier
work with a certain endomorphism algebra in the wrapped Fukaya category of the
symmetric product of a disk.

PSO was partially supported by NSF grant number DMS-1708284, DMS-2104536, and
the Simons Grant New structures in low-dimensional topology.

ZSz was supported by NSF grant number DMS-1904628 and the Simons Grant New
structures in low-dimensional topology.

##  1\. Introduction

In [17], we introduced a differential graded algebra, the pong algebra, which
is an enrichment of the strands algebra from [12]. The aim of this note is to
identify this algebra with the endomorphism algebra in a wrapped, relative
Fukaya category, of a distinguished set of objects.

Consider $\mathbb{C}$, equipped with punctures at the points
${\mathbf{P}}=\\{j+\frac{1}{2}\\}_{j=0}^{m-1}$, and let
$M=\mathbb{C}\setminus{\mathbf{P}}$. Consider $m-1$ disjoint vertical lines
$e_{j}=j\times\mathbb{R}$ for $j=1,\dots,m-1$.

Given any $k$-element subsequence $1\leq x_{1}<\dots<x_{k}\leq m-1$, which we
call a $k$-idempotent state, there is a corresponding Lagrangian
${\Lambda}_{\mathbf{x}}=e_{x_{1}}\times\dots\times
e_{x_{k}}\subset\mathrm{Sym}^{k}(M\subset\mathrm{Sym}^{k}(\mathbb{C})$

The manifold $\mathrm{Sym}^{k}(M)$ is a Liouville manifold, and the
Lagrangians $\Lambda_{\mathbf{x}}$ are exact and conical at infinity. One can
then consider the wrapped Fukaya category of introduced by Abouzaid and Seidel
[2] relative to the divisor
${\mathbf{P}}\times\mathrm{Sym}^{k-1}(\mathbb{C})$; see also [1, 4]. We will
consider the endomorphism algebra of the set of objects
$\\{\Lambda_{\mathbf{x}}\\}_{\mathbf{x}}$ indexed by $k$-idempotent states.

This is an A-infinity algebra $A$ over $\mathbb{F}[v_{1},\dots,v_{m}]$,
equipped with idempotents ${\mathbf{I}}_{\mathbf{x}}$ correspondingto the
idempotent states $\mathbf{x}$, and ${\mathbf{I}}_{\mathbf{x}}\cdot
A\cdot{\mathbf{I}}_{\mathbf{y}}$ is given by the chain complex
$\mathit{CF}(\phi_{H}(\Lambda_{\mathbf{x}}),\Lambda_{\mathbf{y}})$ (again,
relative to the divisor ${\mathbf{P}}$). The composition is given by the chan
map

$\circ\colon\mathrm{Mor}(\Lambda_{x_{2}},\Lambda_{\mathbf{x}_{3}})\otimes\mathrm{Mor}(\Lambda_{\mathbf{x}_{1}},\Lambda_{\mathbf{x}_{2}})\to\mathrm{Mor}(\Lambda_{\mathbf{x}_{1}},\Lambda_{\mathbf{x}_{3}}),$

is specified by the following diagram

(1.1)
$\begin{CD}\mathit{CF}(\phi_{H}(\Lambda_{\mathbf{x}_{1}}),\Lambda_{\mathbf{x}_{2}})\otimes\mathit{CF}(\phi_{H}(\Lambda_{\mathbf{x}_{2}}),\Lambda_{\mathbf{x}_{3}})@>{\circ}>{}>\mathit{CF}(\phi_{H}(\Lambda_{\mathbf{x}_{1}}),\Lambda_{\mathbf{x}_{3}})\\\
@V{(\phi_{H})_{*}\otimes\operatorname{Id}}V{}V@A{\sigma}A{}A\\\
\mathit{CF}(\phi^{2}_{H}(\Lambda_{\mathbf{x}_{1}}),\phi_{H}(\Lambda_{\mathbf{x}_{2}})\otimes\mathit{CF}(\phi_{H}(\Lambda_{\mathbf{x}_{2}}),\Lambda_{\mathbf{x}_{3}})@>{}>{}>\mathit{CF}(\phi^{2}_{H}(\Lambda_{\mathbf{x}_{1}}),\Lambda_{\mathbf{x}_{3}}),\\\
\end{CD}$

where $\sigma$ is a map induced from the Liouville flow, and the bottom arrow
is induced by counting pseudo-holomorphic triangles. The base ring is
$\mathbb{F}[v_{1},\dots,v_{m}]$; and for the map in the bottom arrow, each
holomorphic triangle $u$ is counted as a monomial $v_{1}^{n_{p_{1}}(u)}\dots
v_{m}^{n_{p_{m}}(u)}$.

We can take the homology to obtain an ordinary category, $H({\mathcal{C}})$;
or alternatively, we can consider the $A_{\infty}$ category, where the higher
compositions are defined by counting pseudo-holomorphic polygons.

Our aim is to prove the following:

######  Theorem 1.1.

The pong algebra is isomorphic to the endomorphism algebra, in the wrapped
relative Fukaya category, of the objects
$\\{\Lambda_{\mathbf{x}}\\}_{\mathbf{x}}$.

This proof is in the spirit of Auroux [3]; see also [13]. Indeed, the proof is
a fairly straightforward application of a suitable Heegaard diagram, which can
be thought of as the analogue of the “Auroux-Zarev piece” for the pong
algebra; see [3, 26, 11].

It is interesting to compare the results herein with the results of [8, 14],
and the constructions of [21, 6, 27, 7].

Our interest in the pong algebra stems from our goal of understanding knot
Floer homology, a topic which we do not discuss in the present paper, but hope
to return to in future work [16].

Acknowledgements: The authors wish to thank Denis Auroux, Robert Lipshitz,
Dylan Thurston, and Andy Manion for interesting conversations.

##  2\. The wrapped Fukaya category

###  2.1. The symmetric product

We start with some of the geometric setup, as explained in [2, 5, 4].

######  Definition 2.1.

A Liouville domain is a $2n$-dimensional manifold with boundary, equipped with
a one-form $\lambda$ such that $\omega=d\lambda$ is symplectic, and the dual
vector field, called the Liouville vector field, $Z$ characterized by
$i_{Z}\omega=\lambda$ points strictly outwards along $\partial M$.

A special case of a Liouville domain is a Stein manifold, which is a complex
manifold $(V,J)$, equipped with a proper, smooth function $\phi\colon
V\to\mathbb{R}$, which is strictly psudo-plurisubharmonic; i.e. for which the
two-form $\omega=-dd^{\mathbb{C}}\phi$ is symplectic and $J$-compatible.
(Here, $d^{\mathbb{C}}\phi=d\phi\circ J$.) In this case,
$\lambda=-d^{\mathbb{C}}\phi$; i.e. $d\phi=\lambda\circ J$.

Our basic example is the following. Let $A$ denote the infinite cylinder
$\mathbb{R}\times S^{1}\cong\mathbb{C}\setminus 0$. Let $(t,\theta)$ denote
the coordinates with respect to the parameterization $\mathbb{R}\times S^{1}$;
so that the isomorphism $\mathbb{R}\times S^{1}\cong\mathbb{C}\setminus 0$ is
given by $(t,\theta)\mapsto e^{t+i\theta}$; sometimes we write $r=e^{t}$. The
function $\log(r)^{2}$ is strictly pluri-subharmonic, with
$\omega=2dt\wedge\theta$. Let $H=r^{2}$. The Hamiltonian flow for $H$, written
$\Phi\colon\mathbb{R}\times A\to A$, is given by

(2.1) $\Phi(s,t,\theta)=(t,2st\theta).$

The Liouville flow, written $\Psi\colon\mathbb{R}\times A\to A$, is given by

(2.2) $\Psi(s,t,\theta)=(te^{s},\theta).$

To a Liouville domain Abouzaid and Seidel associate an $A_{\infty}$ category,
the wrapped Fukaya category. Objects are Lagrangian submanifolds $L\subset M$
that intersect $\partial M$ transversely, with the property that
$\theta|L\in\Omega^{1}(L)$ is exact, and $\theta$ vanishes to infinite order
along the boundary $\partial L=L\cap\partial M$.

We will be considering Lagrangians in the symmetric product of $\mathbb{C}$,
$\mathrm{Sym}^{k}(\mathbb{C})$. There is a quotient map
$\pi\colon\mathbb{C}^{k}\to\mathrm{Sym}^{k}(\mathbb{C})$, and also there is a
diffeomorphism $\mathbb{C}^{k}\cong\mathrm{Sym}^{k}(\mathbb{C})$. The
relationship between Lagrangians in a symmetric product of a curve with the
${\mathfrak{S}}_{k}$-invariant Lagrangians in the $k$-fold Cartesian product
is unclear; but there is a nice bridge offered by work of Perutz [19],
building on work of Varouchas [25], who constructs a new symplectic form on
the symmetric product that agrees with group-invariant the symplectic
structure on the Cartesian product on an open set. The case at hand is a
particularly simple, local version. (See Proposition 2.2 below.)

In the interest of concreteness, we find it convenient to have some explicit
parameterizations. Consider the map from the infinite cylinder
$A=\mathbb{C}\setminus\\{0\\}\cong\mathbb{R}\times
S^{1}=\mathbb{R}\times(\mathbb{R}/2\pi\mathbb{Z})$ to $\mathbb{C}$, specified
by

$p(z)=\frac{1}{2}\left(z+\frac{1}{z}\right),$

This map has the following properties:

  * •

$p$ is is a branched double-cover, with two branched points at $1$ and $-1$.

  * •

$p$ is proper

  * •

The image under $p$ of the circle $\\{0\\}\times S^{1}$ is the interval
$[-1,1]\subset\mathbb{C}$.

Given a $k$-element subset $\mathbf{x}\subset S^{1}\setminus\\{\pm 1\\}$, we
can view $\mathbb{R}\times\mathbf{x}$ as a subset of $\mathrm{Sym}^{k}(A)$.
This image is a smooth submanifold, whose image under $\mathrm{Sym}^{k}(p)$ is
a smooth submanifold of $\mathrm{Sym}^{k}(\mathbb{C})$. We denote this
subspace $\Lambda_{\mathbf{x}}$.

More generally, fix ${\mathbf{x}}$ as above and an element
$\phi\in\mathbb{R}^{\geq 0}$. There is a submanifold of
$A\cong\mathbb{R}\times S^{1}$ of elements of the form $\\{(t,e^{i\phi\cdot
t/2}\mathbf{x})\\}_{t\in\mathbb{R}}$, which induces a submanifold
$\Lambda_{\mathbf{x}}^{\phi}\subset\mathrm{Sym}^{k}(\mathbb{C})$. Clearly,
$\Lambda_{\mathbf{x}}^{0}=\Lambda_{\mathbf{x}}$.

Let $t\colon A\to\mathbb{R}$ be projection to the first coordinate or,
equivalently, $z\in\mathbb{C}\setminus\\{0\\}\mapsto\log|z|$. Consider the
function $\delta\colon C^{k}\to\mathbb{R}\geq 0$ defined by

$\delta(z_{1},\dots,z_{k})=\min(\min_{i}t(z_{i}),\min_{i\neq
j}|z_{i}-z_{j}|),$

which descends to a continuous function
$\mathrm{Sym}^{k}(\mathbb{C})\to\mathbb{R}$, so that

$\delta^{-1}(0)=\Delta\cup\left(S^{1}\times\mathrm{Sym}^{k-1}(A)\right).$

The following is an adaptation of a theorem of Perutz [19]; see also Varouchas
[25].

######  Proposition 2.2.

Given any bounded open set $W\subset\mathrm{Sym}^{k}(\mathbb{C})$ containing
$\mathrm{Sym}^{k}[-1,1]$ and any $\eta>0$, there is a smooth plurisubharmonic
function $\psi\colon W\to\mathbb{R}$ with the following properties:

  * •

Given any $k$-elements subset $\mathbf{x}\subset(S^{1}\setminus\\{\pm 1\\})$
with $\delta(\mathbf{x})\geq\eta$, the intersection of $W$ with the
submanifold $\Lambda_{\mathbf{x}}\subset\mathrm{Sym}^{k}(\mathbb{C})$ is
Lagrangian with respect to the symplectic structure $dd^{\mathbb{C}}\psi$.

  * •

Given $s\geq 0$, there is an exact Hamiltonian diffeomorphism
$\Phi^{s}\colon\mathbb{R}\times W\to W$ with the property that
$\Phi^{s}(\Lambda_{\mathbf{x}}\cap W)=\Lambda^{s}_{\mathbf{x}}\cap W$.

Proof. This follows easily from Varouchas’s “Lemme Principal” [25], which
which we state in a slightly simplified form. Given the data:

  * •

Open subsets $U$, $V$, $W$, and $X$ in $\mathbb{C}^{n}$ so that $U$, $V$, and
$W$ are bounded, with ${\overline{U}}\subset V$, ${\overline{V}}\subset W$,
${\overline{W}}\subset X$

  * •

a continuous, strictly pluri-subharmonic function $\phi\colon X\to\mathbb{R}$
so that $\phi|_{X\setminus U}$ is smooth.

there is a smooth, strictly pluri-subharmonic function $\chi\colon
W\to\mathbb{R}$ so that

$\psi|_{W\setminus(V\cap W)}=\chi|_{W\setminus(V\cap W)}.$

The function $t^{2}\colon\mathbb{C}^{k}\to\mathbb{R}$ given by
$t^{2}=\sum_{i=1}^{k}|t_{i}|^{2}$ is a smooth. Let
$X=(\mathbb{C}\setminus\\{0\\})^{\times k}$, and $\Pi\colon\mathbb{C}^{\times
k}\to\mathrm{Sym}^{k}(\mathbb{C})$ be the quotient map. As in [25], since
$t^{2}\colon A\to\mathbb{R}$ is a smooth, strictly pluri-subharmonic function
and $\Pi\circ p^{\times k}\colon A^{k}\to\mathrm{Sym}^{k}(\mathbb{C})$ is a
branched cover, the push-forward $\psi=(\Pi\circ p^{\times k})_{*}(t^{2})$ is
a continuous, strictly pluri-subharmonic function on $X$. Given $W$ as in the
statement of the proposition, apply Varouchas’ lemma to $\psi$,
$U=\delta^{-1}(0,\frac{\eta}{3})$, and $V=\delta^{-1}(0,\frac{\eta}{2})$. The
exact Hamiltonian is associated to the Hamiltonian function $\chi$ coming from
the lemma. Since $\chi$ agrees with $t^{2}$ over the complement of $V$, it is
easy to see that the integral of $\chi$ preserves $\delta$ over that set. The
second point now follows readily; see Equation (2.1).

We fix the following data:

  * •

integers $m$ and $k$ with $0<k<m$

  * •

$m$ basepoints $-1=O_{1},O_{2},\dots,O_{m-1},1=O_{m}$ so that there is a
positively oriented arc in $S^{1}$ from $O_{i+1}$ ot $O_{i}$ containing no
other $O_{j}$.

  * •

$m-1$ additional points $p_{1},\dots,p_{m-1}$, so that $z_{i}$ is on the arc
from $O_{i+1}$ to $O_{i}$.

  * •

$\delta(z_{i},z_{i+1})\geq 2/m-1$.

Choose $W\subset\mathrm{Sym}^{k}(\mathbb{C})$ as in Proposition 2.2, and let
$\psi$ be the function supplied by that proposition. There are
$\binom{m-1}{k}$ Lagrangians $\Lambda_{\mathbf{x}}\cap W$, associated to the
$k$-element subsets of $\\{p_{1},\dots,p_{m-1}\\}$. We will be considering
these as our basic objects in the wrapped Fukaya category of $W$.

Let $\Psi^{c}\colon W\to W$ be the time $\log(c)$ flow of the Liouville vector
field induced from $\psi$. It is an easy consequence of Equation (2.2) that

(2.3) $\Psi^{c}(\Lambda_{\mathbf{x}}^{\phi})=\Lambda_{\mathbf{x}}^{\phi/c}.$

###  2.2. The relative Fukaya category

In our case, the symplectic manifold $W$ is equipped also with $m$ divisors,
of the form $\\{O_{i}\\}\times\mathrm{Sym}^{k-1}(\mathcal{H})$.
Correspondingly, as in [15]; see also, cite [20] for a general construction,
the Floer complexes are to be thought of as modules over a polynomial algebra
$\mathbb{F}[v_{1},\dots,vm]$. Specifically, for
$L_{1}=\Phi_{H}(\Lambda_{\mathbf{x}_{1}})$, $L_{2}=\Lambda_{\mathbf{x}_{2}}$,
the complex $\mathit{CF}(L_{1},L_{2})$ is a module over
$\mathbb{F}[v_{1},\dots,v_{m}]$ freely generated by $L_{1}\cap L_{2}$, with
differential determined by

$\partial\mathbf{x}=\sum_{\mathbf{y}}\sum_{\\{\phi\in\mathcal{M}(\mathbf{x},\mathbf{y})\mid\mu(\phi)=1\\}}\\#\mathcal{M}(\phi)\cdot\mathbf{y}\cdot
v_{1}^{n_{O_{1}}(\phi)}\cdots v_{m}^{n_{O_{m}}(\phi)}.$

Here, $n_{O_{i}}(\phi)$ denotes the (non-negative) algebraic intersection
number of $\phi$ with the divisor
$\\{O_{i}\\}\times\mathrm{Sym}^{k-1}(\mathcal{H})$. The moduli space
$\mathcal{M}(\mathbf{x},\mathbf{y})$ is to be taken with respect to a suitable
perturbation of the Floer equation.

##  3\. Lifted partial permutations and the pong algebra

We recall the construction of the pong algebra from [17, Section 4]; we refer
the reader to that reference for a more leisurely account.

Let $r_{t}\colon\mathbb{R}\to\mathbb{R}$ be the reflection $r_{t}(x)=2t-x$;
and consider the subgroup $G_{m}$ of the reflection group of the real line
generated by $r_{\frac{1}{2}}$ and $r_{m-\frac{1}{2}}$. The quotient of the
integral lattice by this group of rigid motions is naturally an $m-1$ point
set; generated by $\\{1,\dots,m-1\\}$. Let

(3.1) $Q_{1}\colon\mathbb{Z}\to\\{1,\dots,m-1\\}$

denote this quotient map.

Note that $G_{m}$ also acts on the set $\frac{1}{2}+\mathbb{Z}$. The quotient
of $\frac{1}{2}+\mathbb{Z}$ by $G_{m}$ is naturally the $m$-point set,
$\\{\frac{1}{2},\dots,m-\frac{1}{2}\\}$. We think of these points as being in
one-to-one correspondence with the underlying variables in the pong algebra,
where the point $j\in\\{\frac{1}{2},\dots,m-\frac{1}{2}\\}$ corresponds to the
variable $v_{\frac{1}{2}+j}$. Explicitly, we have the map

(3.2) $Q_{2}\colon\mathbb{Z}+\frac{1}{2}\to\\{1,\dots,m\\},$

defined so that $Q_{2}(j-\frac{1}{2})$ is the element $i\in\\{1,\dots,m\\}$
with $i\equiv j\pmod{2m-2}$ or $i\equiv 2-j\pmod{2m-2}$.

A $G_{m}$ invariant subset $\widetilde{S}$ of $\mathbb{Z}$ has a natural
quotient $\widetilde{S}/G_{m}$, which is a subset of $\\{1,\dots,m-1\\}$.

######  Definition 3.1.

A lifted partial permutation on $k$ letters is a pair
$({\widetilde{S}},{\widetilde{f}})$ where:

  * •

${\widetilde{S}}\subset\mathbb{Z}$ is a $G_{m}$-invariant subset

  * •

${\widetilde{f}}\colon{\widetilde{S}}\to\mathbb{Z}$ is a $G_{m}$-equivariant
map;

subject to the following two conditions:

  * •

${\widetilde{S}}/G_{m}$ consists of $k$ elements

  * •

the induced map
${\widetilde{f}}\colon{\widetilde{S}}/G_{m}\to\mathbb{Z}/G_{m}$ is injective.

######  Definition 3.2.

A lifted partial permutation $({\widetilde{S}},{\widetilde{f}})$ has a weight
vector
${\vec{\mathfrak{w}}}=(\mathfrak{w}_{1},\dots,\mathfrak{w}_{m})\in(\frac{1}{2}\mathbb{Z})^{m}$,
specified by

$\mathfrak{w}_{j}({\widetilde{f}})=\frac{1}{2}\\#\\{i\in{\widetilde{S}}\big{|}i<j-\frac{1}{2}<{\widetilde{f}}(i)~{}\text{or}~{}i>j-\frac{1}{2}>{\widetilde{f}}(i)\\}.$

We extend the weight vector to $\mathbb{F}[v_{1},\dots,v_{m}]$ so that
$\mathfrak{w}(v_{i})$ is the $i^{th}$ basis vector in $\mathbb{Z}^{m}$.

######  Definition 3.3.

A crossing in a lifted partial permutation ${\widetilde{f}}$ is an equivalence
class of pairs of integers $(i,j)$ with the property that $i<j$ and
${\widetilde{f}}(i)>{\widetilde{f}}(j)$. We say that $(i,j)$ and
$(i^{\prime},j^{\prime})$ determine the same crossing if there is some $g\in
G_{m}$ so that $\\{g\cdot i,g\cdot j\\}=\\{i^{\prime},j^{\prime}\\}$. We write
$\langle i,j\rangle$ for the equivalence class of the pair of integers$(i,j)$.
Let $\mathrm{Cross}({\widetilde{f}})$ denote the set of crossings in
${\widetilde{f}}$.

Note that $\langle i,j\rangle\in\mathrm{Cross}({\widetilde{f}})$ does not
exclude cases where $[i]=[j]$. Let $\mathrm{cross}({\widetilde{f}})$ denote
the number of crossings in ${\widetilde{f}}$.

Let $({\widetilde{f}},{\widetilde{S}})$ and
$({\widetilde{g}},{\widetilde{T}})$ be two partial permutations with
${\widetilde{T}}={\widetilde{f}}({\widetilde{S}})$. Then, the composite
$({\widetilde{g}}\circ{\widetilde{f}},{\widetilde{S}})$ is a lifted partial
permutation.

It is elementary to verify that

$\displaystyle\mathfrak{w}({\widetilde{g}}\circ{\widetilde{f}})$
$\displaystyle\leq\mathfrak{w}({\widetilde{g}})+\mathfrak{w}({\widetilde{f}})$
$\displaystyle\mathrm{cross}({\widetilde{g}}\circ{\widetilde{f}})$
$\displaystyle\leq\mathrm{cross}({\widetilde{g}})+\mathrm{cross}({\widetilde{f}})$

The pong algebra ${\mathcal{P}}(m,k)$ is the algebra over
$\mathbb{F}[v_{1},\dots,v_{m}]$ freely generated by lifted partial
permutations, with a multiplication map

$\mu_{2}\colon{\mathcal{P}}(m,k)\otimes_{\mathbb{F}[v_{1},\dots,v_{m}]}{\mathcal{P}}(m,k)\to{\mathcal{P}}(m,k)$

characterized by

$\mu_{2}([{\widetilde{f}},{\widetilde{S}}],[{\widetilde{g}},{\widetilde{T}}])]=\begin{cases}0&\text{if
${\widetilde{T}}\neq{\widetilde{f}}({\widetilde{S})}$}\\\ 0&\text{if
$\mathrm{cross}({\widetilde{g}}\circ{\widetilde{f}})<\mathrm{cross}({\widetilde{g}})+\mathrm{cross}({\widetilde{f}})$}\\\
v\cdot[{\widetilde{g}}\circ{\widetilde{f}},{\widetilde{S}}]&{\text{otherwise,}}\end{cases}$

where $v$ is the monomial in $v_{1},\dots,v_{m}$ chosen so that

$\mathfrak{w}(\mu_{2}([{\widetilde{f}},{\widetilde{S}}],[{\widetilde{g}},{\widetilde{T}}]))=\mathfrak{w}[{\widetilde{f}},{\widetilde{S}}]+\mathfrak{w}[{\widetilde{g}},{\widetilde{T}}].$

Given $a,b\in{\mathcal{P}}(m,k)$, we abbreviate $\mu_{2}(a,b)$ by $a\cdot b$.

For each $\langle i,j\rangle\in\mathrm{Cross}({\widetilde{f}})$, there is a
new lifted partial permutation ${\widetilde{f}}_{\langle i,j\rangle}$
characterized as follows:

${\widetilde{f}}_{\langle
i,j\rangle}(k)=\begin{cases}{\widetilde{f}}(k)&{\text{if~{}$[k]\not\in\\{[i],[j]\\}$}}\\\
g\cdot{\widetilde{f}}(j)&{\text{if $k=g\cdot i$}}\\\
g\cdot{\widetilde{f}}(i)&{\text{if $k=g\cdot j$}}\end{cases}.$

It is elementary to verify that

$\displaystyle\mathfrak{w}({\widetilde{f}}_{\langle i,j\rangle})$
$\displaystyle\leq\mathfrak{w}({\widetilde{f}})$
$\displaystyle\mathrm{cross}({\widetilde{f}}_{\langle i,j\rangle})$
$\displaystyle\leq\mathrm{cross}({\widetilde{f}})-1$

Given $\langle i,j\rangle\in\mathrm{Cross}({\widetilde{f}},{\widetilde{S}})$,
let $\partial_{\langle i,j\rangle}{\widetilde{f}}\in{\mathcal{P}}(m,k)$ be the
element defined by

$\partial_{\langle i,j\rangle}{\widetilde{f}}=\begin{cases}0&{\text{if
$\mathrm{cross}({\widetilde{f}}_{\langle
i,j\rangle})<\mathrm{cross}({\widetilde{f}})-1$}}\\\
v\cdot{\widetilde{f}}_{\langle i,j\rangle}&{\text{otherwise,}}\end{cases}$

where now $v$ is the monomial in $v_{1},\dots,v_{m}$ characterized by the
property that

$\mathfrak{w}[{\widetilde{f}},{\widetilde{S}}]=\mathfrak{w}[{\widetilde{f}}_{\langle
i,j\rangle},{\widetilde{S}}].$

Define a map

$\partial\colon{\mathcal{P}}(m,k)\to{\mathcal{P}}(m,k),$

characterized by

$\partial({\widetilde{f}},{\widetilde{S}})=\sum_{\langle
i,j\rangle\in\mathrm{Cross}({\widetilde{f}},{\widetilde{S}})}\partial_{\langle
i,j\rangle}[{\widetilde{f}},{\widetilde{S}}].$

With the above definitions, ${\mathcal{P}}(m,k)$ is a differential graded
algebra over $\mathbb{F}[v_{1},\dots,v_{m}]$; see [17, Proposition
LABEL:P:prop:PongIsAlg] for details.

It will be convenient to have the following:

######  Lemma 3.4.

If
$\mathrm{cross}({\widetilde{f}}\circ{\widetilde{g}})=\mathrm{cross}({\widetilde{f}})+\mathrm{cross}({\widetilde{g}})$
then

$\mathfrak{w}_{1}({\widetilde{f}}\circ{\widetilde{g}})=\mathfrak{w}_{1}({\widetilde{f}})+\mathfrak{w}_{1}({\widetilde{g}})\qquad{\text{and}}\qquad\mathfrak{w}_{m}({\widetilde{f}}\circ{\widetilde{g}})=\mathfrak{w}_{m}({\widetilde{f}})+\mathfrak{w}_{m}({\widetilde{g}}).$

Proof. If
$\mathfrak{w}_{1}({\widetilde{f}}\circ{\widetilde{g}})\neq\mathfrak{w}_{1}({\widetilde{f}}))+\mathfrak{w}_{1}({\widetilde{g}})$,
then there exists some $1<i$ with

(3.3) $1<i\text{~{}such that~{}}{\widetilde{g}}(i)<1;$

and

(3.4) ${\widetilde{f}}\circ{\widetilde{g}}(i)>1.$

On the other hand, Equation (3.3) implies that

${\widetilde{g}}(1-i)=1-{\widetilde{g}}(i)>1-i;$

while Equation (3.4) implies that

${\widetilde{f}}\circ{\widetilde{g}}(1-i)>{\widetilde{g}}(1-i);$

i.e. $i$ and $1-i$ have a crossing in ${\widetilde{g}}$, while
${\widetilde{g}}(i)$ and ${\widetilde{g}}(1-i)$ have a crossing in
${\widetilde{f}}$.

An analogous argument works for $\mathfrak{w}_{m}$.

##  4\. The wrapped diagram

Our aim here is to give a particularly convenient description of the wrapped
diagram for $\mathrm{Sym}^{k}(M)$, equipped with $\Lambda_{\mathbf{x}}$.

Consider the plane $\mathbb{R}^{2}$, decorated with the following data:

  * •

an infinite grid of vertical lines
${\widetilde{\mathbf{\alpha}}}=\\{{\widetilde{\alpha}}_{i}=i\times\mathbb{R}\\}_{i\in\mathbb{Z}}$;

  * •

horizontal lines
${\widetilde{\mathbf{\beta}}}=\\{{\widetilde{\beta}}_{i}=\mathbb{R}\times
i\\}_{i\in\mathbb{Z}}$;

  * •

an infinite set of punctures at the points
$\\{(\frac{1}{2}+i,\frac{1}{2}+i)\\}_{i\in\mathbb{Z}}$, so that
$(\frac{1}{2}+i,\frac{1}{2}+i)$ is labeled by $O_{j}$, where
$j=Q_{2}(\frac{1}{2}+i)$, with $Q_{2}$ as in Equation (3.2).

The symmetry group of this picture is generated by the two
$180^{\circ}$-rotations with fixed point at $(\frac{1}{2},\frac{1}{2})$, and
the one with fixed point at $(m-\frac{1}{2},m-\frac{1}{2})$. Let
${\mathbb{G}}_{m}$ denote this group of rigid motions. Note that
${\mathbb{G}}_{m}\cong G_{m}$, induced by the restriction of
${\mathbb{G}}_{m}$ to the diagonal line in $\mathbb{R}^{2}$.

The quotient space $\mathbb{R}^{2}/{\mathbb{G}}_{m}$ is homeomorphic to the
disk $\mathcal{H}$ with two order $2$ orbifold points, which are the points
marked $O_{1}$ and $O_{m}$. There are an addition $m-2$ marked points, labeled
$O_{i}$ for $i=2,\dots,m-1$.

The vertical lines
$\mathbb{Z}\times\mathbb{R}=\\{{\widetilde{\alpha}}_{i}\\}_{i\in\mathbb{Z}}$
project to $m-1$ embedded lines $\\{\alpha_{i}\\}_{i=1}^{m-1}$ in
$\mathcal{H}$. Similarly, the horizontal lines
$\\{{\widetilde{\beta}}_{i}\\}_{i\in\mathbb{Z}}$, project to $m-1$ embedded
lines $\\{\beta_{i}\\}_{i=1}^{m-1}$ in $\mathcal{H}$. We label the lines in
$\mathcal{H}$ so that $\alpha_{i}$ is the image of ${\widetilde{\alpha}}_{j}$,
for any $j\in\mathbb{Z}$ with $Q_{1}(j)=i$, with $Q_{1}$ as in Equation (3.1);
similarly, $\beta_{i}$ is the image of ${\widetilde{\beta}}_{j}$.

######  Lemma 4.1.

Consider $\mathcal{H}=\mathbb{R}^{2}/{\mathbb{G}}_{m}$, equipped with
$\\{\alpha_{i}\\}_{i=1}^{m-1}$, $\\{\beta_{i}\\}_{i=1}^{m-1}$, and the
markings $\\{O_{i}\\}_{i=1}^{m}$. This is a diagram for the wrapping of
$\\{\alpha_{i}\\}_{i=1}^{m-1}$.

Proof. What we mean is the following. Consider $\mathcal{H}$ as above,
equipped with the vertical circles ${\widetilde{\mathbf{\alpha}}}$.
${\mathbf{G}}_{m}$ contains an index two subgroup of translations, generated
by $(x,y)\mapsto(x+2m,y+2m)$. Consider the cylinder $A$ obtained as the
quotient

(4.1) $\displaystyle
A=\frac{\mathbb{R}\times\mathbb{R}}{(2m,2m)\cdot\mathbb{Z}}.$

Moreover, there is a branched covering map from $A$ to $\mathcal{H}$, with
branching at $O_{1}$ and $O_{m}$.

Equip $\mathrm{Sym}^{k}(\mathcal{H})$ with the symplectic structure from
Proposition 2.2, chosen so that the images in $\mathrm{Sym}^{k}(\mathcal{H})$
of the manifolds
$\alpha_{x_{1}}\times\dots\times\alpha_{x_{k}}\subset\mathcal{H}^{\times k}$
for all subsequences $\mathbf{x}\subset\\{1,\dots,m-1\\}$ are Lagrangian. Note
that our explicit parametrizations here differ from the ones described around
Proposition 2.2 by a linear transformation. In particular, these manifolds are
equivalent to the submanifolds $\Lambda_{\mathbf{x}}$ from that proposition.
Moreover, the Liouville flow carries
$\mathbf{\alpha}_{x_{1}}\times\dots\times\alpha_{x_{k}}$ to
$\beta_{x_{1}}\times\dots\times\beta_{x_{k}}$. after some (positve) time.

See Figures 1 and 2 for examples.

$O_{2}$$O_{1}$$O_{2}$$O_{1}$120102$O_{1}$$O_{2}$

Figure 1.  Heegaard diagrams. The quotient of the infinite grid diagram on the
left is the diagram (with $m=2$) on the right. On the left, we have shaded a
fundamental domain for the ${\mathbb{G}}_{2}$ action.

$O_{2}$$O_{1}$$O_{2}$$O_{3}$120102$O_{1}$$O_{2}$$O_{3}$$O_{1}$$O_{3}$$O_{2}$

Figure 2.  Heegaard diagrams. The case where $m=3$.

As usual, a $k$-fold Heegaard state is a $k$-tuple of points (for some
$0<k<m$), with the property that each point lies on $\alpha_{i}\cap\beta_{j}$,
no two points lie on the same $\alpha_{i}$, and no two points lie on the same
$\beta_{j}$.

Given a lifted partial permutation ${\widetilde{f}}$, we can form its graph

$\Gamma_{\widetilde{f}}=\\{(i,{\widetilde{f}}(i))\mid
i\in{\widetilde{S}}\\}\subset\mathbb{R}^{2}.$

Clearly $\Gamma_{\widetilde{f}}$ is invariant under ${\mathbb{G}}_{m}$; as
such we can form the associated subset
$\mathbf{x}({\widetilde{f}})\subset\mathcal{H}$.

######  Lemma 4.2.

The above map sets up a one-to-one correspondence between ($k$-element) lifted
partial permutations and ($k$-fold) Heegaard states for $\mathcal{H}$.

Proof. The proof is straightforward.

######  Definition 4.3.

Suppose that $\mathcal{H}$ is a surface, equipped with two sets of curves
$\mathbf{\alpha}=\\{\alpha_{i}\\}$ and $\mathbf{\beta}=\\{\beta_{i}\\}$. We
think of the curves as giving $\mathcal{H}$ a $CW$-complex structure, with
$0$-cells the intersection points between $\alpha_{i}$ and $\beta_{j}$,
$1$-cells the arcs in $\alpha_{i}$ and $\beta_{j}$, and two-chains the
components of $\mathcal{H}\setminus(\mathbf{\alpha}\cup\mathbf{\beta})$. Thus,
a two-chain can be thought of as an assignment of integers to each component
of $\mathcal{H}\setminus(\mathbf{\alpha}\cup\mathbf{\beta})$. Fix some
intersection point $x$ of $\alpha_{i}$ with $\beta_{j}$. We say that $x$ is a
corner if the local multiplicities $A$, $B$, $C$, and $D$ around $x$, as
pictured in Figure 3, satisfy $A+D\neq B+C$. In fact, we say that $x$ is an
initial $(\alpha,\beta)$ corner if $B+C=A+D+1$; if $B+C=A+D-1$, we say $x$ is
a terminal $(\alpha,\beta)$-corner. A domain is a cornerless domain if it has
no corner.

$A$$B$$C$$D$

Figure 3.  Corner conventions.

The space of cornerless domains is an abelian group. If $\mathbf{x}$ and
$\mathbf{y}$ are Heegaard states, let $\mathcal{D}(\mathbf{x},\mathbf{y})$
denote the space of domains with initial corner at the components of
$\mathbf{x}\setminus(\mathbf{x}\cap\mathbf{y})$ and terminal corner at the
components of $\mathbf{y}\setminus(\mathbf{x}\cap\mathbf{y})$. (Algebraically,
$\mathcal{D}(\mathbf{x},\mathbf{y})$ is an affine space for the space of
cornerless domains.) As explained in [15], for $k\geq 3$,
$\mathcal{D}(\mathbf{x},\mathbf{y})$ is identified with a space of relative
homotopy classes of Whitney disks connecting $\mathbf{x}$ to $\mathbf{y}$,
denoted there $\pi_{2}(\mathbf{x},\mathbf{y})$.

######  Lemma 4.4.

Given Heegaard states $\mathbf{x}$ and $\mathbf{y}$, corresponding to lifted
partial permutations $({\widetilde{f}},{\widetilde{S}})$ and
$({\widetilde{g}},{\widetilde{T}})$, there is a
$\phi\in\mathcal{D}(\mathbf{x},\mathbf{y})$ with compact support if and only
if ${\widetilde{S}}={\widetilde{T}}$ and
${\widetilde{f}}({\widetilde{S}})={\widetilde{g}}({\widetilde{T}})$. Moreover,
if $\phi$ exists, then it is unique.

Proof. Given $\mathbf{x}$ and $\mathbf{y}$, consider the corresponding lift to
$\mathbb{R}^{2}$. The hypothesis that ${\widetilde{S}}={\widetilde{T}}$ is
equivalent to the condition that we can connect $\mathbf{x}$ to $\mathbf{y}$
by a path $A$ in $\mathbf{\alpha}=\\{\alpha_{i}\\}_{i=1}^{m-1}$. (When the
path exists, its uniqueness, as a relative one-chain in $\mathcal{H}$, is
obvious.) Similarly, the condition that
${\widetilde{f}}({\widetilde{S}})={\widetilde{f}}({\widetilde{T}})$ is
equivalent to the condition that we can connect ${\widetilde{y}}$ to
${\widetilde{\mathbf{x}}}$ (uniquely) by a path $B$ inside $\mathbf{\beta}$.
By construction, $\partial A=\partial B$, so $A-B=\partial D$, for some two-
chain. Uniqueness follows from contractability of $\mathcal{H}$.

Suppose that $({\widetilde{f}},{\widetilde{S}})$ is a lifted partial
permutation with graph ${\widetilde{\mathbf{x}}}$ and Heegaard state
$\mathbf{x}$. Let $A$ be the set of vertical lines in $\mathbb{R}^{2}$ that
connect ${\widetilde{x}}$ to the diagonal line. Given $1\leq i\leq m$, the
weight of ${\widetilde{S}}$ at $i$ can be interpreted as the number of times
$A$ crosses the horizontal line $\mathbb{R}\times(i-\frac{1}{2})$.

######  Lemma 4.5.

Suppose that $({\widetilde{f}},{\widetilde{S}})$ and
$({\widetilde{g}},{\widetilde{T}})$ are lifted partial permutations with
corresponding Heegaard states $\mathbf{x}$ and $\mathbf{y}$, which can be
connected by some $\phi\in\mathcal{D}(\mathbf{x},\mathbf{y})$. Then, the local
multiplicity of $\phi$ at $O_{i}$ coincides with
$\mathfrak{w}(\widetilde{f})-\mathfrak{w}(\widetilde{g})$.

Proof. Let $\mathbf{x}_{0}$ denote the Heegaard states corresponding to the
identity map on ${\widetilde{S}}$. Let $A$ be (oriented) vertical path from
$\mathbf{x}$ to $\mathbf{x}_{0}$, so that $\mathfrak{w}(\mathbf{x})$ counts
half the number of times $A$ crosses $\mathbb{R}\times(i-\frac{1}{2})$. We can
think of $\mathbb{R}\times(i-\frac{1}{2})$ as the union of two rays
${\widetilde{r}}_{i}$ and ${\widetilde{r}}_{i}^{\prime}$ starting at
$(i-\frac{1}{2},i-\frac{1}{2})$. Then, $\mathfrak{w}(\mathbf{x})$ is one half
the oriented intersection number of $r_{i}$ with $A$ plus the oriented
intersection number of $r_{i}^{\prime}$ with $A$.

Let $r_{i}$ resp. $r_{i}^{\prime}$ denote the image in ${\mathbb{H}}$ of
${\widetilde{r}}_{i}$ resp. ${\widetilde{r}}^{\prime}_{i}$. Observe that
$r_{i}$ and $r_{i}^{\prime}$ are paths in ${\mathbb{H}}$ from $O_{i}$ to
infinity that avoid $\mathbf{\beta}$. (Indeed, for $i=1$ and $m$,
$r_{i}=r_{i}^{\prime}$.) Thus,
$\mathfrak{w}(\mathbf{x})-\mathfrak{w}(\mathbf{y})$ is the algebraic
intersection number of $r_{i}$ with $\partial\phi$, which in turn coincides
with the winding number of $\partial\phi$ around $O_{i}$, and hence the local
multiplicity of $\phi$ at $O_{i}$. Since the same remarks apply for
$r_{i}^{\prime}$, the result follows.

3132$O_{2}$2$O_{1}$$O_{2}$$O_{3}$$O_{2}$$O_{3}$$O_{1}$$r_{2}$${\widetilde{r}}_{2}$$O_{2}$1$O_{1}$$O_{3}$

Figure 4.  A lifted bigon. Consider $m=3$ and $k=1$. The black dot corresponds
to the lifted partial permutation sending $2$ to $-1$, while the white dots
corresponds to the map sending $2$ to $2$. There is a (shaded) bigon from the
black to the white dot with multiplicity $1$ at $O_{1}$ and $O_{2}$. The ray
$r_{2}$ is indicated in the picture.

Under this correspondence, $\mathrm{Cross}({\widetilde{f}})$ corresponds
${\mathbb{G}}_{m}$ orbits of pairs of points in $\mathbf{x}$ of the form
$(x_{1},y_{1}),(x_{2},y_{2})$ so that $x_{1}<x_{2}$ and $y_{1}>y_{2}$.

Thus, for each crossing, there is a unique ${\mathbb{G}}_{m}$-orbit of
embedded rectangle $r$ in $\mathbb{R}^{2}$, whose upper left corner is at
$(x_{1},y_{1})$, and whose lower right corner is at $(x_{2},y_{2})$. The graph
of ${\widetilde{f}}_{\langle i,j\rangle}$ is the subset of
$\mathbb{Z}\times\mathbb{Z}$ obtained from ${\widetilde{\mathbf{x}}}$ (the
graph of ${\widetilde{f}}$) by removing the ${\mathbb{G}}_{m}$-orbits of
$(x_{1},y_{1})$ and $(x_{2},y_{2})$, and replacing them with the
${\mathbb{G}}_{m}$ orbits of $(x_{1},y_{2})$ and $(x_{2},y_{1})$.

The condition that $\mathrm{cross}({\widetilde{f}}_{\langle
i,j\rangle})=\mathrm{cross}({\widetilde{f}})-1$ is equivalent to the condition
that $r$ is empty: i.e. it does not contain any points in $\mathbf{x}$ in its
interior. (This is equivalent to the condition that the ${\mathbb{G}}_{m}$
translates of $r$ are a collection of disjoint rectangles, none of which
contains a component of ${\widetilde{x}}$ in its interior.)

######  Lemma 4.6.

Given a lifted partial permutation ${\widetilde{f}}$ and graph
${\widetilde{\mathbf{x}}}$, there is a one-to-one correspondence between the
resolutions of the crossings in ${\widetilde{f}}$ and
${\mathbb{G}}_{m}$-orbits of embedded rectangles in $\mathbb{R}^{2}$ whose
upper left and lower right corners are on ${\widetilde{x}}$. Moreover, the
following conditions are equivalent:

  1. (1)

Each (or any) rectangle in $\mathbb{R}^{2}$ in the ${\mathbb{G}}_{m}$ orbit is
empty.

  2. (2)

The image of the rectangle in $\mathcal{H}$ is an empty bigon (in the case
where ${\mathbb{G}}_{m}$-orbit of the rectangle has isotropy group
${\mathbb{Z}}/2{\mathbb{Z}}$) or an empty rectangle (when the isotropy group
is trivial).

  3. (3)

The number of crossings in the resolution is one less than the number of
crossings in ${\widetilde{f}}$.

Proof. It was already noted that the crossings in ${\widetilde{f}}$ correspond
to embedded rectangles in ${\mathbb{R}}^{2}$, from ${\widetilde{f}}$ to
${\widetilde{g}}$. The rectangle is empty precisely when
$\mathrm{cross}({\widetilde{g}})=\mathrm{cross}({\widetilde{f}})-1$. The
isotropy group of the ${\widetilde{G}}_{m}$-orbit of any rectangle is either
trivial or ${\mathbb{Z}}/{2}{\mathbb{Z}}$. The orbits of empty rectangles
project to either empty rectangles or empty bigons in $\mathcal{H}$. Note that
$\partial r$ is embedded precisely when the interior of $r$ does not contain
any points that are equivalent to the corners of $r$; see Figure 5 for an
example where $\partial r$ is not embedded.

$O_{1}$$O_{2}$$O_{3}$$O_{2}$$O_{2}$$O_{3}$$O_{1}$

Figure 5.  Projection of a nonempty rectangle. We have a non-empty rectangle
on the left which projects to the domain in $\mathcal{H}$ pictured on the
right.

The Heegaard diagram $\mathcal{H}_{m,k}$ is nice in the sense of Sarkar; thus,
by [24], the differentials in ${\rm{CF}}^{-}(\mathcal{H}_{m,k})$ count empty
bigons and rectangles. Thus, we could use this and Lemma 4.6 to deduce an
identification of chain complexes
${\rm{CF}}^{-}(\mathcal{H}_{m,k})\cong{\mathcal{P}}(m,k)$. Instead, we invest
a little more work in understanding the combinatorics of $\mathcal{H}$ to give
an alternative argument.

There is a partial ordering on Heegaard states: we write
$\mathbf{x}\geq\mathbf{y}$ if there is a
$\phi\in\pi_{2}(\mathbf{x},\mathbf{y})$ all of whose local multiplicities are
non-negative; with strict inquality $\mathbf{x}>\mathbf{y}$ if $\phi$ has
positive local multiplicity somewhere.

######  Lemma 4.7.

If $({\widetilde{f}},{\widetilde{S}})$ and $({\widetilde{g}},{\widetilde{T}})$
are two lifted partial permutations, and $\mathbf{x}$ and $\mathbf{y}$ be
their corresponding Heegaard states. The following conditions are equivalent:

  1. (P-1)

$\mathbf{x}\geq\mathbf{y}$

  2. (P-2)

For each $(i,j)$,

$\\#\\{a\in\mathbb{Z}\mid
a<i,{\widetilde{g}}(a)<j<{\widetilde{f}}(a)\\}\geq\\#\\{a\in\mathbb{Z}\mid
a<i,{\widetilde{g}}(a)>j>{\widetilde{f}}(a)\\}.$

  3. (P-3)

There is a sequence of crossings in ${\widetilde{f}}$ with the property that
${\widetilde{g}}$ is obtained from ${\widetilde{f}}$ by resolving those
crossings.

Moreover, if $\mathbf{x}\geq\mathbf{y}$, then

(4.2)
$\mu(\mathbf{x})-\mu(\mathbf{y})=\\#\mathrm{cross}({\widetilde{f}})-\\#\mathrm{cross}({\widetilde{g}});$

Proof. As in the proof of Lemma 4.5, we can interpret

$\\#\\{a\in\mathbb{Z}\mid
a<i,{\widetilde{g}}(a)<i<{\widetilde{f}}(a)\\}-\\#\\{a\in\mathbb{Z}\mid{\widetilde{g}}(a)>j>{\widetilde{f}}(a)\\}$

as the local multiplicity of $\phi$ at the point
$(i+\frac{1}{2},j+\frac{1}{2})/{\mathbb{G}}_{m}$. It follows at once that
Properties (P-1) and (P-2) are equivalent.

Property (P-3) clearly implies Property (P-2). To see that
(P-2)$\Rightarrow$(P-3) we argue as follows. When $\mathbf{x}=\mathbf{y}$, the
result is obvious. Suppose $\mathbf{x}>\mathbf{y}$, then there is some
$i_{1}<i_{2}$ so that:

${\widetilde{f}}(i_{1})>j>{\widetilde{g}}(i_{2})\qquad{\text{and}}\qquad{\widetilde{f}}(i_{1})<j.$

(In particular, $\langle i_{1},i_{2}\rangle$ is a crossing in
${\widetilde{f}}$.) Choose $i_{2}>i_{1}$ minimal with this property, and let
${\widetilde{x}}^{\prime}$ correspond to ${\widetilde{f}}_{\langle
i_{1},i_{2}\rangle}$. Minimality of $i_{2}$ ensures that
$\mathbf{x}>\mathbf{x}^{\prime}$ and $\mathbf{x}^{\prime}\geq\mathbf{y}$. It
is easy to see that $\mathbf{x}^{\prime}\geq\mathbf{y}$. Indeed, the sequence
is constructed Since
$\mathrm{cross}(\mathbf{x}^{\prime})=\mathrm{cross}(\mathbf{x})-1$, this
process must terminate after at most $\mathrm{cross}(\mathbf{x})$ steps.

If $\mathbf{x}^{\prime}$ is obtained from $\mathbf{x}$ by resolving single
crossing, and $\phi\in\pi_{2}(\mathbf{x},\mathbf{x}^{\prime})$ is the
corresponding domain in $\mathcal{H}$, then $\mu(\phi)=1$. This is true
because $\phi$ is a bigon or a rectangle (Lemma 4.6); both of these are easily
seen to have Maslov index one. The Maslov index is additive under
jutapositions, and the algorithm establishing (P-2)$\Rightarrow$(P-3) gave a
sequence $\\{{\widetilde{f}}_{i}\\}_{i=1}^{n}$ with
${\widetilde{f}}_{1}={\widetilde{f}}$, ${\widetilde{f}}_{n}={\widetilde{g}}$,
and $\mu(\mathbf{x}_{i})-\mu(\mathbf{x}_{i+1})=1$,
$\mathrm{cross}({\widetilde{f}}_{i})-\mathrm{cross}({\widetilde{f}}_{i+1})=1$.
Equation (4.2) follows.

######  Proposition 4.8.

There is an isomorphism of chain complexes
${\rm{CF}}^{-}(\mathcal{H}_{m,k})\cong{\mathcal{P}}(m,k)$.

Proof. By definition, the differential on ${\rm{CF}}^{-}(\mathcal{H}_{m,k})$
counts $\phi\in\pi_{2}(\mathbf{x},\mathbf{y})$ with $\phi\geq 0$ and
$\mu(\phi)=1$. Equation (4.2) identifies these with the
${\mathbf{G}}_{m}$-orbits of empty rectangles in $\mathbb{C}$ which in turn,
by Lemma 4.6 identifies such rectangles with crossings in the diagram for
$\mathbf{x}$ whose resolution drops the crossing number by exactly one. Lemma
4.5 identifies the coefficients in $\partial^{-}$ for ${\rm{CF}}^{-}$ with the
coefficients in $\partial$ for ${\mathcal{P}}(m,k)$.

##  5\. Triples

We construct the Heegaard triple corresponding to wrapping. Once again, this
will be drawn as a quotient of $\mathbb{R}^{2}$ by ${\mathbb{G}}$, with the
$O_{i}$ markings along the diagonal line with half-integer coordinates labeled
as before. Now, we have three sets of lines, $\widetilde{\mathbf{\alpha}}$,
$\widetilde{\mathbf{\beta}}$, and $\widetilde{\mathbf{\gamma}}$. As before,
$\\{\widetilde{\alpha}_{i}=i\times\mathbb{R}\\}_{i\in\mathbb{Z}}$ and
$\\{\widetilde{\gamma}_{i}=\mathbb{R}\times i\\}_{i\in\mathbb{Z}}$ (i.e. they
are the $\widetilde{\mathbf{\beta}}$ from before). We choose the
$\widetilde{\mathbf{\beta}}$ with the following properties:

  1. (${\mathcal{H}}$-1)

The set $\widetilde{\mathbf{\beta}}$ is ${\mathbb{G}}_{m}$-invariant.

  2. (${\mathcal{H}}$-2)

The slope of each line in $\widetilde{\mathbf{\beta}}$ is $-1$.

  3. (${\mathcal{H}}$-3)

The ${\mathbb{G}}_{m}$-orbits of the $\widetilde{\mathbf{\beta}}$ consist of
$m$ lines so that $\widetilde{\beta}_{i}$ and $\widetilde{\beta}_{i+1}$ are
separated by $O_{i}$.

  4. (${\mathcal{H}}$-4)

There is no triple-intersection point between $\widetilde{\alpha}_{i}$,
$\widetilde{\beta}_{j}$, and the diagonal.

$O_{2}$$O_{2}$$O_{3}$$O_{4}$$O_{1}$$O_{4}$$O_{1}$$O_{2}$$O_{3}$$O_{3}$$O_{3}$$O_{2}$$O_{1}$$O_{2}$$O_{4}$$O_{3}$

Figure 6.  Heegaard triple. At the left, the (wrapped) Heegaard triple. At the
right, the lift of the diagram on the left to $\mathbb{R}^{2}$. (Note that the
$O$ markings are displayed here as horizontal; this horizontal line is to be
viewed as the diagonal.)

Analogous to Definition 4.3, given Heegaard states $\mathbf{x}$, $\mathbf{y}$,
and $\mathbf{z}$ for ${\mathcal{H}}_{\alpha\beta}$,
${\mathcal{H}}_{\beta\gamma}$, and ${\mathcal{H}}_{\alpha\gamma}$
respectively, we can consider the set of two-chains
$\psi\in\mathcal{D}(\mathbf{x},\mathbf{y},\mathbf{z})$ so that the components
of $\mathbf{x}$ are initial $\alpha-\beta$ corners, components of $\mathbf{y}$
are initial $\beta-\gamma$ corners, and components of $\mathbf{z}$ are
terminal $\alpha-\gamma$ corners.

######  Definition 5.1.

Let $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$ be three Heegaard states. We
say that the states are triangularly connected if their
${\mathbb{G}}_{m}$-equivariant lifts ${\widetilde{\mathbf{x}}}$,
${\widetilde{\mathbf{y}}}$, and ${\widetilde{\mathbf{z}}}$ in $\mathbb{R}^{2}$
admit $k$ triangles in ${\mathbb{\mathbb{R}}}^{2}$, oriented so that the they
have sides in $\alpha$-$\mathbf{\beta}$-$\mathbf{\gamma}$ in counterclockwise
order, whose ${\mathbb{G}}_{m}$ orbits have corners exactly at
$\widetilde{\mathbf{x}}$, $\widetilde{\mathbf{y}}$, and
$\widetilde{\mathbf{z}}$. Taking the quotients of the triangles gives a domain
$\psi\in\mathcal{D}(\mathbf{x},\mathbf{y},\mathbf{z})$.

Note that there is a weaker notion: one can ask whether three Heegaard states
$\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$, can be connected by a Whitney
triangle, a continuous map from the triangle $\psi\colon
T\to\mathrm{Sym}^{k}(\mathcal{H})$, which maps three edges of the triangle to
the tori
${\mathbb{T}}_{\alpha}=\alpha_{1}\times\dots\times\alpha_{k}\subset\mathrm{Sym}^{k}(\mathcal{H})$,
${\mathbb{T}}_{\beta}$, and ${\mathbb{T}}_{\gamma}$, so that the the three
vertices are mapped to $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$. For
example, the elementary region containing $O_{4}$ in Figure 6 is a triangle in
$\mathcal{H}$, and if we think of its three corners as Heegaard states (with
$k=1$), these three states are connected by a Whitney triangle, which is
double-covered by the elementary hexagon containing $O_{4}$ on the right in
Figure 6. Thus, those three states are not triangularly connected in the sense
of Definition 5.1, though they can be connected by a Whitney triangle. Note
that each Whitney triangle gives rise to a domain
$\mathcal{D}(\mathbf{x},\mathbf{y},\mathbf{z})$.

######  Lemma 5.2.

There is a one-to-one correspondence between:

  * •

triples of lifted partial permutations $({\widetilde{f}},{\widetilde{S}})$,
$({\widetilde{g}},{\widetilde{T}})$ with
${\widetilde{T}}={\widetilde{f}}({\widetilde{S}})$ and
$({\widetilde{g}}\circ{\widetilde{f}},{\widetilde{S}})$

  * •

triangularly connected triples of Heegaard states
$\mathbf{x}\in\mathbf{S}({\mathcal{H}}_{\alpha\beta})$,
$\mathbf{y}\in\mathbf{S}({\mathcal{H}}_{\beta\gamma})$,
$\mathbf{z}\in\mathbf{S}({\mathcal{H}}_{\alpha\gamma})$.

Moreover, under this correspondence

(5.1)
$\mathfrak{w}_{i}({\widetilde{f}},{\widetilde{S}})+\mathfrak{w}_{i}({\widetilde{g}},{\widetilde{T}})=\mathfrak{w}_{i}({\widetilde{g}}\circ{\widetilde{f}},{\widetilde{S}})+\\#(O_{i}\cap\psi).$

Proof. Given $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$, their corresponding
lifted partial permutations ${\widetilde{f}}$, ${\widetilde{g}}$, and
${\widetilde{h}}$ are characterized by

${\widetilde{\mathbf{x}}}=\bigcup\widetilde{\alpha}_{s}\cap\widetilde{\beta}_{{\widetilde{f}}(s)},\qquad{\widetilde{\mathbf{y}}}=\bigcup\widetilde{\beta}_{t}\cap\widetilde{\beta}_{{\widetilde{g}}(s)},\qquad{\widetilde{\mathbf{x}}}=\bigcup\widetilde{\alpha}_{s}\cap\widetilde{\beta}_{{\widetilde{h}}(s)}.$

Thus, the condition that the components of $\widetilde{\mathbf{x}}$,
$\widetilde{\mathbf{y}}$, and $\widetilde{\mathbf{z}}$ can be connected by
triangles is precisely the condition that
${\widetilde{h}}(s)={\widetilde{g}}\circ{\widetilde{f}}$.

To establish Equation (5.1), we argue as follows.

Consider a coordinate $x\in\widetilde{\alpha}_{i}\cap\widetilde{\beta}_{j}$.
By Condition (${\mathcal{H}}$-4), the lines $\widetilde{\alpha}_{i}$ and
$\widetilde{\beta}_{j}$, and the diagonal divide $\mathbb{R}^{2}$ into seven
regions, one of which is a compact region – indeed, it is a triangle triangle
$T_{x}$, Let $O_{i}(T_{x})$ be $\frac{1}{2}$ times the number of $O_{i}$
appears in $T_{x}$. (Note that each occurence of $O_{i}$ appears on the
boundary of $T_{i}$, hence the factor of $\frac{1}{2}$.)

Given a Heegaard state ${\widetilde{\mathbf{x}}}$, choose any unordered set of
$m$ points $\\{x_{1},\dots,x_{m}\\}\subset\mathbb{R}^{2}$ whose
${\mathbb{G}}_{m}$-orbit is $\mathbf{x}$. It is elementary to see that

$\mathfrak{w}_{i}(\mathbf{x})=\sum_{j=1}^{m}O_{i}(T_{x_{j}}).$

Equation (5.1) is obtained by counting points in the plane, divided into cases
according to how the diagonal line intersects each triangle. Specifically,
after applying an element of ${\mathbb{G}}_{m}$ if necessary, we can assume
that the $\alpha$-$\gamma$ corner of the triangle is on the upper right. There
are now four remaining cases, according to the number of vertices of the
triangle which lie above the diagonal line: this can be any number between $0$
and $3$. The cases are illustrated in Figure 7.

For the case on the left on that figure (where the triangle is entirely below
the diagonal line), the weight of $\mathbf{x}$ counts the $O$ markings on the
diagonal boundary of $B\cup C$ or, equivalently, $C$; the weight of
$\mathbf{y}$ counts the markings on the diagonal boundary of $A$; and the
weigth of $\mathbf{z}$ counts $O$ markings on the diagonal boundary of $A\cup
B$; more succinctly,

$\mathfrak{w}_{i}(\mathbf{x})=O_{i}(B)+O_{i}(C)\qquad\mathfrak{w}_{i}(\mathbf{y})=O_{i}(A)\qquad\mathfrak{w}_{i}(\mathbf{z})=O_{i}(A)+O_{i}(B).$

Since in this case the number of $O_{i}$ in the triangle is given by
$O_{i}(C)=0$, Equation (5.1) follows. The other three cases are:

$\mathfrak{w}_{i}(\mathbf{x})=O_{i}(B)+O_{i}(C)\qquad\mathfrak{w}_{i}(\mathbf{y})=O_{i}(A)\qquad\mathfrak{w}_{i}(\mathbf{z})=O_{i}(B).$
$\mathfrak{w}_{i}(\mathbf{x})=O_{i}(C)\qquad\mathfrak{w}_{i}(\mathbf{y})=O_{i}(A)+O_{i}(B)\qquad\mathfrak{w}_{i}(\mathbf{z})=O_{i}(B).$
$\mathfrak{w}_{i}(\mathbf{x})=O_{i}(B)\qquad\mathfrak{w}_{i}(\mathbf{y})=O_{i}(A)+O_{i}(C)\qquad\mathfrak{w}_{i}(\mathbf{z})=O_{i}(B)+O_{i}(C);$

and the number of $O_{i}$ markings in the triangles are

$O_{i}(A)+O_{i}(C)\qquad O_{i}(A)+O_{i}(C)\qquad O_{i}(A)=0$

respectively. Thus, in the remaining three cases, Equation (5.1) holds.

$\beta$$A$$C$$B$$\gamma$$\alpha$$\beta$$\beta$$\gamma$$\alpha$$\gamma$$\alpha$$\gamma$$\alpha$$\mathbf{y}$$\mathbf{z}$$\mathbf{y}$$\mathbf{z}$$B$$\mathbf{y}$$\mathbf{x}$$\mathbf{x}$$\mathbf{y}$$\mathbf{z}$$C$$B$$A$$B$$C$$\beta$$\mathbf{x}$$\mathbf{x}$$\mathbf{z}$$B$$A$$A$$C$

Figure 7.  Counting $O$’s in triangles. The diagonal dashed line represents
the diagonal.

Consider two-chains for $\mathcal{H}$ analogous to Definition 4.3, now using
all three sets of curves $\mathbf{\alpha}$, $\mathbf{\beta}$, and
$\mathbf{\gamma}$. These curves divide $\mathcal{H}$ into elementary domains,
which are polygons. As in [15, 22, 9], the euler measure each polygon has an
Euler measure: if the number of sides is $m$, the euler measure of the
corresponding polygon is given by $1-\frac{m}{4}$. Extend the Euler measure
$e$ linearly to all two-chains, and denote the resulting function $e$.

$O_{1}$$O_{3}$$O_{4}$$O_{1}$$O_{2}$$O_{3}$$O_{4}$$O_{2}$

Figure 8.  Triangles. At the left: a triangle with Euler measure $3/4$; at the
right, a triangle with Euler measure $1/4$.

######  Lemma 5.3.

Let $\psi\in\pi_{2}(\mathbf{x},\mathbf{y},\mathbf{z})$ be a positive domain.
Then, the euler measure of $\psi$ is computed by

(5.2) $e(\psi)=\frac{k}{4}+\frac{O_{1}(\psi)}{2}+\frac{O_{m}(\psi)}{2}.$

Proof. Loosely speaking, we claim that any positive domain can be cut along
the $\alpha$, $\beta$, and $\gamma$ lines to give $k$ triangles (each with
Euler measure $1/4$), $O_{i}(\psi)$+$O_{m}(\psi)$ bigons (each with Euler
measure $1/2$), and many rectangles (each with Euler measure $0$). We make
this claim precise as follows.

Consider the group of cornerless $2$-chains with compact support, generalizing
Definition 4.3 in a straightforward way. This group is evidently $0$. But
there is a non-trivial group ${\mathcal{A}}$ of $2$-chains which are allowed
$\beta$-$\gamma$ and $\alpha$-$\gamma$ corners, but no $\alpha$-$\beta$
corners. More precisely, at each intersection of $\alpha$ with $\beta$, we
require that the alternating sum of the local multiplicities of the four
quadrants add up to zero; i.e. $A+D=B+C$, in the conventions of Figure 3.

That group is non-trivial: for example, fix a $\beta$\- or $\gamma$-segment
that connects a pair of consecutive $\beta$-lines; and fix another $\beta$\-
or $\gamma$-segment that connects the same pair of $\beta$-lines. The four
segments enclose some region in the plane: when all four segments are disjoint
that region is a quadrilateral, when they intersect, it is a difference of two
triangles. See Figure 9. When it happens that these four segments are permuted
by some ${\mathbb{G}}_{m}$ action, We call these regions fundamental bigons;
otherwise, we call them fundamental rectangles. Fundamental bigons project to
actual bigons in $\mathcal{H}$; while fundamental rectangles project to
rectangles (or differences of two triangles).

$O_{2}$$O_{3}$$O_{4}$$O_{1}$$O_{4}$$O_{1}$$O_{2}$$O_{3}$$O_{3}$$O_{2}$$O_{2}$

Figure 9.  Fundamental regions. At the left, we have a fundamental bigon, a
fundamental rectangle, and a difference of two triangles. At the right, we
have their lifts.)

It is elementary to verify that ${\mathcal{A}}$ is generated by the
fundamental bigons and rectangles. Moreover, the quantity
$e(D)-\frac{O_{1}(D)}{2}-\frac{O_{m}(D)}{2}$ vanishes on all fundamental
bigons and rectangles.

Let ${\mathcal{B}}$ the space of domains with $\alpha$-$\gamma$-terminal
corner (in the sense of Definition 4.3) at some component of
$\widetilde{\mathbf{z}}$. This is clearly an affine space of ${\mathcal{A}}$.
Moreover, given any $\widetilde{\mathbf{z}}$, we can draw some positive union
of triangles $T_{0}(\mathbf{z})\in B$. Clearly, the Euler measure is $k/4$.
Now, the map
${\widetilde{e}}=e-\frac{O_{1}}{2}-\frac{O_{m}}{2}\colon\mathcal{D}(\mathbf{x},\mathbf{y},\mathbf{z})\to\mathbb{Z}/4$
factors through the space of ${\mathcal{A}}$-orbits in ${\mathcal{B}}$,
${\mathcal{B}}/{\mathcal{A}}$, since ${\widetilde{e}}$ vanishes on
${\mathcal{A}}$. Since $\psi$ and $T_{0}(\mathbf{z})$ have the the same image
in ${\mathcal{B}}/{\mathcal{A}}$, and $e(T_{0}(\mathbf{x}))=k/4$, it follows
that $e(\psi)=k/4$.

The relevance of Lemma 5.3 stems from Sarkar’s computation of the Maslov index
of a triangle (or more generally, a Whitney $n$-gon) [23]. One of his
formulas, generalizing a theorem of Rasmussen when $n=2$ [22], gives

(5.3) $\mu(\psi)=2e(\psi)-\frac{k(n-2)}{2}+\\#(\psi\cap\Delta).$

Here, $\\#(\psi\cap\Delta)$ denotes the algebraic intersection number of of
$\psi$ with the big diagonal in $\mathrm{Sym}^{k}(\mathcal{H})$.

######  Lemma 5.4.

If $\psi\in\mathcal{D}(\mathbf{x},\mathbf{y},\mathbf{z})$ has $\mu(\psi)=0$,
and $\psi$ has a pseudo-holomorphic representative, then $\mathbf{x}$,
$\mathbf{y}$, and $\mathbf{z}$ are triangularly connected, in the sense of
Definition 5.1.

Proof. We begin with some remarks. Let $\mathbf{x}=\\{x_{1},\dots,x_{k}\\}$,
$\mathbf{y}=\\{y_{1},\dots,y_{k}\\}$, $\mathbf{z}=\\{z_{1},\dots,z_{k}\\}$.
Suppose there are $k$ triangles $\psi_{i}\colon T\to\mathcal{H}$ with corners
at $x_{i}$, $y_{i}$, and $z_{i}$, and edges mapping to some $\alpha$-curve,
some $\beta$-curve, and some $\gamma$-curve. We can then form
$\psi=\psi_{1}\times\dots\times\psi_{k}\colon
T\to\mathrm{Sym}^{k}(\mathcal{H})$, to get a Whitney triangle. We call such
Whitney triangles decomposable. Not every Whitney triangle is decomposable,
but Whitney triangles which are disjoint from the diagonal $\Delta$ are.

Now, combining Equation (5.2) with Equation (5.3), we see that for any
triangle,

$\mu(\psi)=O_{1}(\psi)+O_{m}(\psi)+\\#(\psi\cap\Delta).$

If $\psi$ has a pseudo-holomorphic representative, all three of the terms on
the right are non-negative. Indeed, if $\mu(\psi)=0$, then
$\\#(\psi\cap\Delta)=0$; and indeed, $\psi$ is disjoint from the diagonal; and
hence it is decomposable. Next, observe that since
$O_{1}(\psi)=O_{m}(\psi)=0$, each factor $\psi_{i}\colon T\to\mathcal{H}$ in
the factor of decomposition of $\psi$ maps to
$\mathcal{H}\setminus\\{O_{1},O_{m}\\}$; i.e. the locus where the quotient map
$\mathbb{R}^{2}\to\mathcal{H}=\mathbb{R}^{2}/{\mathbb{G}}_{m}$ is a covering
space. Thus, we can lift each factor $\psi_{i}$ to obtain maps
$\\{{\widetilde{\psi}}_{i}\colon T\to\mathbb{R}^{2}\\}_{i=1}^{k}$, showing
that $\mathbf{x}$, $\mathbf{y}$ and $\mathbf{z}$ are triangularly connected.

######  Lemma 5.5.

Suppose that $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$ are triangularly
connected Heegaard states, let
$\psi\in\mathcal{D}(\mathbf{x},\mathbf{y},\mathbf{z})$ be the domain in
$\mathcal{H}$ connecting them, and choose triangles
$\\{\widetilde{\psi}_{i}\colon T\to\mathbb{R}^{2}\\}_{i=1}^{k}$ whose
projection to $\mathcal{H}$ gives
$\psi\in\mathcal{D}(\mathbf{x},\mathbf{y},\mathbf{z})$. Then,

(5.4)
$\\#(\psi\cap\Delta)=\frac{1}{2}\sum_{i,j}^{k}\sum_{g\in{\mathbb{G}}_{m}}\\#\left({\widetilde{\psi}}_{i}\times
g\cdot{\widetilde{\psi}}_{j}\right)\cap\Delta,$

where the right-hand-side is to be interpreted as an intersection with
$\Delta\subset\mathrm{Sym}^{2}(\mathbb{R}^{2})$.

Proof. The intersection of $\psi=\bigtimes_{i=1}^{k}\psi_{i}$ with $\Delta$ in
$\mathrm{Sym}^{k}(\mathcal{H})$ is identified with ${\mathbb{G}}_{m}$-orbits
of data $g_{1},g_{2}\in{\mathbb{G}}_{m}$,
$\tau_{1},\tau_{2}\in{\mathbb{R}}^{2}$ with
$g_{1}\cdot{\widetilde{\psi}}_{i}(\tau_{1})=g_{2}\cdot{\widetilde{\psi}}_{j}(\tau_{2})$,
where the pair of triples $(g_{1},\tau_{1},{\widetilde{\psi}}_{i})$ and
$(g_{2},\tau_{2},{\widetilde{\psi}}_{j})$ is thought of as unordered. Breaking
symmetry, we can think of this as half the count of
$\tau_{1},\tau_{2}\in\mathbb{R}^{2}$ and $g\in{\mathbb{G}}_{m}$ and pairs $i$,
$j$, so that
${\widetilde{\psi}}_{i}(\tau_{1})=g\cdot{\widetilde{\psi}}_{j}(\tau_{2})$.
This is the count on the right-hand-side of Equation (5.4).

######  Lemma 5.6.

Suppose that $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$ are three
triangularly connected Heegaard states, corresponding to lifted partial
permutations Let $({\widetilde{f}},{\widetilde{S}})$,
$({\widetilde{f}},{\widetilde{T}})$, and
$({\widetilde{g}}\circ{\widetilde{f}},{\widetilde{S}})$. Let
$\psi\in\mathcal{D}(\mathbf{x},\mathbf{y},\mathbf{z})$ be the corresponding
domain in ${\mathcal{H}}_{\alpha\beta\gamma}$. Suppose moreover that
$O_{1}(\psi)=O_{m}(\psi)=0$. Then,

(5.5)
$\mathrm{cross}({\widetilde{f}},{\widetilde{S}})+\mathrm{cross}({\widetilde{g}},{\widetilde{T}})-\mathrm{cross}({\widetilde{g}}\circ{\widetilde{f}},{\widetilde{S}})=\mu(\psi).$

Proof. Each strand in ${\widetilde{f}}$ corresponds to a Heegaard state for
${\mathcal{H}}_{\alpha\beta}$; each strand in ${\widetilde{g}}$ corresponds to
a Heegaard state for ${\mathcal{H}}_{\beta\gamma}$, and each composite strand
corresponds to a triangle in ${\mathcal{H}}_{\alpha\beta\gamma}$, which in
turn corresponds to a ${\mathbb{G}}_{m}$-orbit of a triangle in
$\mathbb{R}^{2}$. Up to the action of ${\mathbb{G}}_{m}$, we can assume that
the triangle has its $\alpha-\gamma$ vertex above its $\beta$-line, as in
Figure 10.

$\beta$$\mathbf{x}$$Y-Z$$Y-Z$$\mathbf{y}$$\mathbf{z}$$\gamma$$X-Z$$X+Y$$\alpha$$X-Z$

Figure 10.  Maslov index of triangles.

Extending the lines of each triangle, we see that each triangle splits the
plane into $7$ regions, one of which is the (compact) triangle itself. The
left-hand-side of Equation (5.5) can be interpreted as a sum, over each
triangle $T$ of a count of all the other components of type $X$, $Y$, or $Z$,
in the seven regions, counted with multiplicity $\pm 1$ or $0$, as indicated
in Figure 10. This count, in turn, can be organized according to all other
triangles $T^{\prime}$ connecting three auxiliary generators, taken with given
multiplicity.

Note that $T^{\prime}$ is the image of $T$ under an an affine transformation
$L$ of $\mathbb{R}^{2}$ which is a composition of a real rescaling (which
includes a possible $180^{\circ}$ rotation) composed with a translation. Note
that $L$ is either a translation or it has a unique fixed point.

We claim that $T\cup T^{\prime}$ hits the diagonal precisely when $L$ has a
fixed point, and that fixed point is contained in the interior of $L$ (or
$L^{\prime}$); and . moreover, in that case
$\\#((\psi_{T}\times\psi_{T}^{\prime})\cap\Delta)=2$. To see this, note that
the map to $\mathrm{Sym}^{2}(\mathbb{R}^{2})$ is modeled on the map
$t\mapsto\\{t,L(t)\\}$, which in turn corresponds to the monic polynomial
$z^{2}-(t+L(t))z+tL(t)$, whose discriminant is
$(t+L(t))^{2}-4tL(t)=(t-L(t))^{2}$, which vanishes to order $2$ at the fixed
point of $L$.

We now verify that the contribution of the triangle pair $T,T^{\prime}$ to the
left-hand-side of Equation (5.5) coincides with this intersection number with
the diagonal, by looking at the possible cases for the two triangles. After
possibly switching roles of $T$ and $T^{\prime}$, we can assume that the
number of vertices of $T^{\prime}$ in $T$ is greater or equal to the number of
vertices of $T$ in $T^{\prime}$. There are the following possibilities:

  * •

$T$ contains no vertices of $T^{\prime}$. In this case, either $T^{\prime}$
and $T$ are disjoint, or they can overlap as pictured in the first picture of
Figure 11, in which case the local contribution is $2$.

  * •

$T$ contains exactly one vertex of $T^{\prime}$. This can happen in two
inequivalent ways: either $T^{\prime}$ is a translate of $T$, in which case
the contribution is $0$; or $T^{\prime}$ is not obtained as a translate of
$T$, in which case the local contribution is $2$, as can be seen by
considering the six cases in the second picture of Figure 11

  * •

$T$ contains exactly two vertices of $T^{\prime}$ in its interior, as shown in
the three cases on the third picture of Figure 11.

  * •

$T$ contains all three vertices of $T^{\prime}$ in its interior, as shown in
the fourth picture of Figure 10, in which case the local contribution is $2$.

$X-Z$$Y-Z$$X-Z$$\mathbf{z}$$\mathbf{y}$$X+Y$$\gamma$$\alpha$$Y-Z$$\beta$$\mathbf{x}$$X-Z$$Y-Z$$X-Z$$\mathbf{z}$$\mathbf{y}$$X+Y$$\gamma$$\beta$$\mathbf{x}$$X-Z$$Y-Z$$X-Z$$\mathbf{z}$$\mathbf{y}$$X+Y$$\gamma$$\alpha$$Y-Z$$\beta$$\mathbf{x}$$X-Z$$Y-Z$$X-Z$$\mathbf{z}$$\mathbf{y}$$\gamma$$\alpha$$Y-Z$$\beta$$\mathbf{x}$$\alpha$$Y-Z$$X+Y$

Figure 11.  Triangle pairs. Here are the various combinatorial ways two
triangles can interact, so that their contributions to (both sides of)
Equation (5.5) are non-zero. (Think of $T$ as the large triangle; various
choices of the other triangle $T^{\prime}$ are indicated by the smaller
triangles.)

In view of Lemma 5.5, we have verified that

$\mathrm{cross}({\widetilde{f}},{\widetilde{S}})+\mathrm{cross}({\widetilde{g}},{\widetilde{T}})-\mathrm{cross}({\widetilde{g}}\circ{\widetilde{f}},{\widetilde{S}})=\\#(\Delta\cap\psi).$

Combining Equations (5.2) and 5.3, we see that

$\mu(\psi)=O_{1}(\psi)+O_{m}(\psi)+\\#(\psi\cap\Delta);$

since $O_{1}(\psi)=O_{m}(\psi)=0$, Equation (5.5).

######  Proposition 5.7.

Under the identification from Proposition 4.8, the triangle map corresponds to
composition in the pong algebra.

Proof. The triangle map counts index zero triangles, which corresponds to
triangularly connected lifted partial permutations by Lemma 5.4. Lemma 5.6 in
turn identifies the counts of index zero triangles with compositions in the
pong algebra, in view of Lemma 3.4. Moreover, Equation (5.1) identifies the
coefficients of those counts.

Implicit in the above identification is the identification
$(\mathcal{H},\mathbf{\alpha},\mathbf{\beta},\\{O_{1},\dots,O_{m}\\})\cong(\mathcal{H},\mathbf{\alpha},\mathbf{\beta},\\{O_{1},\dots,O_{m}\\})$,
which can be viewed as the identification coming from the Liouville flow.
(Compare Equation (2.3).) The corresponding map on the wrapped complexes was
denoted $\sigma$ in Equation (1.1). To promote this to a chain map, in
general, following [1], one must use a continuation map interpolating between
the complex structure used on $\mathcal{H}$, and its pull-back under the
Liouville flow.

For the Heegaard diagram
$(\mathcal{H},\mathbf{\alpha},\mathbf{\beta},\\{O_{i}\\}_{i=1}^{m})$, though
the continuation map between any two admissible paths of almost-complex
structures is simply the identity map, since the continuation is supported on
non-negative domains with index zero and, following Lemma 4.7 (i.e. Property
(P-3) combined with Equation (4.2)), the only such domain is the constant
domain (with $\mathbf{x}=\mathbf{y}$).

##  6\. Polygons with $n>3$ sides

We now generalize the Heegaard diagram and the Heegaard triple as follows.
Consider $\mathbb{R}^{2}$, with the $O_{i}$ marking the diagonal line with
half-integer coordinates as before. Choose $n$ sets
$\widetilde{\mathbf{\alpha}}^{1},\dots,\widetilde{\mathbf{\alpha}}^{n}$ of
lines, as follows. Choose an increasing set of angles
$\frac{\pi}{4}<\theta_{1}<\dots<\theta_{n}<\frac{5\pi}{4}$, and let
$\widetilde{\mathbf{\alpha}}^{j}$ be parallel lines in $\mathbb{R}^{n}$
forming angle $\theta_{j}$ with respect to the real axis $\mathbb{R}\times 0$,
so that two consecutive lines are separated by some $O$ marking, and so that
the intersections between the various lines $\widetilde{\alpha}^{j}_{\ell}$
within the set $\widetilde{\mathbf{\alpha}}^{j}$ are in general position.
Explicitly,

$\widetilde{\mathbf{\alpha}}^{j}_{\ell}=\\{(\ell+\epsilon^{j}(\ell)+t\cos\theta_{j},\ell+\epsilon^{j}(\ell)+t\sin\theta_{j})\\}_{t\in\mathbb{R}},$

where $\\{\epsilon^{j}\colon\mathbb{Z}\to\mathbb{R}\\}_{j=1}^{n}$ are
$G_{m}$-invariant functions with $\epsilon^{j}(\ell)<\frac{1}{2}$, so that
$\epsilon^{s}(\ell)\neq\epsilon^{t}(\ell)$ for all $s\neq t$. See Figure 12
for a non-generic picture (i.e. $\epsilon^{j}\equiv 0$ in the above formulas)
in $\mathbb{R}^{2}$, when $n=4$. Note that the picture is also rotated
90$\circ$: the line through the $O$ markings should have slope $1$.

$O_{1}$$O_{2}$$O_{3}$$O_{4}$$O_{3}$$O_{2}$$\widetilde{\mathbf{\alpha}}^{2}$$\widetilde{\mathbf{\alpha}}^{3}$$\widetilde{\mathbf{\alpha}}^{4}$$\widetilde{\mathbf{\alpha}}^{1}$

Figure 12.  Heegaard quadruple. This is the (unperturbed) multi-diagram with
$n=4$ lifted to $\mathbb{R}^{2}$. Two elements of ${\mathcal{A}}$ are shaded.

When $n=2$, a linear transformation carries this to the Heegaard diagram from
Section 4; and when $n=3$, a linear transformation carries this to the
Heegaard triple from Section 5. As in Lemma 4.1, this is the Heegaard diagram
for the $A_{\infty}$ actions on the wrapped Fukaya category.

Lemma 5.3, which was stated earlier for triangles, actually holds for
arbitrary $n$-gons:

######  Lemma 6.1.

Let $\psi_{n}\in\pi_{2}(\mathbf{x}_{1},\dots,\mathbf{x}_{n})$ be a positive
domain, then

$e(\psi_{n})=\frac{k(n-2)}{4}+\frac{O_{1}(\psi_{n})}{2}+\frac{O_{m}(\psi)}{2}.$

Proof. The argument used in the proof of Lemma 5.3 when $n=3$ can be adapted
to $n>2$, as follows.

Let ${\mathcal{A}}$ denote the Abelian group of compactly supported $2$-chains
which are required to be cornerless at all the intersections of
$\widetilde{\alpha}^{j}_{\ell}$ with
${\mathfrak{L}}^{j^{\prime}}_{\ell^{\prime}}$, provided that $j,j^{\prime}\neq
2$.

Fix Heegaard states $\mathbf{x}^{i,i+1}$ for
$(\mathcal{H},\mathbf{\alpha}^{i},\mathbf{\alpha}^{i+1})$ and
$\mathbf{x}^{n,1}$ for
$(\mathcal{H},\mathbf{\alpha}^{n},\mathbf{\alpha}^{1})$. Let ${\mathcal{B}}$
be the affine space for ${\mathcal{A}}$ with initial corners at the lifts of
$\mathbf{x}^{3,4},\mathbf{x}^{4,5},\dots,\mathbf{x}^{n-1,n},\mathbf{x}^{n,1}$;
i.e. there are no constraints placed on the intersections with
$\mathbf{\alpha}^{2}$. There are, once again, fundamental regions, which are
rectangles (possibly with a self-intersection) formed now by two segments in
$\widetilde{\mathbf{\alpha}}^{2}$ and two other segments in
$\widetilde{\mathbf{\alpha}}^{j}$ and
$\widetilde{\mathbf{\alpha}}^{j^{\prime}}$. The regions are called fundamental
bigons if the four segments are permuted by the ${\mathbb{G}}_{m}$ action,
otherwise they are called fundamental rectangles.

Like for the case of triangles, the fundamental rectangles and bigons generate
${\mathcal{A}}$; the function
${\widetilde{e}}=e-\frac{O_{1}}{2}-\frac{O_{m}}{2}$, which is defined on all
(finite) $2$-chains, vanishes on ${\mathcal{A}}$. Consider next
${\mathcal{B}}$, which is the set of (finite) $2$-chains with initial corners
at $\mathbf{x}^{3,4},\mathbf{x}^{4,5},\dots,\mathbf{x}^{n-1,n}$, and terminal
corner at $\mathbf{x}^{n,1}$. We can find representatives of
${\mathcal{B}}/{\mathcal{A}}$, which are a union of $n-2$ triangles that miss
$O_{1}$ and $O_{m}$. It is an easy computation to see that this representative
has ${\widetilde{e}}=\frac{(n-2)}{k}$.

The result is also true when $n=2$. We do not explicitly need it here, and we
leave it to the reader to supply the details.

Recall that a rigid holomorphic $n$-gon has $\mu(\phi_{n})=3-n$.

######  Proposition 6.2.

There are no rigid, holomorphic Whitney $n$-gons with $n>3$; in particular,
for any pseudo-holomorphic $n$-gon with $n>3$, $\mu(\psi_{n})\geq 0$.

Proof. Combining Lemma 6.1 with Equation (5.3), we find that

$\mu(\psi_{n})=\frac{O_{1}(\psi_{n})}{2}+\frac{O_{m}(\psi_{n})}{2}+\\#(\psi_{n}\cap\Delta).$

For pseudo-holomorphic $\psi_{n}$, all three terms on the right-hand-side are
non-negative.

Proof. [of Theorem 1.1] The identification from Theorem 1.1 consists of three
statements:

  * •

The endomorphism algebra is isomorphic with the pong algebra, as a chain
complex. This is Proposition 4.8.

  * •

The composition law on the endomorphism algebra is identified with the
multiplication on the pong algebra. This is Proposition 5.7.

  * •

Like the pong algebra, the endomorphism algebra has vanishing $\mu_{n}$ with
$n>3$. This is Proposition 6.2.

##  7\. Further

There are several variants on the constructions considered here. For example,
we could consider the cylinder $A$, equipped with $m$ vertical lines separated
by $O$-markings. The endomorphism algebra, in $\mathrm{Sym}^{k}(A)$ of the
corresponding Lagrangians can be thought of as a more symmetric, circular
analogue of the pong algebra (i.e. one without left and right walls),
discovered by Manion and Rouqier [14].

In our notation, to each integer $m\geq 1$ and $1\leq k\leq m$, one can
consider a differential graded algebra $A(m,k)$ over
$\mathbb{F}[v_{1},\dots,v_{m}]$, which could be called the asteroids algebra,
defined as follows.111The choice of terminology for our algebras can be taken
as evidence for misspent youth. (This algebra was fist considered in
unpublished joint work of the first author with Robert Lipshitz and Dylan
Thurston, when consider knot Floer homology for toroidal grid diagrams;
compare [10, 21]. It is also closely related to the “differential graded nil
Hecke algebras associated to the extended affine symmetric groups” of [14].)
Consider the circle $\mathbb{R}/m\mathbb{Z}$, equipped with with basepoints
$\\{\frac{1}{2},\dots,\frac{1}{2}+m-1\\}$ corresponding to
$v_{1},\dots,v_{m}$. An idempotent state corresponds to a $k$-element subset
of $\\{1,\dots,m\\}\subset\mathbb{R}/m\mathbb{Z}$; or, equivalently, a a
$m\mathbb{Z}$-invariant subset ${\widetilde{S}}\subset\mathbb{Z}$. A
($m\mathbb{Z}$-lifted) partial permutation, now, consists of such a subset
${\widetilde{S}}$, and a map $f\colon{\widetilde{S}}\to\mathbb{Z}$ satisfying
$f(x+m)=f(x)+m$. A crossing consists of a pair of integers $i<j$ so that
$f(i)>f(j)$. Once again, there is a Maslov grading that counts the number of
crossings. There is a multiplication map induced from composition of
$m\mathbb{Z}$-lifed partial permutations, which is set to $0$ if the Maslov
grading of the product is smaller than the sum of the Maslov gradings of the
factors. Similarly, there is a differential that resolves crossings,
containing only those terms whose resolutions have exactly one fewer crossing.
Verifying that the result is a differential graded algebra is straightforward,
and slightly simpler than the corresponding verification for the pong algebra.
(Compare [17, Section 4].) Weights are at points in
$\frac{\frac{1}{2}+\mathbb{Z}}{m\mathbb{Z}}$, with the weight of
$a\in\frac{1}{2}+\mathbb{Z}$ given by

$\frac{1}{2}(\\#\\{i\big{|}i<a<f(i)\\}+\\#\\{i\big{|}f(i)<a<i\\}).$

For example, consider the $3\mathbb{Z}$-lifted partial permutation $f$ with
domain $\\{1+3\mathbb{Z},2+3\mathbb{Z}\\}$, which is determined by

(7.1) $f(1)=6\qquad{\text{and}}\qquad f(2)=1.$

This has two crossings: the equivalence class of the pair of strands starting
at $1$ and $2$; and the pair of strands starting at $-2$ and $3$. The weight
vector is given by $(1/2,3/2,1)$. See Figure 13 for a picture.

Note also that when $m=4$ and $k=1$, this is the “peculiar algebra” of
Zibrowius [27].

$\frac{1}{2}$$\frac{3}{2}$$1$$\frac{1}{2}$$\frac{1}{2}$$1$$\frac{1}{2}$$\frac{1}{2}$$1$

Figure 13.  Asteroids diagram. At the left, an asteroids diagram for $f$ from
Equation (7.1). The other two pictures represent the terms in the differential
of the first term, both taken with multiplicity $v_{2}$.

The Heegaard diagram for the wrapped Fukaya category (analogous to the diagram
from Section 4) is a quotient of $\mathbb{R}^{2}$ by a group of translations
(rather than the group of motions ${\mathbb{G}}_{m}$ considered above).
Proposition 6.2 holds also in this case.

One can look at other configurations of Lagrangians in punctured spheres. For
instance, for a linear chain of spheres, the endomorphism algebra in the
wrapped Fukaya category does have a higher multiplication (and indeed it can
be computed, for example, with the methods of [17]).

Finally, building on [17], the pong algebra can be thought of as governing the
bordered invariants for certain types of Heegaard diagrams associated to
tangles; see [16]; compare also [12, 18]. In light of this, the present work
can be thought of as analogous to Auroux’s interpretation of bordered Floer
homology [3].

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