# Chain level loop bracket and pseudo-holomorphic disks

Kei Irie  Research Institute for Mathematical Sciences, Kyoto University,
Kyoto 606-8502, Japan / Simons Center for Geometry and Physics, State
University of New York, Stony Brook NY 11794-3636, USA (on visit)
<EMAIL_ADDRESS>
###### Abstract.

Let $\displaystyle L$ be a Lagrangian submanifold in a symplectic vector space
which is closed, oriented and spin. Using virtual fundamental chains of moduli
spaces of nonconstant pseudo-holomorphic disks with boundaries on
$\displaystyle L$, one can define a Maurer-Cartan element of a Lie bracket
operation in string topology (the loop bracket) defined at chain level. This
observation is due to Fukaya, who also pointed out its important consequences
in symplectic topology. The goal of this paper is to work out details of this
observation. Our argument is based on a string topology chain model previously
introduced by the author, and the theory of Kuranishi structures on moduli
spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-
Ohta-Ono.

###### 2010 Mathematics Subject Classification:

53D12, 53D40, 55P50

##  1\. Introduction

The study of Lagrangian submanifolds is one of central topics in symplectic
topology. In the monumental paper [11], Gromov proved that the first Betti
number of a closed Lagrangian submanifold in $\displaystyle{\mathbb{C}}^{n}$
(with the standard symplectic structure) is nonzero, using moduli spaces of
(perturbed) pseudo-holomorphic disks with boundaries on the Lagrangian
submanifold. On the other hand, string topology is the study of algebraic
structures on (homology of) loop spaces, introduced by Chas-Sullivan [1].

In this paper we discuss an application of chain level string topology
operations to the pseudo-holomorphic curve theory in symplectic topology. This
idea is due to Fukaya [6], who also pointed out its important consequences,
including a proof of Audin’s conjecture for (closed, oriented and spin)
aspherical Lagrangian submanifolds in $\displaystyle{\mathbb{C}}^{n}$, and a
complete classification of orientable, closed, prime three-manifolds admitting
Lagrangian embeddings into $\displaystyle{\mathbb{C}}^{3}$.

Let us briefly sketch the key argument in [6]. Let $\displaystyle L$ be a
closed, oriented and spin Lagrangian submanifold in
$\displaystyle{\mathbb{C}}^{n}$, and
$\displaystyle{\mathcal{L}}L:=C^{\infty}(S^{1},L)$ be the space of free loops
on $\displaystyle L$. Let $\displaystyle D:=\\{z\in{\mathbb{C}}\mid|z|\leq
1\\}$, and let $\displaystyle{\mathcal{M}}$ denote the (compactified) moduli
space of nonconstant holomorphic maps $\displaystyle(D,\partial
D)\to({\mathbb{C}}^{n},L)$ modulo $\displaystyle\text{Aut}(D,1)$. Then one can
define a map $\displaystyle\text{\rm
ev}\,_{{\mathcal{M}}}:{\mathcal{M}}\to{\mathcal{L}}L$ by
$\displaystyle\text{\rm ev}\,_{{\mathcal{M}}}(u):=u|_{\partial D}$ (strictly
speaking, this “definition” has an ambiguity up to parametrizations of loops,
however we omit this issue for the moment). Considering virtual fundamental
chain of the moduli space $\displaystyle{\mathcal{M}}$, the pair
$\displaystyle({\mathcal{M}},\text{\rm ev}\,_{{\mathcal{M}}})$ defines a
“chain” $\displaystyle x\in{\mathcal{C}}_{*}({\mathcal{L}}L)$. Here
$\displaystyle{\mathcal{C}}_{*}({\mathcal{L}}L)$ denotes the complex of
“chains” on $\displaystyle{\mathcal{L}}L$, on which the loop bracket is
defined and makes $\displaystyle{\mathcal{C}}_{*}({\mathcal{L}}L)$ a dg Lie
algebra. Since the codimension $\displaystyle 1$ boundary of
$\displaystyle{\mathcal{M}}$ consists of configurations of two disks glued at
a point, one sees that the chain $\displaystyle x$ satisfies the Maurer-Cartan
equation

(1) $\partial x-\frac{1}{2}[x,x]=0$

where $\displaystyle[\,,\,]$ denotes the loop bracket defined at chain level.
Nextly, we take a time-dependent Hamiltonian $\displaystyle H$ on
$\displaystyle{\mathbb{C}}^{n}$ which displaces $\displaystyle L$, and define
a moduli space $\displaystyle{\mathcal{N}}$ which consists of solutions of the
Cauchy-Riemann equation perturbed by $\displaystyle H$. Then, the associated
chain $\displaystyle y:=({\mathcal{N}},\text{\rm
ev}\,_{{\mathcal{N}}})\in{\mathcal{C}}_{*}({\mathcal{L}}L)$ satisfies

(2) $\partial y-[x,y]=z$

where $\displaystyle z$ is another chain whose “symplectic area zero part” is
a cycle representing the fundamental class of $\displaystyle L$. Once we
obtain chains $\displaystyle x$, $\displaystyle y$, $\displaystyle z$
satisfying equations (1) and (2), using the homotopy transfer theorem for
$\displaystyle L_{\infty}$-algebras, one can formulate an equivalent result on
homology of the free loop space (Theorem 3.2). This result has the following
remarkable consequences:

  * (i):

If $\displaystyle L$ is aspherical, then there exists $\displaystyle a\in
H_{1}(L:{\mathbb{Z}})$ with Maslov index $\displaystyle 2$ and positive
symplectic action (Corollary 3.5).

  * (ii):

If $\displaystyle n=3$ and $\displaystyle L$ is prime as a three-manifold,
then $\displaystyle L$ is diffeomorphic to $\displaystyle S^{1}$ times a
closed surface (Corollary 3.6).

(i) in particular confirms Audin’s conjecture for Lagrangian tori in
$\displaystyle{\mathbb{C}}^{n}$, and (ii) gives a complete classification of
orientable, closed, prime three-manifolds admitting Lagrangian embeddings into
$\displaystyle{\mathbb{C}}^{3}$; see Section 3 for details and previous
related works.

In the above argument, $\displaystyle{\mathcal{C}}_{*}({\mathcal{L}}L)$
denotes the complex of “chains” on $\displaystyle{\mathcal{L}}L$, on which the
loop bracket is defined and makes it a dg Lie algebra. It is a highly
nontrivial technical problem to define such chain models of the free loop
space, in particular those compatible with virtual techniques in the pseudo-
holomorphic curve theory. Partly due to this issue, in spite of the importance
of its consequences, full details of the above argument have not been
available so far.

In [12], the author developed foundations for part of chain level algebraic
structures (specifically, Batalin-Vilkovisky structure) in string topology,
using de Rham chains on spaces of Moore loops with arbitrarily many marked
points. The goal of this paper is to combine techniques from [12] with the
theory of Kuranishi structures [10] to work out details of the argument
sketched above.

Now we describe the structure of this paper. The goal of the first part
(Sections 2–6) is to state the main result and introduce our setup in string
topology. Section 2 explains some preliminaries on $\displaystyle
L_{\infty}$-algebras, in particular the homotopy transfer theorem. Section 3
states the main result (Theorem 3.2) on homology of the free loop space. We
also recall a few applications in symplectic topology from [6]. In Sections
4–6, we introduce our setup in string topology, following [12] with minor
modifications, and reduce Theorem 3.2 to a chain-level statement (Theorem
6.1). Further details will be explained in the last paragraph of Section 3.

The goal of the second part (Sections 7–10) is to prove Theorem 6.1. The plan
of the proof will be explained at the beginning of Section 7. Our proof uses
the theory of Kuranishi structures on moduli spaces of (perturbed) pseudo-
holomorphic disks. In particular, our arguments heavily rely on [10] by
Fukaya-Oh-Ohta-Ono. Section 10 very briefly explains some notions in the
theory of Kuranishi structures, mainly to fix notations.

Conventions. Throughout this paper all manifolds are assumed to be of
$\displaystyle C^{\infty}$. All vector spaces are over
$\displaystyle{\mathbb{R}}$, unless otherwise specified.

Acknowledgements. The author appreciates Kenji Fukaya for sharing his time and
insights into virtual techniques in the pseudo-holomorphic curve theory, and
his comments on an early version of this paper. The author also appreciates
the Simons Center for Geometry and Physics for a great work environment. This
work is supported by JSPS Postdoctoral Fellowship for Research Abroad.

##  2\. Preliminaries on $\displaystyle L_{\infty}$-algebras

We briefly recall basics of $\displaystyle L_{\infty}$-algebras, partially
following [14].

###  2.1. Bar construction

Let $\displaystyle C=\bigoplus_{i\in{\mathbb{Z}}}C_{i}$ be a
$\displaystyle{\mathbb{Z}}\,$-graded vector space. For every integer
$\displaystyle k\geq 1$, let $\displaystyle\mathbb{S}\,_{k}$ denote the
$\displaystyle k$-th symmetric group, and let us define an
$\displaystyle\mathbb{S}\,_{k}$ -action on $\displaystyle C^{\otimes k}$ by

$\displaystyle\rho\cdot(c_{1}\otimes\cdots\otimes
c_{k}):=\varepsilon(\rho:c_{1},\ldots,c_{k})\cdot
c_{\rho(1)}\otimes\cdots\otimes c_{\rho(k)}$

where
$\displaystyle\varepsilon(\rho:c_{1},\ldots,c_{k}):=\prod_{\begin{subarray}{c}i<j\\\
\rho(i)>\rho(j)\end{subarray}}(-1)^{|c_{i}||c_{j}|}$. Let $\displaystyle
S^{k}C$ denote the quotient of $\displaystyle C^{\otimes k}$ by the
$\displaystyle\mathbb{S}\,_{k}$-action, and set $\displaystyle
SC:=\bigoplus_{1\leq k\leq\infty}S^{k}C$.

We define a coproduct $\displaystyle\Delta:SC\to SC^{\otimes 2}$ by

$\displaystyle\Delta(c_{1}\ldots
c_{k}):=\sum_{\begin{subarray}{c}k_{1}+k_{2}=k\\\
\rho\in\mathbb{S}\,_{k}\end{subarray}}\frac{\varepsilon(\rho:c_{1},\ldots,c_{k})}{k_{1}!k_{2}!}\cdot
c_{\rho(1)}\cdots c_{\rho(k_{1})}\otimes c_{\rho(k_{1}+1)}\cdots c_{\rho(k)}.$

Then $\displaystyle\Delta$ is coassociative, namely
$\displaystyle(1\otimes\Delta)\circ\Delta=(\Delta\otimes 1)\circ\Delta$. We
denote $\displaystyle\Delta$ by $\displaystyle\Delta_{C}$ when we need to
specify $\displaystyle C$.

###  2.2. $\displaystyle L_{\infty}$-algebras and $\displaystyle
L_{\infty}$-homomorphisms

For any $\displaystyle{\mathbb{Z}}$-graded vector space $\displaystyle C$ and
$\displaystyle n\in{\mathbb{Z}}$, we define a shifted complex $\displaystyle
C[n]$ by $\displaystyle C[n]_{d}:=C_{n+d}\,(\forall d\in{\mathbb{Z}})$.

######  Definition 2.1.

  1. (i):

An $\displaystyle L_{\infty}$-algebra is a pair of a graded vector space
$\displaystyle C$ and a linear map $\displaystyle l:S(C[-1])\to S(C[-1])$ such
that

     * •

$\displaystyle|l|=-1$.

     * •

$\displaystyle l$ is a coderivation; $\displaystyle\Delta\circ l=(l\otimes
1+1\otimes l)\circ\Delta$.

     * •

$\displaystyle l^{2}=0$.

For each integer $\displaystyle k\geq 1$, we define $\displaystyle
l_{k}:S^{k}(C[-1])\to C[-1]$ by $\displaystyle l_{k}:=\text{\rm pr}_{1}\circ
l|_{S^{k}(C[-1])}$, where $\displaystyle\text{\rm pr}_{1}:S(C[-1])\to
S^{1}(C[-1])\cong C[-1]$ denotes the projection.

  2. (ii):

Let $\displaystyle(C,l)$ and $\displaystyle(C^{\prime},l^{\prime})$ be
$\displaystyle L_{\infty}$-algebras. An $\displaystyle
L_{\infty}$-homomorphism from $\displaystyle(C,l)$ to
$\displaystyle(C^{\prime},l^{\prime})$ is a linear map $\displaystyle
f:S(C[-1])\to S(C^{\prime}[-1])$ such that

     * •

$\displaystyle|f|=0$.

     * •

$\displaystyle f$ is a coalgebra map;
$\displaystyle\Delta_{C^{\prime}[-1]}\circ f=(f\otimes f)\circ\Delta_{C[-1]}$.

     * •

$\displaystyle\l^{\prime}\circ f=f\circ l$.

For each integer $\displaystyle k\geq 1$, we define $\displaystyle
f_{k}:S^{k}(C[-1])\to C^{\prime}[-1]$ by $\displaystyle f_{k}:=\text{\rm
pr}_{1}\circ f|_{S^{k}(C[-1])}$.

Here is another definition of $\displaystyle L_{\infty}$-algebras and
$\displaystyle L_{\infty}$-homomorphisms using exterior products. For each
integer $\displaystyle k\geq 1$, let $\displaystyle\Lambda^{k}C$ denote the
quotient of $\displaystyle C^{\otimes k}$ by the
$\displaystyle\mathbb{S}\,_{k}$-action defined by

$\displaystyle\rho\cdot(c_{1}\otimes\cdots\otimes c_{k}):=\text{\rm
sgn}(\rho)\cdot\varepsilon(\rho:c_{1},\ldots,c_{k})\cdot
c_{\rho(1)}\otimes\cdots\otimes c_{\rho(k)}.$

Then there exists a natural isomorphism

$\displaystyle\sigma_{k}:(\Lambda^{k}C)[-k]\to S^{k}(C[-1]);\quad
c_{1}\wedge\cdots\wedge c_{k}\mapsto(-1)^{\sum_{i}(k-i)|c_{i}|}c_{1}\cdots
c_{k}.$

We define $\displaystyle\lambda_{k}:\Lambda^{k}C\to C$ by
$\displaystyle\lambda_{k}:=\sigma_{1}^{-1}\circ l_{k}\circ\sigma_{k}$.

Then one can define an $\displaystyle L_{\infty}$-structure on $\displaystyle
C$ as a sequence $\displaystyle(\lambda_{k})_{k\geq 1}$ such that each
$\displaystyle\lambda_{k}:\Lambda^{k}C\to C$ is of degree $\displaystyle k-2$
and satisfies the equation

$\displaystyle\sum_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
\rho\in\mathbb{S}\,_{k}\end{subarray}}\pm\frac{1}{k_{1}!(k-k_{1})!}\lambda_{k_{2}}(\lambda_{k_{1}}(c_{\rho(1)}\wedge\cdots\wedge
c_{\rho(k_{1})})\wedge c_{\rho(k_{1}+1)}\wedge\cdots\wedge c_{\rho(k)})=0$

where $\displaystyle\pm$ stands for appropriate signs. Similarly, one can
define an $\displaystyle L_{\infty}$-homomorphism from
$\displaystyle(C,(\lambda_{k})_{k})$ to
$\displaystyle(C^{\prime},(\lambda^{\prime}_{k})_{k})$ as a sequence
$\displaystyle(\varphi_{k})_{k\geq 1}$ such that each
$\displaystyle\varphi_{k}:\Lambda^{k}C\to C^{\prime}$ is of degree
$\displaystyle k-1$ and satisfies the equation

$\displaystyle\displaystyle\sum_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
\rho\in\mathbb{S}\,_{k}\end{subarray}}\pm\frac{1}{k_{2}!(k_{1}-1)!}\cdot\varphi_{k_{1}}(\lambda_{k_{2}}(c_{\rho(1)}\wedge\cdots\wedge
c_{\rho(k_{2})})\wedge c_{\rho(k_{2}+1)}\wedge\cdots\wedge c_{\rho(k)})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{r}=k\\\
\rho\in\mathbb{S}\,_{k}\end{subarray}}\pm\frac{1}{r!k_{1}!\cdots
k_{r}!}\cdot\lambda^{\prime}_{r}(\varphi_{k_{1}}(c_{\rho(1)}\wedge\cdots\wedge
c_{\rho(k_{1})})\wedge\cdots\wedge\varphi_{k_{r}}(c_{\rho(k-k_{r}+1)}\wedge\cdots\wedge
c_{\rho(k)}))$

where $\displaystyle\pm$ stands for appropriate signs.

######  Remark 2.2.

For later purposes we need to specify signs for dg Lie algebras. A dg Lie
algebra is an $\displaystyle L_{\infty}$-algebra such that
$\displaystyle\lambda_{k}=0$ for every $\displaystyle k\geq 3$. Setting
$\displaystyle\partial x:=\lambda_{1}(x)$ and
$\displaystyle[x,y]:=\lambda_{2}(x,y)$, there holds

$\displaystyle\displaystyle[x,y]+(-1)^{|x||y|}[y,x]=0,$
$\displaystyle\displaystyle\partial[x,y]=[\partial x,y]+(-1)^{|x|}[x,\partial
y],$
$\displaystyle\displaystyle[x,[y,z]]+(-1)^{|x|(|y|+|z|)}[y,[z,x]]+(-1)^{|z|(|x|+|y|)}[z,[x,y]]=0.$

We also introduce the following notions for later purposes.

######  Definition 2.3.

  1. (i):

Let $\displaystyle V$ be a vector space and $\displaystyle A$ be a commutative
semigroup. A decomposition of $\displaystyle V$ over $\displaystyle A$ is a
decomposition $\displaystyle V=\bigoplus_{a\in A}V(a)$, where each
$\displaystyle V(a)$ is a subspace of $\displaystyle V$, and $\displaystyle
V(a)+V(a^{\prime})\subset V(a+a^{\prime})$ for every $\displaystyle
a,a^{\prime}\in A$.

  2. (ii):

Let $\displaystyle V$ and $\displaystyle W$ be vector spaces with
decompositions over $\displaystyle A$. A linear map $\displaystyle
f:V^{\otimes k}\to W$ respects these decompositions if

$\displaystyle f(V(a_{1})\otimes\cdots\otimes V(a_{k}))\subset
W(a_{1}+\cdots+a_{k})$

for every $\displaystyle a_{1},\ldots,a_{k}\in A$.

  3. (iii):

An $\displaystyle L_{\infty}$-algebra structure $\displaystyle l=(l_{k})_{k}$
on $\displaystyle V$ respects the decomposition of $\displaystyle V$ if
$\displaystyle l_{k}$ respects the decomposition for every $\displaystyle
k\geq 1$.

###  2.3. Homotopy transfer theorem

Finally we state the homotopy transfer theorem for $\displaystyle
L_{\infty}$-algebras. The proof is only sketched since it is now standard
(perhaps goes back to the paper by Kadeishvili [13] on $\displaystyle
A_{\infty}$-algebras).

######  Theorem 2.4.

Let $\displaystyle(C,l)$ be an $\displaystyle L_{\infty}$-algebra, and
$\displaystyle H(C):=\text{\rm Ker}\,l_{1}/\text{\rm Im}\,l_{1}$. Suppose that
there are linear maps

$\displaystyle\iota:H_{*}(C)\to C_{*},\quad\pi:C_{*}\to
H_{*}(C),\quad\kappa:C_{*}\to C_{*+1}$

such that

$\displaystyle l_{1}\circ\iota=0,\quad\pi\circ
l_{1}=0,\quad\pi\circ\iota=\text{\rm id}_{H(C)},\quad\text{\rm
id}_{C}-\iota\circ\pi=l_{1}\circ\kappa+\kappa\circ l_{1}.$

Then there exist an $\displaystyle L_{\infty}$-algebra structure
$\displaystyle l^{H}$ on $\displaystyle H(C)$ and an $\displaystyle
L_{\infty}$-homomorphism $\displaystyle p:(C,l)\to(H(C),l^{H})$, such that
$\displaystyle l^{H}_{1}=0$ and $\displaystyle p_{1}=\pi$.

When $\displaystyle C$ has a decomposition over a commutative semigroup
$\displaystyle A$ (thus $\displaystyle H(C)$ also has a decomposition over
$\displaystyle A$), and linear maps $\displaystyle\iota$, $\displaystyle\pi$
and $\displaystyle\kappa$ respect these decompositions, then one can take
$\displaystyle l^{H}$ and $\displaystyle p$ so that they respect the
decompositions over $\displaystyle A$.

###### Proof.

It is sufficient to define sequences $\displaystyle(l^{H}_{\leq k})_{k\geq 1}$
and $\displaystyle(p_{\leq k})_{k\geq 1}$ satisfying the following conditions:

  * •

$\displaystyle p_{\leq k}$ is a coalgebra map of degree $\displaystyle 0$ from
$\displaystyle S(C[-1])$ to $\displaystyle S(H[-1])$ which respects
decompositions over $\displaystyle A$.

  * •

$\displaystyle l^{H}_{\leq k}$ is a coderivation of degree $\displaystyle-1$
from $\displaystyle S(H[-1])$ to $\displaystyle S(H[-1])$ which respects
decompositions over $\displaystyle A$.

  * •

$\displaystyle l^{H}_{\leq k}\circ l^{H}_{\leq k}=0$ on $\displaystyle S^{\leq
k+1}(H[-1]):=\bigoplus_{1\leq i\leq k+1}S^{i}(H[-1])$.

  * •

$\displaystyle l^{H}_{\leq k}\circ p_{\leq k}-p_{\leq k}\circ l=0$ on
$\displaystyle S^{\leq k}(C[-1]):=\bigoplus_{1\leq i\leq k}S^{i}(C[-1])$.

  * •

$\displaystyle p_{\leq k}=p_{\leq k+1}$ on $\displaystyle S^{\leq k}(C[-1])$.

  * •

$\displaystyle l^{H}_{\leq k}=l^{H}_{\leq k+1}$ on $\displaystyle S^{\leq
k}(H[-1])$.

  * •

$\displaystyle p_{\leq 1}=\pi$, $\displaystyle l^{H}_{\leq 1}=0$.

Once we obtain these sequences, the limits $\displaystyle
p:=\lim_{k\to\infty}p_{\leq k}$ and $\displaystyle
l^{H}:=\lim_{k\to\infty}l^{H}_{\leq k}$ satisfy the conditions in the theorem.

We can define $\displaystyle(l^{H}_{\leq k})_{k\geq 1}$ and
$\displaystyle(p_{\leq k})_{k\geq 1}$ by upward induction on $\displaystyle
k$. For $\displaystyle k=1$, $\displaystyle p_{\leq 1}$ and $\displaystyle
l^{H}_{\leq 1}$ are defined by the last condition. We assume that we have
defined $\displaystyle p_{\leq k}$, $\displaystyle l^{H}_{\leq k}$ and are
going to define $\displaystyle p_{\leq k+1}$, $\displaystyle l^{H}_{\leq
k+1}$. Let $\displaystyle\text{\rm Hom}_{A}(S^{k+1}C,H)$ denote the space of
linear maps from $\displaystyle S^{k+1}C$ to $\displaystyle H$ preserving
decompositions over $\displaystyle A$. Namely:

$\displaystyle\displaystyle\text{\rm Hom}_{A}(S^{k+1}C,H)$
$\displaystyle\displaystyle:=\\{f\in\text{\rm Hom}(S^{k+1}C,H)\mid$
$\displaystyle\displaystyle f(C(a_{1})\cdots C(a_{k+1}))\subset
H(a_{1}+\cdots+a_{k+1})\,(a_{1},\ldots,a_{k+1}\in A)\\}.$

We define a boundary operator $\displaystyle\partial$ on
$\displaystyle\text{\rm Hom}(S^{k+1}C,H)$ by $\displaystyle\partial f:=f\circ
l_{1}$, then the homology is isomorphic to $\displaystyle\text{\rm
Hom}_{A}(S^{k+1}H,H)$. Our induction assumption shows that $\displaystyle
l^{H}_{\leq k}\circ p_{\leq k}-p_{\leq k}\circ l$ is a cycle in
$\displaystyle\text{\rm Hom}_{A}(S^{k+1}C,H)$, thus one can define

$\displaystyle l^{H}_{k+1}:=[p_{\leq k}\circ l-l^{H}_{\leq k}\circ p_{\leq
k}]\in\text{\rm Hom}_{A}(S^{k+1}H,H).$

Then $\displaystyle l^{H}_{k+1}\circ p_{\leq 1}+l^{H}_{\leq k}\circ p_{\leq
k}-p_{\leq k}\circ l$ is a null-homologous cycle, thus there exists
$\displaystyle p_{k+1}\in\text{\rm Hom}_{A}(S^{k+1}C,H)$ such that

$\displaystyle p_{k+1}\circ l_{1}=l^{H}_{k+1}\circ p_{\leq 1}+l^{H}_{\leq
k}\circ p_{\leq k}-p_{\leq k}\circ l.$

Then we define a coderivation $\displaystyle l^{H}_{\leq k+1}$ so that
$\displaystyle l^{H}_{\leq k+1}=l^{H}_{\leq k}$ on $\displaystyle S^{\leq
k}H$, and

$\displaystyle l^{H}_{\leq k+1}|_{S^{i}H\to
H}=\begin{cases}l^{H}_{k+1}&(i=k+1)\\\ 0&(i>k+1).\end{cases}$

Similarly, we define a coalgebra map $\displaystyle p_{\leq k+1}$ so that
$\displaystyle p_{\leq k+1}=p_{\leq k}$ on $\displaystyle S^{\leq k}C$, and

$\displaystyle p_{\leq k+1}|_{S^{i}C\to H}=\begin{cases}p_{k+1}&(i=k+1)\\\
0&(i>k+1).\end{cases}$

It is easy to check $\displaystyle l^{H}_{\leq k+1}\circ l^{H}_{\leq k+1}=0$
on $\displaystyle S^{\leq k+2}H$. ∎

##  3\. Main result

We state the main result (Theorem 3.2) and explain a few applications to
symplectic topology of Lagrangian submanifolds. Let $\displaystyle\omega_{n}$
denote the standard symplectic form on $\displaystyle{\mathbb{C}}^{n}$, namely
$\displaystyle\omega_{n}:=\sum_{j=1}^{n}dx_{j}\wedge dy_{j}$, and
$\displaystyle L$ be a Lagrangian submanifold in
$\displaystyle({\mathbb{C}}^{n},\omega_{n})$. We assume that $\displaystyle L$
is closed (compact and $\displaystyle\partial L=\emptyset$), connected,
oriented and spin. Let $\displaystyle\mu\in H^{1}(L:{\mathbb{Z}})$ denote the
Maslov class. Since $\displaystyle L$ is oriented,
$\displaystyle\mu(H_{1}(L:{\mathbb{Z}}))\subset 2{\mathbb{Z}}$.

######  Remark 3.1.

Let us explicitly define the Maslov class $\displaystyle\mu$ as follows. Let
$\displaystyle\Lambda(n):=U(n)/O(n)$ denote the unoriented Lagrangian
Grassmannian, and consider maps

$\displaystyle\tau:L\to\Lambda(n);\,x\mapsto
T_{x}L,\qquad{\det}^{2}:\Lambda(n)=U(n)/O(n)\to U(1).$

Then we define $\displaystyle\mu:=(\text{det}^{2}\circ\tau)^{*}[U(1)]$, where
$\displaystyle U(1)$ is identified with
$\displaystyle\\{e^{\sqrt{-1}\theta}|\theta\in{\mathbb{R}}/2\pi{\mathbb{Z}}\\}$,
and $\displaystyle[U(1)]$ is defined as $\displaystyle[U(1)]:=[d\theta]/2\pi$.

Let $\displaystyle S^{1}:={\mathbb{R}}/{\mathbb{Z}}$, and
$\displaystyle{\mathcal{L}}L:=C^{\infty}(S^{1},L)$. We will often abbreviate
$\displaystyle{\mathcal{L}}L$ by $\displaystyle{\mathcal{L}}$. For every
$\displaystyle a\in H_{1}(L:{\mathbb{Z}})$, we set
$\displaystyle{\mathcal{L}}(a):=\\{\gamma\in{\mathcal{L}}\mid[\gamma]=a\\}$.
Obviously $\displaystyle{\mathcal{L}}=\bigsqcup_{a\in
H_{1}(L:{\mathbb{Z}})}{\mathcal{L}}(a)$. For each $\displaystyle a\in
H_{1}(L:{\mathbb{Z}})$, we consider the $\displaystyle C^{\infty}$-topology on
$\displaystyle{\mathcal{L}}(a)$ and set

$\displaystyle H^{{\mathcal{L}}}(a)_{*}:=H^{\text{\rm
sing}}_{*+n+\mu(a)-1}({\mathcal{L}}(a):{\mathbb{R}}),$

where the RHS is the singular homology with respect to the $\displaystyle
C^{\infty}$-topology on $\displaystyle{\mathcal{L}}(a)$ (in the following we
often omit the superscript “sing” ). Now we consider the direct product

$\displaystyle H^{\mathcal{L}}_{*}:=\bigoplus_{a\in
H_{1}(L:{\mathbb{Z}})}H^{\mathcal{L}}(a)_{*}$

equipped with the energy filtration; for each $\displaystyle
E\in{\mathbb{R}}$, we set

$\displaystyle
F^{E}H^{\mathcal{L}}_{*}:=\bigoplus_{\omega_{n}(\bar{a})>E}H^{\mathcal{L}}(a)_{*}$

where $\displaystyle\bar{a}$ denotes the unique element in $\displaystyle
H_{2}({\mathbb{C}}^{n},L)$ satisfying $\displaystyle\partial\bar{a}=a$.
Finally $\displaystyle\widehat{H}^{\mathcal{L}}_{*}$ denotes the completion by
the energy filtration:

$\displaystyle\widehat{H}^{\mathcal{L}}_{*}:=\varprojlim_{E\to\infty}H^{\mathcal{L}}_{*}/F^{E}H^{\mathcal{L}}_{*}.$

Now let us state the main result of this paper:

######  Theorem 3.2.

Let $\displaystyle L$ be a Lagrangian submanifold in
$\displaystyle({\mathbb{C}}^{n},\omega_{n})$ which is closed, oriented and
spin. Then, there exist an $\displaystyle L_{\infty}$-structure
$\displaystyle(l^{H}_{k})_{k\geq 1}$ on $\displaystyle H^{\mathcal{L}}$ and
$\displaystyle X\in\widehat{H}^{\mathcal{L}}_{-1}$, $\displaystyle
Y\in\widehat{H}^{\mathcal{L}}_{2}$, satisfying the following conditions:

  1. (i):

$\displaystyle l^{H}_{1}=0$.

  2. (ii):

The $\displaystyle L_{\infty}$-structure $\displaystyle(l^{H}_{k})_{k\geq 1}$
respects the decomposition of $\displaystyle H^{\mathcal{L}}$ over
$\displaystyle H_{1}(L:{\mathbb{Z}})$. In particular, the $\displaystyle
L_{\infty}$-structure extends to the completion
$\displaystyle\widehat{H}^{\mathcal{L}}$.

  3. (iii):

There exists $\displaystyle c>0$ such that $\displaystyle X\in
F^{c}\widehat{H}^{\mathcal{L}}_{-1}$.

  4. (iv):

$\displaystyle X$ and $\displaystyle Y$ satisfy the following equations:

(3) $\sum_{k\geq 2}\frac{1}{k!}l^{H}_{k}(X,\ldots,X)=0,$ (4)
$\biggl{(}\sum_{k\geq
2}\frac{1}{(k-1)!}l^{H}_{k}(Y,X,\ldots,X)\biggr{)}_{a=0}=(-1)^{n+1}[L].$

Note that infinite sums in the LHS make sense by the condition (iii).
$\displaystyle[L]$ in the RHS of (4) denotes the image of the fundamental
class $\displaystyle[L]\in H_{n}(L:{\mathbb{R}})$ by the embedding map
$\displaystyle H_{*}(L:{\mathbb{R}})\to H_{*}({\mathcal{L}}(0):{\mathbb{R}})$
which is induced by

$\displaystyle L\to{\mathcal{L}}(0);\quad x\mapsto\text{constant loop at
$\displaystyle x$}.$

######  Remark 3.3.

It will be possible to show that $\displaystyle l^{H}_{2}$ coincides (up to
sign) with the Chas-Sullivan loop bracket [1], and the full $\displaystyle
L_{\infty}$-algebra structure is homotopy equivalent to the dg Lie algebra
defined by the chain level loop bracket in [12]. However, we do not give
complete proofs of these claims in this paper.

######  Remark 3.4.

If $\displaystyle l^{H}_{k}=0$ for $\displaystyle k\geq 3$, assuming that
$\displaystyle l^{H}_{2}$ coincides with the loop bracket up to sign, (4)
implies $\displaystyle[Y(-a),X(a)]\neq 0$ for some $\displaystyle a\in
H_{1}(L:{\mathbb{Z}})$. When $\displaystyle L$ is diffeomorphic to
$\displaystyle S^{1}\times S^{2}$, this equation implies nonvanishing of the
Maslov class $\displaystyle\mu$, contradicting a result by Ekholm-Eliashberg-
Murphy-Smith (Corollary 1.6 in [4]). Therefore, if $\displaystyle L$ is
diffeomorphic to $\displaystyle S^{1}\times S^{2}$, there exists at least one
nonvanishing higher term in the LHS of (4).

Let us quickly recall two applications of Theorem 3.2 from [6]. For further
results, consult the original paper [6].

######  Corollary 3.5.

Suppose that $\displaystyle L$ is aspherical. Then there exists $\displaystyle
a\in H_{1}(L:{\mathbb{Z}})$ such that $\displaystyle\mu(a)=2$,
$\displaystyle\omega_{n}(\bar{a})>0$ and $\displaystyle
H_{n}({\mathcal{L}}(a):{\mathbb{R}})\neq 0$.

######  Proof.

By the equation (4), there exist $\displaystyle a_{1},\ldots,a_{k-1}\in
H_{1}(L:{\mathbb{Z}})$ such that

$\displaystyle
l^{H}_{k}(Y(-(a_{1}+\cdots+a_{k-1})),X(a_{1}),\ldots,X(a_{k-1}))\neq 0.$

On the other hand, the assumption that $\displaystyle L$ is aspherical implies
that $\displaystyle H_{i}({\mathcal{L}})\neq 0$ only if $\displaystyle 0\leq
i\leq n$ (Lemma 12.11 in [6]). Therefore we obtain

$\displaystyle 1\leq\mu(a_{1}+\cdots+a_{k-1})\leq n+1,\qquad
2-n\leq\mu(a_{j})\leq 2\quad(1\leq\forall j\leq k-1).$

Since $\displaystyle\mu(a_{1}+\cdots+a_{k-1})>0$, there exists $\displaystyle
j$ such that $\displaystyle\mu(a_{j})>0$. Since $\displaystyle\mu$ takes
values in $\displaystyle 2{\mathbb{Z}}$, we obtain
$\displaystyle\mu(a_{j})=2$. Since $\displaystyle X(a_{j})\neq 0$ we obtain
$\displaystyle\omega_{n}(\bar{a_{j}})>0$ and $\displaystyle
H_{n}({\mathcal{L}}(a_{j}):{\mathbb{R}})\neq 0$. ∎

Corollary 3.5 in particular confirms Audin’s conjecture: every Lagrangian
torus in $\displaystyle{\mathbb{C}}^{n}$ bounds a disk with positive
symplectic area and Maslov index $\displaystyle 2$. Note that Cieliebak-Mohnke
[2] proved Audin’s conjecture by an approach different from ours. For other
previous results on this conjecture see [2].

Another important application is a complete classification of orientable,
closed, prime three-manifolds admitting Lagrangian embeddings into
$\displaystyle{\mathbb{C}}^{3}$:

######  Corollary 3.6.

A closed, connected, orientable and prime three-manifold $\displaystyle M$
admits a Lagrangian embedding into
$\displaystyle({\mathbb{C}}^{3},\omega_{3})$ if and only if $\displaystyle M$
is diffeomorphic to $\displaystyle S^{1}\times\Sigma$ where
$\displaystyle\Sigma$ is a closed orientable two-manifold.

The “if” part in Corollary 3.6 is elementary and classically known. The “only
if” part follows from Corollary 3.5 and some classical results in three-
dimensional topology. See [6] Section 11 or [14] Section 5 for details. Note
that Evans-Kȩdra [5] and Damian [3] proved the same conclusion for monotone
Lagrangian submanifolds in $\displaystyle{\mathbb{C}}^{3}$ which are
orientable but not necessarily prime.

The proof of Theorem 3.2 occupies the rest of this paper. In Sections 4–6 we
reduce Theorem 3.2 to Theorem 6.1 (see Section 6), which will be proved in
Sections 7–9 using the pseudo-holomorphic curve theory. In Section 4 we
introduce the space of Moore loops with marked points, and the notion of de
Rham chains on these spaces, following [12] with minor modifications. Then we
define the chain complex of de Rham chains and study its basic properties. In
Section 5, we reduce Theorem 3.2 to Theorem 5.1, which asserts the existence
of a solution of (3), (4) at chain level. In Section 6, we reduce Theorem 5.1
to Theorem 6.1, which asserts the existence of a sequence of approximate
solutions connected by “gauge equivalences”.

##  4\. de Rham chains on the space of loops with marked points

In Section 4.1, we introduce the space of Moore loops with $\displaystyle k+1$
marked points (where $\displaystyle k\in{\mathbb{Z}}_{\geq 0}$) which we
denote by $\displaystyle{\mathcal{L}}_{k+1}$. In Section 4.2 we fix our
conventions on signs. In Section 4.3, we define the chain complex
$\displaystyle C^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1})$ which consists of
“de Rham chains” on $\displaystyle{\mathcal{L}}_{k+1}$. In Section 4.4, we
introduce a chain model of $\displaystyle[-1,1]\times{\mathcal{L}}_{k+1}$. In
Section 4.5, we introduce a natural dg Lie algebra by taking direct products
of de Rham chain complexes introduced in Sections 4.3 and 4.4.

###  4.1. Space of Moore loops with marked points

First we consider the space of Moore paths

$\displaystyle\Pi:=\\{(T,\gamma)\mid T\in{\mathbb{R}}_{>0},\,\gamma\in
C^{\infty}([0,T],L),\,\partial_{t}^{m}\gamma(0)=\partial_{t}^{m}\gamma(T)=0\,(\forall
m\geq 1)\\}.$

We define evaluation maps $\displaystyle\text{\rm ev}\,_{0},\text{\rm
ev}\,_{1}:\Pi\to L$ by

$\displaystyle\text{\rm ev}\,_{0}(T,\gamma):=\gamma(0),\qquad\text{\rm
ev}\,_{1}(T,\gamma):=\gamma(T)$

and a concatenation map

$\displaystyle\Pi\,_{\text{\rm ev}\,_{1}}\\!\times_{\text{\rm
ev}\,_{0}}\Pi\to\Pi;\quad(\Gamma_{0},\Gamma_{1})\mapsto\Gamma_{0}*\Gamma_{1}$

by

$\displaystyle(T_{0},\gamma_{0})*(T_{1},\gamma_{1}):=(T_{0}+T_{1},\gamma_{0}*\gamma_{1})$

where

$\displaystyle(\gamma_{0}*\gamma_{1})(t):=\begin{cases}\gamma_{0}(t)&(0\leq
t\leq T_{0}),\\\ \gamma_{1}(t-T_{0})&(T_{0}\leq t\leq
T_{0}+T_{1}).\end{cases}$

Next we consider the space of Moore loops with marked points. For every
$\displaystyle k\in{\mathbb{Z}}_{\geq 0}$, we define the space
$\displaystyle{\mathcal{L}}_{k+1}$ which consists of
$\displaystyle(T,\gamma,t_{1},\ldots,t_{k})$ such that

  * •

$\displaystyle T>0$ and $\displaystyle\gamma\in
C^{\infty}({\mathbb{R}}/T{\mathbb{Z}},L)$.

  * •

$\displaystyle 0<t_{1}<\cdots<t_{k}<T$. We set $\displaystyle
t_{0}:=0=T\in{\mathbb{R}}/T{\mathbb{Z}}$.

  * •

$\displaystyle\partial_{t}^{m}\gamma(t_{j})=0$ for every $\displaystyle
m\in{\mathbb{Z}}_{\geq 1}$ and $\displaystyle j\in\\{0,\ldots,k\\}$.

For every $\displaystyle j\in\\{0,\ldots,k\\}$, we define
$\displaystyle\text{\rm ev}\,^{{\mathcal{L}}}_{j}:{\mathcal{L}}_{k+1}\to L$ by

$\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{j}(T,\gamma,t_{1},\ldots,t_{k}):=\gamma(t_{j}).$

$\displaystyle\text{\rm ev}\,^{{\mathcal{L}}}_{j}$ will be abbreviated as
$\displaystyle\text{\rm ev}\,_{j}$ when there is no risk of confusion.

For $\displaystyle k\in{\mathbb{Z}}_{\geq 1}$, $\displaystyle
k^{\prime}\in{\mathbb{Z}}_{\geq 0}$ and $\displaystyle j\in\\{1,\ldots,k\\}$,
we define a concatenation map

$\displaystyle\text{\rm con}\,_{j}:{\mathcal{L}}_{k+1}\,{}_{\text{\rm
ev}\,^{{\mathcal{L}}}_{j}}\\!\times_{\text{\rm
ev}\,^{{\mathcal{L}}}_{0}}{\mathcal{L}}_{k^{\prime}+1}\to{\mathcal{L}}_{k+k^{\prime}}$

as follows. Notice that one can identify $\displaystyle{\mathcal{L}}_{k+1}$
with

$\displaystyle\\{(\Gamma_{0},\ldots,\Gamma_{k})\in\Pi^{k+1}\mid\text{\rm
ev}\,_{1}(\Gamma_{i})=\text{\rm ev}\,_{0}(\Gamma_{i+1})\,(0\leq i\leq
k-1),\,\text{\rm ev}\,_{1}(\Gamma_{k})=\text{\rm ev}\,_{0}(\Gamma_{0})\\}.$

Then we define $\displaystyle\text{\rm con}\,_{j}$ by

$\displaystyle\displaystyle\text{\rm
con}\,_{j}((\Gamma_{0},\ldots,\Gamma_{k}),(\Gamma^{\prime}_{0},\ldots,\Gamma^{\prime}_{k^{\prime}}))$
$\displaystyle\displaystyle:=\begin{cases}(\Gamma_{0},\ldots,\Gamma_{j-2},\Gamma_{j-1}*\Gamma^{\prime}_{0},\Gamma^{\prime}_{1},\ldots,\Gamma^{\prime}_{k^{\prime}-1},\Gamma^{\prime}_{k^{\prime}}*\Gamma_{j},\Gamma_{j+1},\ldots,\Gamma_{k})&(k^{\prime}\geq
1)\\\
(\Gamma_{0},\ldots,\Gamma_{j-2},\Gamma_{j-1}*\Gamma^{\prime}_{0}*\Gamma_{j},\Gamma_{j+1},\ldots,\Gamma_{k})&(k^{\prime}=0).\end{cases}$

For every $\displaystyle a\in H_{1}(L:{\mathbb{Z}})$, let
$\displaystyle{\mathcal{L}}_{k+1}(a)$ denote the subset of
$\displaystyle{\mathcal{L}}_{k+1}$ which consists of
$\displaystyle(T,\gamma,t_{1},\ldots,t_{k})$ such that
$\displaystyle[\gamma]=a$. Obviously
$\displaystyle{\mathcal{L}}_{k+1}=\bigsqcup_{a\in
H_{1}(L:{\mathbb{Z}})}{\mathcal{L}}_{k+1}(a)$, and the concatenation map
$\displaystyle\text{\rm con}\,_{j}$ satisfies

$\displaystyle\text{\rm con}\,_{j}({\mathcal{L}}_{k+1}(a)\,_{\text{\rm
ev}\,^{{\mathcal{L}}}_{j}}\\!\times_{\text{\rm
ev}\,^{{\mathcal{L}}}_{0}}{\mathcal{L}}_{k^{\prime}+1}(a^{\prime}))\subset{\mathcal{L}}_{k+k^{\prime}}(a+a^{\prime}).$

###  4.2. Signs

Here we summarize some conventions on signs. Our sign conventions for
direct/fiber products follow [8] Section 8.2, and for pushout of differntial
forms we follow [10] Section 7.1. Note that these conventions are different
from those in [12].

Direct and fiber products of manifolds

Let $\displaystyle X_{1}$ and $\displaystyle X_{2}$ be oriented manifolds.
Their direct product $\displaystyle X_{1}\times X_{2}$ is oriented so that

$\displaystyle T(X_{1}\times X_{2})\cong TX_{1}\oplus TX_{2}$

preserves orientations.

Next we consider fiber product. Let $\displaystyle M$ be an oriented manifold
and $\displaystyle\pi_{i}:X_{i}\to M\,(i=1,2)$ be $\displaystyle
C^{\infty}$-maps. We assume that $\displaystyle\pi_{2}$ is a submersion.
$\displaystyle\ker d\pi_{2}$ is oriented so that the isomorphism

$\displaystyle TX_{2}\cong TM\oplus\ker d\pi_{2}$

preserves orientations. Then we orient $\displaystyle
X_{1}\,{}_{\pi_{1}}\\!\times_{\pi_{2}}X_{2}$ so that

$\displaystyle T(X_{1}\,{}_{\pi_{1}}\\!\times_{\pi_{2}}X_{2})\cong
TX_{1}\oplus\ker d\pi_{2}$

preserves orientations.

Direct and fiber products of K-spaces

For later use we also fix sign conventions for direct and fiber products of
K-spaces (see Section 10 for basic notions in the theory of Kuranishi
structures). Let $\displaystyle X_{1}$, $\displaystyle X_{2}$ be topological
spaces with K-structures, and
$\displaystyle{\mathcal{U}}_{i}=(U_{i},{\mathcal{E}}_{i},s_{i},\psi_{i})$ be a
K-chart on $\displaystyle X_{i}$, for each $\displaystyle i=1,2$.

Then the direct product
$\displaystyle{\mathcal{U}}_{1}\times{\mathcal{U}}_{2}$, which is a K-chart of
$\displaystyle X_{1}\times X_{2}$, is oriented by

$\displaystyle{\mathcal{U}}_{1}\times{\mathcal{U}}_{2}:=(-1)^{\text{\rm
rk}\,{\mathcal{E}}_{2}(\text{\rm dim}\,U_{1}-\text{\rm
rk}\,{\mathcal{E}}_{1})}(U_{1}\times
U_{2},{\mathcal{E}}_{1}\times{\mathcal{E}}_{2},s_{1}\times
s_{2},\psi_{1}\times\psi_{2}),$

where $\displaystyle U_{1}\times U_{2}$ is oriented as before, and
$\displaystyle{\mathcal{E}}_{1}\times{\mathcal{E}}_{2}$ is oriented so that
the isomorphism

$\displaystyle({\mathcal{E}}_{1}\times{\mathcal{E}}_{2})_{(x_{1},x_{2})}\cong({\mathcal{E}}_{1})_{x_{1}}\oplus({\mathcal{E}}_{2})_{x_{2}}\qquad(x_{1}\in
U_{1},\,x_{2}\in U_{2})$

preserves orientations.

Next we consider the fiber product. Let $\displaystyle M$ be an oriented
$\displaystyle C^{\infty}$-manifold and $\displaystyle\pi_{i}:X_{i}\to
M\,(i=1,2)$ be strongly smooth maps. We assume that $\displaystyle\pi_{2}$ is
weakly submersive. Then the fiber product
$\displaystyle{\mathcal{U}}_{1}\,{}_{\pi_{1}}\\!\times_{\pi_{2}}{\mathcal{U}}_{2}$,
which is a K-chart of $\displaystyle X\,_{\pi_{1}}\\!\times_{\pi_{2}}X_{2}$,
is oriented by

$\displaystyle{\mathcal{U}}_{1}\,{}_{\pi_{1}}\\!\times_{\pi_{2}}{\mathcal{U}}_{2}:=(-1)^{\text{\rm
rk}\,{\mathcal{E}}_{2}(\text{\rm dim}\,U_{1}-\text{\rm dim}\,M-\text{\rm
rk}\,{\mathcal{E}}_{1})}(U_{1}\,{}_{\pi_{1}}\\!\times_{\pi_{2}}U_{2},{\mathcal{E}}_{1}\times{\mathcal{E}}_{2},s_{1}\times
s_{2},\psi_{1}\times\psi_{2}),$

where $\displaystyle U_{1}\,{}_{\pi_{1}}\\!\times_{\pi_{2}}U_{2}$ is oriented
as before.

Pushout of differential forms

For any manifold $\displaystyle X$ and $\displaystyle j\in{\mathbb{Z}}$, let
$\displaystyle{\mathcal{A}}^{j}(X)$ denote the space of degree $\displaystyle
j$ differential forms on $\displaystyle X$, and
$\displaystyle{\mathcal{A}}^{j}_{c}(X)$ denote its subspace which consists of
compactly supported differential forms. We set
$\displaystyle{\mathcal{A}}^{j}(X)=0$ when $\displaystyle j<0$ or
$\displaystyle j>\text{\rm dim}\,X$.

Suppose $\displaystyle X$ and $\displaystyle Y$ are oriented manifolds and
$\displaystyle\pi:X\to Y$ is a $\displaystyle C^{\infty}$-submersion. We
define a pushout (or integration along fibers)

$\displaystyle\pi_{!}:{\mathcal{A}}^{*}_{c}(X)\to{\mathcal{A}}^{*-\text{\rm
dim}\,\pi}_{c}(Y)$

(here $\displaystyle\text{\rm dim}\,\pi:=\text{\rm dim}\,X-\text{\rm dim}\,Y$)
so that the formula

$\displaystyle\int_{Y}\pi_{!}\omega\wedge\eta=\int_{X}\omega\wedge\pi^{*}\eta$

holds for any $\displaystyle\omega\in{\mathcal{A}}^{*}_{c}(X)$ and
$\displaystyle\eta\in{\mathcal{A}}^{*}(Y)$. Simple computations show

$\displaystyle\displaystyle d(\pi_{!}\omega)=(-1)^{\text{\rm
dim}\,\pi}\pi_{!}(d\omega)\qquad\qquad(\omega\in{\mathcal{A}}^{*}_{c}(X)),$
$\displaystyle\displaystyle\pi_{!}(\omega\wedge\pi^{*}\omega^{\prime})=\pi_{!}\omega\wedge\omega^{\prime}\qquad\qquad\,(\omega\in{\mathcal{A}}^{*}_{c}(X),\,\omega^{\prime}\in{\mathcal{A}}^{*}(Y)).$

###  4.3. de Rham chain complex of $\displaystyle{\mathcal{L}}_{k+1}$

Let us define the “de Rham chain complex” of
$\displaystyle{\mathcal{L}}_{k+1}(a)$. First we need the following definition.

######  Definition 4.1.

Let $\displaystyle U$ be a $\displaystyle C^{\infty}$-manifold and
$\displaystyle\varphi:U\to{\mathcal{L}}_{k+1}$. We set

$\displaystyle\varphi(u)=(T(u),\gamma(u),t_{1}(u),\ldots,t_{k}(u)).$

We say that $\displaystyle\varphi$ is of $\displaystyle C^{\infty}$, if the
map

$\displaystyle
U\to{\mathbb{R}}^{k+1};\,u\mapsto(T(u),t_{1}(u),\ldots,t_{k}(u))$

is of $\displaystyle C^{\infty}$ and

$\displaystyle\\{(u,t)\mid u\in U,\,0\leq t\leq T(u)\\}\to
L;\,(u,t)\mapsto\gamma(u)(t)$

is of $\displaystyle C^{\infty}$, namely it extends to a $\displaystyle
C^{\infty}$-map from an open neighborhood of the LHS in $\displaystyle
U\times{\mathbb{R}}$ to $\displaystyle L$. We say that $\displaystyle\varphi$
is smooth, if $\displaystyle\varphi$ is of $\displaystyle C^{\infty}$ and
$\displaystyle\text{\rm ev}\,^{{\mathcal{L}}}_{0}\circ\varphi:U\to L$ is a
submersion.

For every $\displaystyle N\in{\mathbb{Z}}_{\geq 1}$, let
$\displaystyle{\mathfrak{U}}_{N}$ denote the set of oriented submanifolds in
$\displaystyle{\mathbb{R}}^{N}$, and let
$\displaystyle{\mathfrak{U}}:=\bigsqcup_{N\geq 1}{\mathfrak{U}}_{N}$. Let
$\displaystyle{\mathcal{P}}({\mathcal{L}}_{k+1}(a))$ denote the set of pairs
$\displaystyle(U,\varphi)$ such that $\displaystyle U\in{\mathfrak{U}}$ and
$\displaystyle\varphi:U\to{\mathcal{L}}_{k+1}(a)$ is a smooth map.

For every $\displaystyle N\in{\mathbb{Z}}$, let us consider the vector space

(5)
$\bigoplus_{(U,\varphi)\in{\mathcal{P}}({\mathcal{L}}_{k+1}(a))}{\mathcal{A}}^{\text{\rm
dim}\,U-N}_{c}(U).$

For any $\displaystyle(U,\varphi)\in{\mathcal{P}}({\mathcal{L}}_{k+1}(a))$ and
$\displaystyle\omega\in{\mathcal{A}}^{\text{\rm dim}\,U-N}_{c}(U)$, let
$\displaystyle(U,\varphi,\omega)$ denote the vector in (5) such that its
$\displaystyle(U,\varphi)$-component is $\displaystyle\omega$ and the other
components are $\displaystyle 0$.

Let $\displaystyle Z_{N}$ denote the subspace of (5) which is generated by

$\displaystyle\displaystyle\\{(U,\varphi,\pi_{!}\omega)-(U^{\prime},\varphi\circ\pi,\omega)\mid(U,\varphi)\in{\mathcal{P}}({\mathcal{L}}_{k+1}(a)),\quad
U^{\prime}\in{\mathfrak{U}},$
$\displaystyle\displaystyle\quad\omega\in{\mathcal{A}}_{c}^{\text{\rm
dim}\,U^{\prime}-N}(U^{\prime}),\quad\pi:U^{\prime}\to U\,\text{is a
$\displaystyle C^{\infty}$-submersion}\\}.$

Then we define

$\displaystyle C^{\text{\rm
dR}}_{N}({\mathcal{L}}_{k+1}(a)):=\biggl{(}\bigoplus_{(U,\varphi)\in{\mathcal{P}}({\mathcal{L}}_{k+1}(a))}{\mathcal{A}}^{\text{\rm
dim}\,U-N}_{c}(U)\biggr{)}/Z_{N}.$

We often abbreviate $\displaystyle[(U,\varphi,\omega)]\in C^{\text{\rm
dR}}_{N}({\mathcal{L}}_{k+1}(a))$ by $\displaystyle(U,\varphi,\omega)$.

We define a boundary operator $\displaystyle\partial:C^{\text{\rm
dR}}_{*}({\mathcal{L}}_{k+1}(a))\to C^{\text{\rm
dR}}_{*-1}({\mathcal{L}}_{k+1}(a))$ by

(6) $\partial(U,\varphi,\omega):=(-1)^{|\omega|+1}(U,\varphi,d\omega).$

It is easy to check that $\displaystyle\partial$ is well-defined and
$\displaystyle\partial^{2}=0$. We call this chain complex the de Rham chain
complex of $\displaystyle{\mathcal{L}}_{k+1}(a)$, and denote its homology by
$\displaystyle H^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1}(a))$.

######  Remark 4.2.

Here are slight differences between the presentation in this section and that
in [12].

  * •

The sign for the boundary operator in (6) is different from that in [12].

  * •

In the definition of $\displaystyle{\mathcal{P}}({\mathcal{L}}_{k+1}(a))$ we
only require that $\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{0}\circ\varphi:U\to L$ is a submersion. On the other
hand, in Section 7.2 in [12], we consider maps
$\displaystyle\varphi:U\to{\mathcal{L}}_{k+1}$ such that
$\displaystyle\text{\rm ev}\,^{{\mathcal{L}}}_{j}\circ\varphi:U\to L$ are
submersions for all $\displaystyle j\in\\{0,\ldots,k\\}$. However the
resulting chain complexes are quasi-isomorphic.

######  Lemma 4.3.

  1. (i):

The forgetting map

$\displaystyle{\mathcal{L}}_{k+1}(a)\to{\mathcal{L}}_{1}(a);\qquad(T,\gamma,t_{1},\ldots,t_{k})\mapsto(T,\gamma)$

induces an isomorphism $\displaystyle H^{\text{\rm
dR}}_{*}({\mathcal{L}}_{k+1}(a))\cong H^{\text{\rm
dR}}_{*}({\mathcal{L}}_{1}(a))$. In particular $\displaystyle H^{\text{\rm
dR}}_{*}({\mathcal{L}}_{k+1}(a))$ does not depend on $\displaystyle k$.

  2. (ii):

$\displaystyle H^{\text{\rm dR}}_{*}({\mathcal{L}}_{1}(a))\cong H^{\text{\rm
sing}}_{*}({\mathcal{L}}(a):{\mathbb{R}})$, where the RHS denotes the singular
homology with respect to the $\displaystyle C^{\infty}$-topology on
$\displaystyle{\mathcal{L}}(a)$.

######  Proof.

For each $\displaystyle k\in{\mathbb{Z}}_{\geq 0}$, let
$\displaystyle\Delta^{k}$ denote the $\displaystyle k$-dimensional simplex:

$\displaystyle\Delta^{k}:=\begin{cases}{\mathbb{R}}^{0}&(k=0),\\\
\\{(t_{1},\ldots,t_{k})\in{\mathbb{R}}^{k}\mid 0\leq t_{1}\leq\cdots\leq
t_{k}\leq 1\\}&(k\geq 1).\end{cases}$

Then we define $\displaystyle{\mathcal{P}}({\mathcal{L}}(a)\times\Delta^{k})$
to be the set which consists of $\displaystyle(U,\varphi)$ such that
$\displaystyle U\in{\mathfrak{U}}$ and
$\displaystyle\varphi:U\to{\mathcal{L}}(a)\times\Delta^{k}$ is of
$\displaystyle C^{\infty}$ (i.e. projections to each components are of
$\displaystyle C^{\infty}$). Then we can define a chain complex $\displaystyle
C^{\text{\rm dR}}_{*}({\mathcal{L}}(a)\times\Delta^{k})$ in exactly the same
manner as $\displaystyle C^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1}(a))$.
Moreover, there exists a zig-zag of quasi-isomorphisms connecting
$\displaystyle C^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1}(a))$ and
$\displaystyle C^{\text{\rm dR}}_{*}({\mathcal{L}}(a)\times\Delta^{k})$ (see
the last part of Section 7.2 in [12], where $\displaystyle C^{\text{\rm
dR}}_{*}({\mathcal{L}}_{k+1}(a))$ is denoted by $\displaystyle C^{\text{\rm
dR}}_{*}(\bar{{\mathcal{L}}}^{a}_{k,\text{reg}})$). Then, to prove (i) it is
sufficient to show that the map

(7)
${\mathcal{L}}(a)\times\Delta^{k}\to{\mathcal{L}}(a);\qquad(\gamma,t_{1},\ldots,t_{k})\mapsto\gamma$

induces an isomorphism on $\displaystyle H^{\text{\rm dR}}_{*}$, which follows
from homotopy invariance of $\displaystyle H^{\text{\rm dR}}_{*}$ (see
Proposition 4.7 in [12]). (ii) follows from $\displaystyle H^{\text{\rm
dR}}_{*}({\mathcal{L}}_{1}(a))\cong H^{\text{\rm dR}}_{*}({\mathcal{L}}(a))$
(apply the zig-zag mentioned above for $\displaystyle k=0$), and
$\displaystyle H^{\text{\rm dR}}_{*}({\mathcal{L}}(a))\cong H^{\text{\rm
sing}}_{*}({\mathcal{L}}(a):{\mathbb{R}})$, which follows from Theorem 6.1 in
[12]. ∎

Next we define the fiber product on de Rham chain complexes. For every
$\displaystyle k\in{\mathbb{Z}}_{\geq 1}$, $\displaystyle
k^{\prime}\in{\mathbb{Z}}_{\geq 0}$, $\displaystyle j\in\\{1,\ldots,k\\}$ and
$\displaystyle a,a^{\prime}\in H_{1}(L:{\mathbb{Z}})$, we define a linear map

(8) $\circ_{j}:C^{\text{\rm dR}}_{n+d}({\mathcal{L}}_{k+1}(a))\otimes
C^{\text{\rm dR}}_{n+d^{\prime}}({\mathcal{L}}_{k^{\prime}+1}(a^{\prime}))\to
C^{\text{\rm
dR}}_{n+d+d^{\prime}}({\mathcal{L}}_{k+k^{\prime}}(a+a^{\prime}));\,x\otimes
y\mapsto x\circ_{j}y$

in the following way. Setting

$\displaystyle x:=(U,\varphi,\omega),\qquad
y:=(U^{\prime},\varphi^{\prime},\omega^{\prime})$

let $\displaystyle\varphi_{j}:=\text{\rm
ev}\,^{{\mathcal{L}}}_{j}\circ\varphi$ and
$\displaystyle\varphi^{\prime}_{0}:=\text{\rm
ev}\,^{{\mathcal{L}}}_{0}\circ\varphi^{\prime}$. Then we define

$\displaystyle x\circ_{j}y:=(-1)^{(\text{\rm
dim}\,U-|\omega|-n)|\omega^{\prime}|}(U\,_{\varphi_{j}}\\!\times_{\varphi^{\prime}_{0}}U^{\prime},\text{\rm
con}\,_{j}\circ(\varphi_{j}\times\varphi^{\prime}_{0}),\omega\times\omega^{\prime}),$

where the fiber product $\displaystyle
U\,_{\varphi_{j}}\\!\times_{\varphi^{\prime}_{0}}U^{\prime}$ is oriented as in
Section 4.2, and $\displaystyle\text{\rm con}\,_{j}$ denotes the concatenation
map defined in Section 4.1. It is straightforward to check that the fiber
product (8) is a chain map, i.e. it satisfies the Leibniz rule

$\displaystyle\partial(x\circ_{j}y)=\partial
x\circ_{j}y+(-1)^{d}x\circ_{j}\partial y.$

It is also easy to check the associativity: given $\displaystyle x_{i}\in
C^{\text{\rm dR}}_{n+d_{i}}({\mathcal{L}}_{k_{i}+1}(a_{i}))\,(i=1,2,3)$, there
holds

$\displaystyle\displaystyle(x_{1}\circ_{i_{1}}x_{2})\circ_{k_{2}+i_{2}-1}x_{3}$
$\displaystyle\displaystyle=(-1)^{d_{2}d_{3}}(x_{1}\circ_{i_{2}}x_{3})\circ_{i_{1}}x_{2}\quad\,(1\leq
i_{1}<i_{2}\leq k_{1}),$
$\displaystyle\displaystyle(x_{1}\circ_{i_{1}}x_{2})\circ_{i_{1}+i_{2}-1}x_{3}$
$\displaystyle\displaystyle=x_{1}\circ_{i_{1}}(x_{2}\circ_{i_{2}}x_{3})\qquad\qquad\qquad(1\leq
i_{1}\leq k_{1},\,1\leq i_{2}\leq k_{2}).$

On homology level, the fiber product corresponds to the Chas-Sullivan loop
product, which was originally defined in [1]:

######  Lemma 4.4.

The fiber product (8) induces a linear map

$\displaystyle H^{\text{\rm dR}}_{n+d}({\mathcal{L}}_{k+1}(a))\otimes
H^{\text{\rm dR}}_{n+d^{\prime}}({\mathcal{L}}_{k^{\prime}+1}(a^{\prime}))\to
H^{\text{\rm
dR}}_{n+d+d^{\prime}}({\mathcal{L}}_{k+k^{\prime}}(a+a^{\prime})).$

Via isomorphisms in Lemma 4.3, this map corresponds to the loop product

$\displaystyle H_{n+d}({\mathcal{L}}(a))\otimes
H_{n+d^{\prime}}({\mathcal{L}}(a^{\prime}))\to
H_{n+d+d^{\prime}}({\mathcal{L}}(a+a^{\prime})).$

###### Proof.

By Lemma 4.3 (i), it is sufficient to prove the case $\displaystyle k=1$ and
$\displaystyle k^{\prime}=0$, which follows from Proposition 8.7 in [12]. ∎

###  4.4. Chain model of $\displaystyle[-1,1]\times{\mathcal{L}}_{k+1}$

In this subsection, we define another chain complex
$\displaystyle\bar{C}^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1}(a))$. Roughly
speaking, it consists of chains on
$\displaystyle[-1,1]\times{\mathcal{L}}_{k+1}(a)$ relative to
$\displaystyle\\{-1,1\\}\times{\mathcal{L}}_{k+1}(a)$. In Section 6, we use
this chain complex to define “gauge-equivalence” of (approximate) solutions of
the Maurer-Cartan equation of loop bracket.

Let $\displaystyle\bar{{\mathcal{P}}}$ denote the set consists of tuple
$\displaystyle(U,\varphi,\tau_{+},\tau_{-})$ such that the following
conditions are satisfied:

  * •

$\displaystyle U\in{\mathfrak{U}}$ and
$\displaystyle\varphi:U\to{\mathbb{R}}\times{\mathcal{L}}_{k+1}(a)$. We denote
$\displaystyle\varphi:=(\varphi_{\mathbb{R}},\varphi_{{\mathcal{L}}})$, and
for every interval $\displaystyle I\subset{\mathbb{R}}$ we denote
$\displaystyle U_{I}:=(\varphi_{\mathbb{R}})^{-1}(I)$.

  * •

$\displaystyle\varphi_{\mathbb{R}}$ and $\displaystyle\varphi_{{\mathcal{L}}}$
are of $\displaystyle C^{\infty}$. Moreover, $\displaystyle
U\to{\mathbb{R}}\times L;\,u\mapsto(\varphi_{\mathbb{R}}(u),\text{\rm
ev}\,_{0}\circ\varphi_{{\mathcal{L}}}(u))$ is a submersion.

  * •

$\displaystyle\tau_{+}:U_{\geq 1}\to{\mathbb{R}}_{\geq 1}\times U_{1}$ is a
diffeomorphism ($\displaystyle U_{\geq 1}$ is an abbreviation of
$\displaystyle U_{{\mathbb{R}}_{\geq 1}}$) such that

$\displaystyle\varphi|_{U_{\geq 1}}=(i_{\geq
1}\times\varphi_{{\mathcal{L}}}|_{U_{1}})\circ\tau_{+}$

where $\displaystyle i_{\geq 1}:{\mathbb{R}}_{\geq 1}\to{\mathbb{R}}$ is the
inclusion map.

  * •

$\displaystyle\tau_{-}:U_{\leq-1}\to{\mathbb{R}}_{\leq-1}\times U_{-1}$ is a
diffeomorphism $\displaystyle(U_{\leq-1}$ is an abbreviation of $\displaystyle
U_{{\mathbb{R}}_{\leq-1}}$) such that

$\displaystyle\varphi|_{U_{\leq-1}}=(i_{\leq-1}\times\varphi_{{\mathcal{L}}}|_{U_{-1}})\circ\tau_{-}$

where $\displaystyle i_{\leq-1}:{\mathbb{R}}_{\leq-1}\to{\mathbb{R}}$ is the
inclusion map.

######  Remark 4.5.

$\displaystyle U_{\geq 1}$ and $\displaystyle U_{\leq-1}$ may be the empty
set.

For any $\displaystyle(U,\varphi,\tau_{+},\tau_{-})\in\bar{{\mathcal{P}}}$ and
$\displaystyle N\in{\mathbb{Z}}$, let
$\displaystyle{\mathcal{A}}^{N}(U,\varphi,\tau_{+},\tau_{-})$ denote the
vector space which consists of $\displaystyle\omega\in{\mathcal{A}}^{N}(U)$
satisfying the following conditions:

  * •

$\displaystyle\omega|_{U_{[-1,1]}}$ is compactly supported.

  * •

$\displaystyle\omega|_{U_{\geq 1}}=(\tau_{+})^{*}(1\times\omega|_{U_{1}})$.

  * •

$\displaystyle\omega|_{U_{\leq-1}}=(\tau_{-})^{*}(1\times\omega|_{U_{-1}})$.

For $\displaystyle N\in{\mathbb{Z}}$, let us define

$\displaystyle\bar{C}^{\text{\rm
dR}}_{N}({\mathcal{L}}_{k+1}(a)):=\biggl{(}\bigoplus_{(U,\varphi,\tau_{+},\tau_{-})\in\bar{{\mathcal{P}}}}{\mathcal{A}}^{\text{\rm
dim}\,U-N-1}(U,\varphi,\tau_{+},\tau_{-})\biggr{)}/Z_{N}$

where $\displaystyle Z_{N}$ is a subspace generated by vectors

$\displaystyle(U,\varphi,\tau_{+},\tau_{-},\omega)-(U^{\prime},\varphi^{\prime},\tau^{\prime}_{+},\tau^{\prime}_{-},\omega^{\prime})$

such that there exists a submersion $\displaystyle\pi:U^{\prime}\to U$
satisfying

$\displaystyle\displaystyle\varphi^{\prime}$
$\displaystyle\displaystyle=\varphi\circ\pi,$
$\displaystyle\displaystyle\omega$
$\displaystyle\displaystyle=\pi_{!}\omega^{\prime},$
$\displaystyle\displaystyle\tau_{+}\circ\pi|_{U^{\prime}_{\geq 1}}$
$\displaystyle\displaystyle=(\text{\rm id}_{{\mathbb{R}}_{\geq
1}}\times\pi|_{U^{\prime}_{1}})\circ\tau^{\prime}_{+},$
$\displaystyle\displaystyle\tau_{-}\circ\pi|_{U^{\prime}_{\leq-1}}$
$\displaystyle\displaystyle=(\text{\rm
id}_{{\mathbb{R}}_{\leq-1}}\times\pi|_{U^{\prime}_{-1}})\circ\tau^{\prime}_{-}.$

Let us define $\displaystyle\partial:\bar{C}^{\text{\rm
dR}}_{*}({\mathcal{L}}_{k+1}(a))\to\bar{C}^{\text{\rm
dR}}_{*-1}({\mathcal{L}}_{k+1}(a))$ by

$\displaystyle\partial(U,\varphi,\tau_{+},\tau_{-},\omega):=(-1)^{|\omega|+1}(U,\varphi,\tau_{+},\tau_{-},d\omega).$

It is easy to check that $\displaystyle\partial$ is well-defined and
$\displaystyle\partial^{2}=0$, thus we obtain a chain complex.

In the following argument of this subsection, we abbreviate
$\displaystyle\bar{C}^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1}(a))$ and
$\displaystyle C^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1}(a))$ by
$\displaystyle\bar{C}_{*}$ and $\displaystyle C_{*}$, respectively. Let us
define $\displaystyle i:C_{*}\to\bar{C}_{*}$ by

$\displaystyle i(U,\varphi,\omega):=(-1)^{\text{\rm
dim}\,U}({\mathbb{R}}\times U,\text{\rm
id}_{\mathbb{R}}\times\varphi,\tau_{+},\tau_{-},1\times\omega)$

where $\displaystyle\tau_{+}$ and $\displaystyle\tau_{-}$ are defined in the
obvious way. Also, we define $\displaystyle e_{+}:\bar{C}_{*}\to C_{*}$ and
$\displaystyle e_{-}:\bar{C}_{*}\to C_{*}$ by

$\displaystyle\displaystyle e_{+}(U,\varphi,\tau_{+},\tau_{-},\omega)$
$\displaystyle\displaystyle:=(-1)^{\text{\rm
dim}\,U-1}(U_{1},\varphi|_{U_{1}},\omega|_{U_{1}}),$
$\displaystyle\displaystyle e_{-}(U,\varphi,\tau_{+},\tau_{-},\omega)$
$\displaystyle\displaystyle:=(-1)^{\text{\rm
dim}\,U-1}(U_{-1},\varphi|_{U_{-1}},\omega|_{U_{-1}}),$

where $\displaystyle U_{1}$ (resp. $\displaystyle U_{-1}$) is oriented so that
$\displaystyle\tau_{+}:U_{\geq 1}\to{\mathbb{R}}_{\geq 1}\times U_{1}$ (resp.
$\displaystyle\tau_{-}:U_{\leq-1}\to{\mathbb{R}}_{\leq-1}\times U_{-1}$) is
orientation-preserving, where $\displaystyle{\mathbb{R}}_{\geq 1}$ (resp.
$\displaystyle{\mathbb{R}}_{\leq-1}$) is oriented so that
$\displaystyle\partial/\partial t$ is of positive direction ($\displaystyle t$
denotes the standard coordinate on $\displaystyle{\mathbb{R}}$). Then it is
easy to see that $\displaystyle i$, $\displaystyle e_{+}$, $\displaystyle
e_{-}$ are well-defined chain maps, and there holds $\displaystyle e_{+}\circ
i=e_{-}\circ i=\text{\rm id}_{C}$.

######  Lemma 4.6.

$\displaystyle(e_{+},e_{-}):\bar{C}_{*}\to C_{*}\oplus C_{*}$ is surjective.

###### Proof.

Let us take $\displaystyle\chi\in C^{\infty}({\mathbb{R}},[0,1])$ so that
$\displaystyle\chi(t)=1$ for every $\displaystyle t\geq 1$ and
$\displaystyle\chi(t)=0$ for every $\displaystyle t\leq-1$. Given
$\displaystyle x=(U,\varphi,\omega)\in C_{*}$, let

$\displaystyle\bar{x}:=(-1)^{\text{\rm dim}\,U}({\mathbb{R}}\times U,\text{\rm
id}_{\mathbb{R}}\times\varphi,\tau_{+},\tau_{-},\chi\times\omega),$

where $\displaystyle\tau_{+}$ and $\displaystyle\tau_{-}$ are defined in the
obvious manner. Then $\displaystyle e_{+}(\bar{x})=x$ and $\displaystyle
e_{-}(\bar{x})=0$, thus we have proved that the image of
$\displaystyle(e_{+},e_{-})$ contains $\displaystyle C_{*}\oplus 0$. A similar
argument shows that the image contains $\displaystyle 0\oplus C_{*}$. This
completes the proof. ∎

######  Lemma 4.7.

$\displaystyle i\circ e_{+}$ and $\displaystyle i\circ e_{-}$ are chain
homotopic to $\displaystyle\text{\rm id}_{\bar{C}}$.

###### Proof.

We only prove that $\displaystyle i\circ e_{+}$ is chain homotopic to
$\displaystyle\text{\rm id}_{\bar{C}}$ by explicitly defining a linear map
$\displaystyle K:\bar{C}_{*}\to\bar{C}_{*+1}$ which satisfies

$\displaystyle K\partial+\partial K=\text{\rm id}_{\bar{C}}-i\circ e_{+}.$

The proof for $\displaystyle e_{-}$ is completely parallel.

Step 1. Let us take $\displaystyle C^{\infty}$-functions
$\displaystyle\alpha:{\mathbb{R}}^{2}\to{\mathbb{R}}$ and
$\displaystyle\chi:{\mathbb{R}}^{2}\to[0,1]$ so that the following conditions
are satisfied:

  * •

$\displaystyle x\leq 0\implies\alpha(x,y)=y$.

  * •

$\displaystyle x\geq 1,\,y\geq-1\implies\alpha(x,y)=-x$.

  * •

$\displaystyle\nabla\alpha(x,y)\neq 0$ for every
$\displaystyle(x,y)\in{\mathbb{R}}^{2}$.

  * •

$\displaystyle\nu_{\alpha}:=\nabla\alpha/|\nabla\alpha|$ is proper. Namely,
for any $\displaystyle p\in{\mathbb{R}}^{2}$, there exists $\displaystyle
c:{\mathbb{R}}\to{\mathbb{R}}^{2}$ such that $\displaystyle c(0)=p$ and
$\displaystyle\dot{c}(t)=\nu_{\alpha}(c(t))$ for any $\displaystyle
t\in{\mathbb{R}}$.

  * •

$\displaystyle\\{\alpha\geq 1\\}\subset\\{y\geq 1\\}$.

  * •

$\displaystyle\\{\alpha\leq-1\\}\subset\\{y\leq-1\\}\cup\\{x\geq 1\\}$.

  * •

$\displaystyle\chi\equiv 1$ on a neighborhood of
$\displaystyle\\{x=0\\}\cup\\{x\geq 0,y\leq 1\\}$.

  * •

$\displaystyle d\chi(\nabla\alpha)=0$ on $\displaystyle\\{\alpha\geq
1\\}\cup\\{\alpha\leq-1\\}$.

  * •

$\displaystyle\text{\rm supp}\,\chi\cap\\{-1\leq\alpha\leq 1\\}$ is compact.

$\displaystyle x$$\displaystyle
y$$\displaystyle(0,-1)$$\displaystyle(1,-1)$$\displaystyle\\{\alpha=-1\\}$$\displaystyle\\{\alpha=1\\}$$\displaystyle(1,1)$$\displaystyle(0,1)$

Step 2. We define a linear map $\displaystyle K:\bar{C}_{*}\to\bar{C}_{*+1}$
by

$\displaystyle
K(U,\varphi,\tau_{+},\tau_{-},\omega):=(-1)^{|\omega|+1}({\mathbb{R}}\times
U,\bar{\varphi},\bar{\tau}_{+},\bar{\tau}_{-},\bar{\omega})$

where $\displaystyle\bar{\varphi}$, $\displaystyle\bar{\tau}_{\pm}$ and
$\displaystyle\bar{\omega}$ are defined as follows. $\displaystyle\bar{C}_{*}$
is defined by taking a quotient, however well-definedness of $\displaystyle K$
is easy to check.

  * •

Let us denote
$\displaystyle\varphi:U\to{\mathbb{R}}\times{\mathcal{L}}_{k+1}(a)$ by
$\displaystyle\varphi=(\varphi_{\mathbb{R}},\varphi_{{\mathcal{L}}})$. We
define $\displaystyle\tilde{\alpha}:{\mathbb{R}}\times U\to{\mathbb{R}}$ and
$\displaystyle\bar{\varphi}:{\mathbb{R}}\times
U\to{\mathbb{R}}\times{\mathcal{L}}_{k+1}(a)$ by

$\displaystyle\tilde{\alpha}(r,u):=\alpha(r,\varphi_{\mathbb{R}}(u)),\qquad\bar{\varphi}(r,u):=(\tilde{\alpha}(r,u),\varphi_{{\mathcal{L}}}(u)).$

We have to check that

$\displaystyle{\mathbb{R}}\times U\to{\mathbb{R}}\times
L;\quad(r,u)\mapsto(\tilde{\alpha}(r,u),\text{\rm
ev}\,_{0}\circ\varphi_{{\mathcal{L}}}(u))$

is a submersion. This is because $\displaystyle U\to{\mathbb{R}}\times
L;\,u\mapsto(\varphi_{\mathbb{R}}(u),\text{\rm
ev}\,_{0}\circ\varphi_{{\mathcal{L}}}(u))$ is a submersion, and $\displaystyle
d\alpha(x,y)\neq 0$ for every $\displaystyle(x,y)\in{\mathbb{R}}^{2}$.

  * •

To define $\displaystyle\bar{\tau}_{+}$ and $\displaystyle\bar{\tau}_{-}$, we
first define a vector field

$\displaystyle V\in{\mathfrak{X}}({\mathbb{R}}\times(U_{\geq 1}\cup
U_{\leq-1})\cup{\mathbb{R}}_{\geq 1}\times U_{[-1,1]})$

as follows (recall $\displaystyle\nu_{\alpha}:=\nabla\alpha/|\nabla\alpha|$
from Step 1):

    * –

On $\displaystyle{\mathbb{R}}\times U_{\geq
1}\cong{\mathbb{R}}\times{\mathbb{R}}_{\geq 1}\times U_{1}$, we set
$\displaystyle V(x,y,u):=(\nu_{\alpha}(x,y),0)$.

    * –

On $\displaystyle{\mathbb{R}}\times
U_{\leq-1}\cong{\mathbb{R}}\times{\mathbb{R}}_{\leq-1}\times U_{-1}$, we set
$\displaystyle V(x,y,u):=(\nu_{\alpha}(x,y),0)$.

    * –

On $\displaystyle{\mathbb{R}}_{\geq 1}\times U_{[-1,1]}$, we set
$\displaystyle V(x,u):=(-1,0)$.

In particular, $\displaystyle V$ is defined on
$\displaystyle\\{\tilde{\alpha}\geq 1\\}\cup\\{\tilde{\alpha}\leq-1\\}$, and
$\displaystyle V$ is forward (resp. backward) complete on
$\displaystyle\\{\tilde{\alpha}\geq 1\\}$ (resp.
$\displaystyle\\{\tilde{\alpha}\leq-1\\}$). Then we define a diffeomorphism
$\displaystyle\bar{\tau}_{+}:\\{\tilde{\alpha}\geq 1\\}\cong{\mathbb{R}}_{\geq
1}\times\\{\tilde{\alpha}=1\\}$ by
$\displaystyle\bar{\tau}_{+}(p):=(\tilde{\alpha}(p),q)$, where $\displaystyle
q$ is a unique point in $\displaystyle\\{\tilde{\alpha}=1\\}$ which is
connected to $\displaystyle p$ by an integral curve of $\displaystyle V$.
Similarly, we define a diffeomorphism
$\displaystyle\bar{\tau}_{-}:\\{\tilde{\alpha}\leq-1\\}\cong{\mathbb{R}}_{\leq-1}\times\\{\tilde{\alpha}=-1\\}$
by $\displaystyle\bar{\tau}_{-}(p):=(\tilde{\alpha}(p),q)$, where
$\displaystyle q$ is a unique point in $\displaystyle\\{\tilde{\alpha}=-1\\}$
which is connected to $\displaystyle p$ by an integral curve of $\displaystyle
V$.

  * •

We define $\displaystyle\bar{\omega}$ by
$\displaystyle\bar{\omega}(r,u):=\chi(r,\varphi_{\mathbb{R}}(u))\cdot(\text{pr}_{U})^{*}\omega$.

Step 3. We prove that $\displaystyle K\partial+\partial K=\text{\rm
id}_{\bar{C}}-i\circ e_{+}$. It is easy to see that

$\displaystyle(K\partial+\partial
K)(U,\varphi,\tau_{+},\tau_{-},\omega)=({\mathbb{R}}\times
U,\bar{\varphi},\bar{\tau}_{+},\bar{\tau}_{-},d\chi(r,\varphi_{\mathbb{R}}(u))\wedge(\text{\rm
pr}_{U})^{*}\omega).$

Let us denote $\displaystyle d\chi:=(d\chi)_{+}+(d\chi)_{-}$ where
$\displaystyle(d\chi)_{+}$ is supported on $\displaystyle\\{x>0,y>1\\}$ and
$\displaystyle(d\chi)_{-}$ is supported on $\displaystyle\\{x<0\\}$. Then

(9) $({\mathbb{R}}\times
U,\bar{\varphi},\bar{\tau}_{+},\bar{\tau}_{-},(d\chi)_{+}\wedge(\text{pr}_{U})^{*}\omega)=-(i\circ
e_{+})(U,\varphi,\tau_{+},\tau_{-},\omega).$

To check (9), let us consider a submersion

$\displaystyle{\mathbb{R}}_{>0}\times
U_{>1}\cong{\mathbb{R}}_{>0}\times({\mathbb{R}}_{>1}\times
U_{1})\to{\mathbb{R}}\times U_{1}$

where the first map (diffeomorphism) is $\displaystyle\text{\rm
id}_{{\mathbb{R}}_{>0}}\times\tau_{+}|_{U_{>1}}$, and the second map is
$\displaystyle(x,y,u)\mapsto(\alpha(x,y),u)$. Then
$\displaystyle\pi:{\mathbb{R}}_{>0}\times{\mathbb{R}}_{>1}\to{\mathbb{R}};\,(x,y)\mapsto\alpha(x,y)$
satisfies $\displaystyle\pi_{!}((d\chi)_{+})=-1$, and this shows (9). On the
other hand

$\displaystyle({\mathbb{R}}\times
U,\bar{\varphi},\bar{\tau}_{+},\bar{\tau}_{-},(d\chi)_{-}\wedge\text{pr}_{U}^{*}\omega)=(U,\varphi,\tau_{+},\tau_{-},\omega)$

can be proved by the projection $\displaystyle{\mathbb{R}}_{<0}\times U\to U$,
which pushes $\displaystyle(d\chi)_{-}$ to $\displaystyle 1$. ∎

Let us define the fiber product on $\displaystyle\bar{C}_{*}$. For every
$\displaystyle k\in{\mathbb{Z}}_{\geq 1}$, $\displaystyle
k^{\prime}\in{\mathbb{Z}}_{\geq 0}$, $\displaystyle j\in\\{1,\ldots,k\\}$ and
$\displaystyle a,a^{\prime}\in H_{1}(L:{\mathbb{Z}})$, we define

(10) $\circ_{j}:\bar{C}^{\text{\rm
dR}}_{n+d}({\mathcal{L}}_{k+1}(a))\otimes\bar{C}^{\text{\rm
dR}}_{n+d^{\prime}}({\mathcal{L}}_{k^{\prime}+1}(a^{\prime}))\to\bar{C}^{\text{\rm
dR}}_{n+d+d^{\prime}}({\mathcal{L}}_{k+k^{\prime}}(a+a^{\prime}));\,x\otimes
y\mapsto x\circ_{j}y$

in the following way. Setting

$\displaystyle x=(U,\varphi,\tau_{+},\tau_{-},\omega),\qquad
y=(U^{\prime},\varphi^{\prime},\tau^{\prime}_{+},\tau^{\prime}_{-},\omega^{\prime}),$

let us define $\displaystyle C^{\infty}$-maps
$\displaystyle\varphi_{j},\,\varphi^{\prime}_{0}:U\to{\mathbb{R}}\times L$ by

$\displaystyle\varphi_{j}:=(\varphi_{\mathbb{R}},\text{\rm
ev}\,^{{\mathcal{L}}}_{j}\circ\varphi_{{\mathcal{L}}}),\qquad\varphi^{\prime}_{0}:=(\varphi^{\prime}_{\mathbb{R}},\text{\rm
ev}\,^{{\mathcal{L}}}_{0}\circ\varphi^{\prime}_{{\mathcal{L}}}).$

Note that $\displaystyle\varphi^{\prime}_{0}$ is a submersion, thus the fiber
product $\displaystyle
U\,_{\varphi_{j}}\\!\times_{\varphi^{\prime}_{0}}U^{\prime}$ is a
$\displaystyle C^{\infty}$-manifold. Then we define

$\displaystyle x\circ_{j}y:=(-1)^{(\text{\rm
dim}\,U-|\omega|-n-1)|\omega^{\prime}|+n}(U\,_{\varphi_{j}}\\!\times_{\varphi^{\prime}_{0}}U^{\prime},\varphi^{\prime\prime},\tau^{\prime\prime}_{+},\tau^{\prime\prime}_{-},\omega\times\omega^{\prime})$

where $\displaystyle\varphi^{\prime\prime}$ is defined by

$\displaystyle\varphi^{\prime\prime}(u,u^{\prime}):=(\varphi_{\mathbb{R}}(u),\text{\rm
con}\,_{j}(\varphi_{{\mathcal{L}}}(u),\varphi^{\prime}_{{\mathcal{L}}}(u^{\prime}))),$

and $\displaystyle\tau^{\prime\prime}_{+}$,
$\displaystyle\tau^{\prime\prime}_{-}$ are defined as follows:

$\displaystyle\displaystyle\rho_{+}(u,u^{\prime})$
$\displaystyle\displaystyle:=\text{\rm pr}_{{\mathbb{R}}_{\geq
1}}\circ\tau_{+}(u)=\text{\rm pr}_{{\mathbb{R}}_{\geq
1}}\circ\tau^{\prime}_{+}(u^{\prime}),$
$\displaystyle\displaystyle\tau^{\prime\prime}_{+}(u,u^{\prime})$
$\displaystyle\displaystyle:=(\rho_{+}(u,u^{\prime}),(\text{\rm
pr}_{U_{1}}\circ\tau_{+}(u),\text{\rm
pr}_{U^{\prime}_{1}}\circ\tau^{\prime}_{+}(u^{\prime}))),$
$\displaystyle\displaystyle\rho_{-}(u,u^{\prime})$
$\displaystyle\displaystyle:=\text{\rm
pr}_{{\mathbb{R}}_{\leq-1}}\circ\tau_{-}(u)=\text{\rm
pr}_{{\mathbb{R}}_{\leq-1}}\circ\tau^{\prime}_{-}(u^{\prime}),$
$\displaystyle\displaystyle\tau^{\prime\prime}_{-}(u,u^{\prime})$
$\displaystyle\displaystyle:=(\rho_{-}(u,u^{\prime}),(\text{\rm
pr}_{U_{-1}}\circ\tau_{-}(u),\text{\rm
pr}_{U^{\prime}_{-1}}\circ\tau^{\prime}_{-}(u^{\prime}))).$

It is easy to see that the fiber product (10) is a chain map, i.e. it
satisfies the Leibniz rule:

$\displaystyle\partial(x\circ_{j}y)=\partial
x\circ_{j}y+(-1)^{d}x\circ_{j}\partial y.$

Moreover, chain maps $\displaystyle i$, $\displaystyle e_{+}$, $\displaystyle
e_{-}$ intertwine fiber products on $\displaystyle C_{*}$ and
$\displaystyle\bar{C}_{*}$.

###  4.5. dg Lie algebras $\displaystyle C^{{\mathcal{L}}}$,
$\displaystyle\bar{C}^{{\mathcal{L}}}$ and their completions

For every $\displaystyle a\in H_{1}(L:{\mathbb{Z}})$ and $\displaystyle
k\in{\mathbb{Z}}_{\geq 0}$, we define

$\displaystyle C^{{\mathcal{L}}}(a,k)_{*}:=C^{\text{\rm
dR}}_{*+n+\mu(a)+k-1}({\mathcal{L}}_{k+1}(a))$

and set

$\displaystyle C^{{\mathcal{L}}}_{*}:=\bigoplus_{\begin{subarray}{c}a\in
H_{1}(L:{\mathbb{Z}})\\\ k\in{\mathbb{Z}}_{\geq
0}\end{subarray}}C^{{\mathcal{L}}}(a,k)_{*}.$

To define the action filatration on $\displaystyle C^{{\mathcal{L}}}_{*}$, for
each $\displaystyle E\in{\mathbb{R}}$ we set

$\displaystyle
F^{E}C^{{\mathcal{L}}}_{*}:=\bigoplus_{\begin{subarray}{c}\omega_{n}(\bar{a})>E\\\
k\in{\mathbb{Z}}_{\geq 0}\end{subarray}}C^{{\mathcal{L}}}(a,k)_{*}$

and take completion:

$\displaystyle\widehat{C}^{{\mathcal{L}}}_{*}:=\varprojlim_{E\to\infty}C^{{\mathcal{L}}}_{*}/F^{E}C^{{\mathcal{L}}}_{*}.$

$\displaystyle C^{{\mathcal{L}}}_{*}$ has a dg Lie algebra structure (see
Remark 2.2 for our convention) defined as follows:

$\displaystyle\displaystyle(\partial x)(a,k)$
$\displaystyle\displaystyle:=\partial(x(a,k)),$
$\displaystyle\displaystyle(x\circ y)(a,k)$
$\displaystyle\displaystyle:=\sum_{\begin{subarray}{c}k^{\prime}+k^{\prime\prime}=k+1\\\
1\leq i\leq k^{\prime}\\\
a^{\prime}+a^{\prime\prime}=a\end{subarray}}(-1)^{(i-1)(k^{\prime\prime}-1)+(k^{\prime}-1)(|y|+1+k^{\prime\prime})}x(a^{\prime},k^{\prime})\circ_{i}y(a^{\prime\prime},k^{\prime\prime}),$
$\displaystyle\displaystyle[x,y]$ $\displaystyle\displaystyle:=x\circ
y-(-1)^{|x||y|}y\circ x.$

For the sign in the RHS of the second formula, see Theorem 2.8 (ii) in [12].
The Jacobi identity follows from the associativity of the fiber product. Since
this dg Lie algebra structure respects the decomposition $\displaystyle
C^{{\mathcal{L}}}_{*}\cong\bigoplus_{(a,k)\in
H_{1}(L:{\mathbb{Z}})\times{\mathbb{Z}}_{\geq 0}}C^{{\mathcal{L}}}(a,k)_{*}$,
it extends to the completion $\displaystyle\widehat{C}^{{\mathcal{L}}}_{*}$.

We also consider
$\displaystyle\bar{C}^{{\mathcal{L}}}_{*}:=\bigoplus_{(a,k)\in
H_{1}(L:{\mathbb{Z}})\times{\mathbb{Z}}_{\geq 0}}\bar{C}^{\text{\rm
dR}}_{*+n+\mu(a)+k-1}({\mathcal{L}}_{k+1}(a))$ and its completion
$\displaystyle\widehat{\bar{C}}^{{\mathcal{L}}}_{*}$. One can naturally define
a dg Lie algebra structure on $\displaystyle\bar{C}^{{\mathcal{L}}}_{*}$,
$\displaystyle C^{{\mathcal{L}}}_{*}$, and it extends to the completion. One
can define morphisms of dg Lie algebras

$\displaystyle i:C^{{\mathcal{L}}}_{*}\to\bar{C}^{{\mathcal{L}}}_{*},\quad
e_{+}:\bar{C}^{{\mathcal{L}}}_{*}\to C^{{\mathcal{L}}}_{*},\quad
e_{-}:\bar{C}^{{\mathcal{L}}}_{*}\to C^{{\mathcal{L}}}_{*},$

so that the following properties hold:

  * •

$\displaystyle e_{+}\circ i=e_{-}\circ i=\text{\rm id}_{C^{{\mathcal{L}}}}$.

  * •

$\displaystyle i\circ e_{+}$ and $\displaystyle i\circ e_{-}$ are chain
homotopic to $\displaystyle\text{\rm id}_{\bar{C}^{{\mathcal{L}}}}$. One can
take chain homotopies to respect decompositions over $\displaystyle
H_{1}(L:{\mathbb{Z}})\times{\mathbb{Z}}_{\geq 0}$.

  * •

$\displaystyle(e_{+},e_{-}):\bar{C}^{{\mathcal{L}}}_{*}\to
C^{{\mathcal{L}}}_{*}\oplus C^{{\mathcal{L}}}_{*}$ is surjective.

##  5\. Chain level statement

The goal of this section is to reduce Theorem 3.2 to Theorem 5.1, which is
formulated on the dg Lie algebra $\displaystyle\widehat{C}^{{\mathcal{L}}}$.
Throughout this section, $\displaystyle{\mathbb{R}}$-coefficient singular
homology $\displaystyle H^{\text{\rm sing}}_{*}(\,\cdot\,:{\mathbb{R}})$ is
abbreviated as $\displaystyle H_{*}(\,\cdot\,)$.

######  Theorem 5.1.

There exist $\displaystyle x\in\widehat{C}^{{\mathcal{L}}}_{-1}$,
$\displaystyle y\in\widehat{C}^{{\mathcal{L}}}_{2}$, $\displaystyle
z\in\widehat{C}^{{\mathcal{L}}}_{1}$ and $\displaystyle\varepsilon>0$ such
that:

  1. (i):

$\displaystyle\partial x-\frac{1}{2}[x,x]=0$, namely $\displaystyle x$ is a
Maurer-Cartan element of $\displaystyle\widehat{C}^{{\mathcal{L}}}$.

  2. (ii):

$\displaystyle\partial y-[x,y]=z$.

  3. (iii):

$\displaystyle x(a,k)\neq 0$ only if $\displaystyle\omega(\bar{a})\geq
2\varepsilon$ or $\displaystyle a=0$, $\displaystyle k\geq 2$. Moreover,
$\displaystyle x(0,2)\in C^{\text{\rm dR}}_{n}({\mathcal{L}}_{3}(0))$ is a
cycle such that $\displaystyle[x(0,2)]\in H^{\text{\rm
dR}}_{n}({\mathcal{L}}_{3}(0))\cong H_{n}({\mathcal{L}}(0))$ corresponds to
$\displaystyle(-1)^{n+1}[L]$.

  4. (iv):

$\displaystyle z(a,k)\neq 0$ only if $\displaystyle\omega(\bar{a})\geq
2\varepsilon$ or $\displaystyle a=0$. Moreover, $\displaystyle z(0,0)\in
C^{\text{\rm dR}}_{n}({\mathcal{L}}_{1}(0))$ is a cycle such that
$\displaystyle[z(0,0)]\in H^{\text{\rm dR}}_{n}({\mathcal{L}}_{1}(0))\cong
H_{n}({\mathcal{L}}(0))$ corresponds to $\displaystyle(-1)^{n+1}[L]$.

In (iii) and (iv), $\displaystyle[L]\in H_{n}({\mathcal{L}}(0))$ is defined as
in Theorem 3.2.

Conditions (i) and (iii) imply that

$\displaystyle x^{0}:=\sum_{k\geq 2}x(0,k)\in\widehat{C}^{{\mathcal{L}}}_{-1}$

is a Maurer-Cartan element of $\displaystyle\widehat{C}^{{\mathcal{L}}}$. For
every $\displaystyle a\in H_{1}(L:{\mathbb{Z}})$, let $\displaystyle
C^{{\mathcal{L}}}(a)_{*}:=\prod_{k\geq 0}C^{\mathcal{L}}(a,k)_{*}$. We define
$\displaystyle\partial_{x^{0}}:C^{{\mathcal{L}}}(a)_{*}\to
C^{{\mathcal{L}}}(a)_{*-1}$ by $\displaystyle\partial_{x^{0}}(u):=\partial
u-[x^{0},u]$.

######  Lemma 5.2.

$\displaystyle H_{*}(C^{{\mathcal{L}}}(a)_{*},\partial_{x^{0}})\cong
H_{*+n+\mu(a)-1}({\mathcal{L}}(a))$.

######  Proof.

Let us consider the exact sequence

$\displaystyle 0\to\prod_{k\geq 1}C^{{\mathcal{L}}}(a,k)_{*}\to
C^{{\mathcal{L}}}(a)_{*}\to C^{{\mathcal{L}}}(a,0)_{*}\to 0$

where the first map is inclusion and the second map is projection. Since
$\displaystyle H_{*}(C^{{\mathcal{L}}}(a,0))\cong
H_{*+n+\mu(a)-1}({\mathcal{L}}(a))$ by Lemma 4.3, it is sufficient to show
that $\displaystyle\prod_{k\geq 1}C^{{\mathcal{L}}}(a,k)_{*}$ is acyclic with
the boundary operator $\displaystyle\partial_{x^{0}}$.

For every positive integer $\displaystyle N$, the operator
$\displaystyle\partial_{x^{0}}$ preserves
$\displaystyle\prod_{k>2N}C^{{\mathcal{L}}}(a,k)_{*}$. Thus
$\displaystyle\partial_{x^{0}}$ acts on $\displaystyle\prod_{1\leq k\leq
2N}C^{{\mathcal{L}}}(a,k)_{*}$, preserving the filtration
$\displaystyle\biggl{(}\prod_{i\leq k\leq
2N}C^{{\mathcal{L}}}(a,k)_{*}\biggr{)}_{1\leq i\leq 2N}$.

Then the $\displaystyle E_{1}$-term is

$\displaystyle H_{*}(C^{{\mathcal{L}}}(a,k))\cong H^{\text{\rm
dR}}_{*+n+\mu(a)+k-1}({\mathcal{L}}_{k+1}(a))\cong
H_{*+n+\mu(a)+k-1}({\mathcal{L}}(a)).$

Let us compute $\displaystyle d_{1}:H_{*}(C^{{\mathcal{L}}}(a,k))\to
H_{*-1}(C^{{\mathcal{L}}}(a,k+1))$. By direct computations

$\displaystyle\displaystyle[x^{0},y](a,k+1)=$
$\displaystyle\displaystyle\qquad(-1)^{|y|}\biggl{(}x(0,2)\circ_{2}y(a,k)+\sum_{1\leq
i\leq
k}(-1)^{i}y(a,k)\circ_{i}x(0,2)+(-1)^{k+1}x(0,2)\circ_{1}y(a,k)\biggr{)}.$

Since $\displaystyle[x(0,2)]\in H^{\text{\rm dR}}_{n}({\mathcal{L}}_{3}(0))$
corresponds to $\displaystyle(-1)^{n+1}[L]\in H_{n}({\mathcal{L}})$ and the
fiber product on homology level corresponds to the loop product (Lemma 4.4),
$\displaystyle d_{1}$ coincides with $\displaystyle\pm\sum_{0\leq i\leq
k+1}(-1)^{i}$. Then all $\displaystyle E_{2}$-terms vanish, thus
$\displaystyle\prod_{1\leq k\leq 2N}C^{{\mathcal{L}}}(a,k)_{*}$ is acyclic.
Finally, by taking an inverse limit (Theorem 3.5.8 in [15]) we have shown that
$\displaystyle\prod_{k\geq 1}C^{{\mathcal{L}}}(a,k)_{*}$ is acyclic. ∎

######  Lemma 5.3.

$\displaystyle x^{+}:=x-x^{0}$ satisfies equations

$\displaystyle\partial_{x^{0}}x^{+}-\frac{1}{2}[x^{+},x^{+}]=0,\qquad\partial_{x^{0}}y-[x^{+},y]=z.$

###### Proof.

Straightforward from $\displaystyle\partial x-\frac{1}{2}[x,x]=0$,
$\displaystyle\partial y-[x,y]=z$ and $\displaystyle x=x^{0}+x^{+}$. ∎

Let $\displaystyle H^{{\mathcal{L}}}_{*}=\bigoplus_{a\in
H_{1}(L:{\mathbb{Z}})}H_{*+n+\mu(a)-1}({\mathcal{L}}(a))$ as in Section 3.
Then there exist linear maps

$\displaystyle\iota:H^{{\mathcal{L}}}_{*}\to
C^{{\mathcal{L}}}_{*},\quad\pi:C^{{\mathcal{L}}}_{*}\to
H^{{\mathcal{L}}}_{*},\quad\kappa:C^{{\mathcal{L}}}_{*}\to
C^{{\mathcal{L}}}_{*+1}$

such that

$\displaystyle\partial_{x^{0}}\circ\iota=0,\quad\pi\circ\partial_{x^{0}}=0,\quad\pi\circ\iota=\text{\rm
id}_{H^{{\mathcal{L}}}},\quad\kappa\circ\partial_{x^{0}}+\partial_{x^{0}}\circ\kappa=\text{\rm
id}_{C^{{\mathcal{L}}}}-\iota\circ\pi.$

One can take these maps so that they preserve decompositions of $\displaystyle
H^{{\mathcal{L}}}$ and $\displaystyle C^{{\mathcal{L}}}$ over $\displaystyle
H_{1}(L:{\mathbb{Z}})$, thus they extend to the completions. Also, one can
take $\displaystyle\pi$ so that it maps $\displaystyle\sum_{k\geq 0}z(0,k)$ to
$\displaystyle(-1)^{n+1}[L]\in H_{n}({\mathcal{L}}(0))$.

Using homotopy transfer theorem (Thoerem 2.4), one can define an
$\displaystyle L_{\infty}$-structure $\displaystyle l^{H}$ on $\displaystyle
H^{{\mathcal{L}}}_{*}$ and an $\displaystyle L_{\infty}$-homomorphism
$\displaystyle p$ from
$\displaystyle(C^{{\mathcal{L}}}_{*},\partial_{x^{0}},[\,,\,])$ to
$\displaystyle(H^{{\mathcal{L}}}_{*},l^{H})$ so that $\displaystyle
l^{H}_{1}=0$ and $\displaystyle p_{1}=\pi$. We can take $\displaystyle l^{H}$
and $\displaystyle p$ so that they respect decompositions over $\displaystyle
H_{1}(L:{\mathbb{Z}})$, thus they extend to the completions. Applying
Proposition 4.9 in [14], elements in
$\displaystyle\widehat{H}^{{\mathcal{L}}}$

$\displaystyle\displaystyle X$
$\displaystyle\displaystyle:=\sum_{k=1}^{\infty}\frac{1}{k!}p_{k}(x^{+},\ldots,x^{+})\in\widehat{H}^{{\mathcal{L}}}_{-1}$
$\displaystyle\displaystyle Y$
$\displaystyle\displaystyle:=\sum_{k=1}^{\infty}\frac{1}{(k-1)!}p_{k}(y,x^{+},\ldots,x^{+})\in\widehat{H}^{{\mathcal{L}}}_{2},$
$\displaystyle\displaystyle Z$
$\displaystyle\displaystyle:=\sum_{k=1}^{\infty}\frac{1}{(k-1)!}p_{k}(z,x^{+},\ldots,x^{+})\in\widehat{H}^{{\mathcal{L}}}_{1},$

satisfy

$\displaystyle\displaystyle\sum_{k=2}^{\infty}\frac{1}{k!}l^{H}_{k}(X,\ldots,X)=0,$
$\displaystyle\displaystyle\sum_{k=2}^{\infty}\frac{1}{(k-1)!}l^{H}_{k}(Y,X,\ldots,X)=Z.$

Notice that infinite sums in the definitions of $\displaystyle X$,
$\displaystyle Y$, $\displaystyle Z$ make sense since $\displaystyle
x^{+}(a)\neq 0$ only if $\displaystyle\omega(\bar{a})\geq 2\varepsilon$ and
$\displaystyle p$ preserves decompositions over $\displaystyle
H_{1}(L:{\mathbb{Z}})$. Moreover $\displaystyle X(a)\neq 0$ only if
$\displaystyle\omega(\bar{a})\geq 2\varepsilon$, thus Theorem 3.2 (iii) holds
with $\displaystyle c:=2\varepsilon$. To complete the proof of Theorem 3.2, we
need to show that $\displaystyle Z(0)=(-1)^{n+1}[L]$. Since $\displaystyle p$
respects decompositions over $\displaystyle H_{1}(L:{\mathbb{Z}})$ and
$\displaystyle z(a,k)\neq 0$ only if $\displaystyle\omega(\bar{a})\geq
2\varepsilon$ or $\displaystyle a=0$, we obtain

$\displaystyle
Z(0)=\pi\biggl{(}\sum_{k=0}^{\infty}z(0,k)\biggr{)}=(-1)^{n+1}[L].$

##  6\. Sequence of approximate solutions

To prove Theorem 5.1, we have to define chains $\displaystyle x$,
$\displaystyle y$, $\displaystyle z$ satisfying the equations

(11) $\partial x-\frac{1}{2}[x,x]=0,\qquad\partial y-[x,y]=z.$

These chains are defined from virtual fundamental chains of moduli spaces of
pseudo-holomorphic disks with boundaries on $\displaystyle L$. However, it is
difficult to get such chains in one step since it will involve simultaneous
perturbations of Kuranishi maps of infinitely many moduli spaces. Due to this
difficulty, we first define a sequence $\displaystyle(x_{i},y_{i},z_{i})_{i}$
such that the following conditions hold for every $\displaystyle i$:

  * •

The tuple $\displaystyle(x_{i},y_{i},z_{i})$ satisfies equations (11) up to
certain energy level which goes to $\displaystyle\infty$ as $\displaystyle
i\to\infty$.

  * •

Tuples $\displaystyle(x_{i},y_{i},z_{i})$ and
$\displaystyle(x_{i+1},y_{i+1},z_{i+1})$ are “gauge equivalent” up to certain
energy level which goes to $\displaystyle\infty$ as $\displaystyle
i\to\infty$.

By the second condition, limits $\displaystyle x:=\lim_{i\to\infty}x_{i}$,
$\displaystyle y:=\lim_{i\to\infty}y_{i}$, $\displaystyle
z:=\lim_{i\to\infty}z_{i}$ exist, and by the first condition these limits
satisfy (11). This procedure is similar to the construction of the
$\displaystyle A_{\infty}$-structure in Lagrangian Floer theory as a limit of
$\displaystyle A_{K}$-structures; see Section 7.2.3 in [8], and Remark 22.27
in [10].

The goal of this section is to explain details of the algebraic procedure
sketched above; we reduce Theorem 5.1 to Theorem 6.1, which asserts the
existence of a sequence of approximate solutions connected by gauge
equivalences. The proof of Theorem 6.1 involves construction of virtual
fundamental chains using the theory of Kuranishi structures, which is carried
out in Sections 7–9.

Let $\displaystyle J$ be the standard complex structure on
$\displaystyle{\mathbb{C}}^{n}$, and take $\displaystyle\varepsilon>0$ so that
$\displaystyle 2\varepsilon$ is less than the minimal symplectic area of
nonconstant $\displaystyle J$-holomorphic disks with boundaries on
$\displaystyle L$. We fix such $\displaystyle\varepsilon$ in the following
arguments.

For each $\displaystyle m\in{\mathbb{Z}}$, we set

$\displaystyle F^{m}C^{{\mathcal{L}}}_{*}:=\bigoplus_{\begin{subarray}{c}a\in
H_{1}(L:{\mathbb{Z}})\\\ k\in{\mathbb{Z}}_{\geq 0}\\\
\omega_{n}(\bar{a})\geq\varepsilon(m+1-k)\end{subarray}}C^{{\mathcal{L}}}(a,k)_{*}.$

Obviously $\displaystyle\partial F^{m}\subset F^{m}$. We can also show
$\displaystyle[F^{m},F^{m^{\prime}}]\subset F^{m+m^{\prime}}$ since

$\displaystyle\omega_{n}(\bar{a})\geq\varepsilon(m+1-k),\,\omega_{n}(\bar{a^{\prime}})\geq\varepsilon(m^{\prime}+1-k^{\prime})\implies\omega_{n}(\bar{a}+\bar{a^{\prime}})\geq\varepsilon(m+m^{\prime}+1-(k+k^{\prime}-1)).$

The filtration on $\displaystyle\bar{C}^{{\mathcal{L}}}$ (see Section 4.5) is
defined in a similar manner. These filtrations extend to the completions. In
the following we abbreviate $\displaystyle C^{{\mathcal{L}}}$ and
$\displaystyle\bar{C}^{{\mathcal{L}}}$ by $\displaystyle C$ and
$\displaystyle\bar{C}$.

######  Theorem 6.1.

There exist integers $\displaystyle I,U\geq 2$ and a sequence
$\displaystyle(x_{i},y_{i},z_{i},\bar{x}_{i},\bar{y}_{i},\bar{z}_{i})_{i\geq
I}$ satisfying the following conditions for every $\displaystyle i\geq I$:

$\displaystyle x_{i}\in F^{1}C_{-1},\,\bar{x}_{i}\in
F^{1}\bar{C}_{-1},\,y_{i}\in F^{-U}C_{2},\,\bar{y}_{i}\in
F^{-U}\bar{C}_{2},\,z_{i}\in F^{-1}C_{1},\,\bar{z}_{i}\in F^{-1}\bar{C}_{1}.$
$\displaystyle x_{i}=e_{-}(\bar{x}_{i}),\quad y_{i}=e_{-}(\bar{y}_{i}),\quad
z_{i}=e_{-}(\bar{z}_{i}).$
$\displaystyle\partial\bar{x}_{i}-\frac{1}{2}[\bar{x}_{i},\bar{x}_{i}]\in
F^{i}\bar{C}_{-2},\quad\partial\bar{y}_{i}-[\bar{x}_{i},\bar{y}_{i}]-\bar{z}_{i}\in
F^{i-U-1}\bar{C}_{1},\quad\partial\bar{z}_{i}-[\bar{x}_{i},\bar{z}_{i}]\in
F^{i-2}\bar{C}_{0}.$ $\displaystyle x_{i+1}-e_{+}(\bar{x}_{i})\in
F^{i}C_{-1},\quad y_{i+1}-e_{+}(\bar{y}_{i})\in F^{i-U-1}C_{2},\quad
z_{i+1}-e_{+}(\bar{z}_{i})\in F^{i-2}C_{1}.$

  * •

$\displaystyle x_{i}(a,k)\neq 0$ only if $\displaystyle\omega(\bar{a})\geq
2\varepsilon$ or $\displaystyle a=0$, $\displaystyle k\geq 2$. Moreover,
$\displaystyle x_{i}(0,2)$ is a cycle in $\displaystyle C^{\text{\rm
dR}}_{n}({\mathcal{L}}_{3}(0))$ such that $\displaystyle[x_{i}(0,2)]$
corresponds to $\displaystyle(-1)^{n+1}[L]$ via the isomorphism $\displaystyle
H^{\text{\rm dR}}_{n}({\mathcal{L}}_{3}(0))\cong H_{n}({\mathcal{L}})$.

  * •

$\displaystyle z_{i}(a,k)\neq 0$ only if $\displaystyle\omega(\bar{a})\geq
2\varepsilon$ or $\displaystyle a=0$. Moreover, $\displaystyle z_{i}(0,0)$ is
a cycle in $\displaystyle C^{\text{\rm dR}}_{n}({\mathcal{L}}_{1}(0))$ such
that $\displaystyle[z_{i}(0,0)]$ corresponds to $\displaystyle(-1)^{n+1}[L]$
via the isomorphism $\displaystyle H^{\text{\rm
dR}}_{n}({\mathcal{L}}_{1}(0))\cong H_{n}({\mathcal{L}})$.

######  Remark 6.2.

$\displaystyle\partial z-[x,z]=0$ follows from $\displaystyle\partial
x-\frac{1}{2}[x,x]=0$ and $\displaystyle\partial y-[x,y]=z$, thus the
assumption $\displaystyle\partial\bar{z}_{i}-[\bar{x}_{i},\bar{z}_{i}]\in
F^{i-2}$ in Theorem 6.1 may seem redundant. However, this assumption is
necessary to carry out the induction argument.

The rest of this section is devoted to the proof of Theorem 5.1 assuming
Theorem 6.1. We first need the following elementary lemma.

######  Lemma 6.3.

Let $\displaystyle V_{*}$, $\displaystyle W_{*}$ be chain complexes and
$\displaystyle e:V_{*}\to W_{*}$ be a surjective quasi-isomorphism. For any
$\displaystyle x\in V_{*}$ and $\displaystyle y\in W_{*+1}$ such that

$\displaystyle\partial x=0,\qquad e(x)=\partial y$

there exists $\displaystyle\bar{y}\in V_{*+1}$ such that $\displaystyle
e(\bar{y})=y$ and $\displaystyle\partial\bar{y}=x$.

###### Proof.

Since $\displaystyle e$ is surjective, there exists $\displaystyle z\in
V_{*+1}$ such that $\displaystyle e(z)=y$. Then $\displaystyle e(x-\partial
z)=0$ and $\displaystyle\partial(x-\partial z)=0$. Thus $\displaystyle
x-\partial z$ is a cycle in $\displaystyle\text{\rm Ker}\,e$. Since
$\displaystyle\text{\rm Ker}\,e$ is acyclic, there exists $\displaystyle
w\in\text{\rm Ker}\,e$ such that $\displaystyle\partial w=x-\partial z$. Then,
$\displaystyle\bar{y}:=z+w$ satisfies the condition of the lemma. ∎

Key arguments are summarized in Lemma 6.4 below.

######  Lemma 6.4.

Let $\displaystyle I$, $\displaystyle U$ be as in Theorem 6.1. There exists a
sequence

$\displaystyle(x_{i,j},y_{i,j},z_{i,j},\bar{x}_{i,j},\bar{y}_{i,j},\bar{z}_{i,j})_{i\geq
I,\,j\geq 0}$

satisfying the following conditions for every $\displaystyle i\geq I$ and
$\displaystyle j\geq 0$:

(12) $x_{i,0}=x_{i},\quad y_{i,0}=y_{i},\quad
z_{i,0}=z_{i},\quad\bar{x}_{i,0}=\bar{x}_{i},\quad\bar{y}_{i,0}=\bar{y}_{i},\quad\bar{z}_{i,0}=\bar{z}_{i}.$
(13) $\displaystyle\displaystyle x_{i,j}\in F^{1}C_{-1},\quad\bar{x}_{i,j}\in
F^{1}\bar{C}_{-1},\quad y_{i,j}\in F^{-U}C_{2},$
$\displaystyle\displaystyle\bar{y}_{i,j}\in F^{-U}\bar{C}_{2},\quad z_{i,j}\in
F^{-1}C_{1},\quad\bar{z}_{i,j}\in F^{-1}\bar{C}_{1}.$ (14)
$x_{i,j}=e_{-}(\bar{x}_{i,j}),\quad y_{i,j}=e_{-}(\bar{y}_{i,j}),\quad
z_{i,j}=e_{-}(\bar{z}_{i,j}).$ (15)
$\displaystyle\displaystyle\partial\bar{x}_{i,j}-\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}]\in
F^{i+j}\bar{C}_{-2},\qquad\partial\bar{y}_{i,j}-[\bar{x}_{i,j},\bar{y}_{i,j}]-\bar{z}_{i,j}\in
F^{i+j-U-1}\bar{C}_{1}.$
$\displaystyle\displaystyle\partial\bar{z}_{i,j}-[\bar{x}_{i,j},\bar{z}_{i,j}]\in
F^{i+j-2}\bar{C}_{0}.$ (16) $\displaystyle\displaystyle
x_{i+1,j}-e_{+}(\bar{x}_{i,j})\in F^{i+j}C_{-1},\qquad
y_{i+1,j}-e_{+}(\bar{y}_{i,j})\in F^{i+j-U-1}C_{2},$
$\displaystyle\displaystyle z_{i+1,j}-e_{+}(\bar{z}_{i,j})\in F^{i+j-2}C_{1}.$
(17) $\bar{x}_{i,j+1}-\bar{x}_{i,j}\in
F^{i+j}\bar{C}_{-1},\quad\bar{y}_{i,j+1}-\bar{y}_{i,j}\in
F^{i+j-U-1}\bar{C}_{2},\quad\bar{z}_{i,j+1}-\bar{z}_{i,j}\in
F^{i+j-2}\bar{C}_{1}.$

Moreover we require the following conditions:

  * •

$\displaystyle\bar{x}_{i,j}(a,k)\neq 0$ only if
$\displaystyle\omega(\bar{a})\geq 2\varepsilon$ or $\displaystyle a=0$,
$\displaystyle k\geq 2$. Moreover, $\displaystyle\bar{x}_{i,j}(0,2)$ is a
cycle in $\displaystyle C^{\text{\rm dR}}_{n}({\mathcal{L}}_{3}(0))$ such that
$\displaystyle[\bar{x}_{i,j}(0,2)]$ corresponds to
$\displaystyle(-1)^{n+1}[L]$.

  * •

$\displaystyle\bar{z}_{i,j}(a,k)\neq 0$ only if
$\displaystyle\omega(\bar{a})\geq 2\varepsilon$ or $\displaystyle a=0$.
Moreover, $\displaystyle\bar{z}_{i,j}(0,0)$ is a cycle in $\displaystyle
C^{\text{\rm dR}}_{n}({\mathcal{L}}_{1}(0))$ such that
$\displaystyle[\bar{z}_{i,j}(0,0)]$ corresponds to
$\displaystyle(-1)^{n+1}[L]$.

###### Proof.

The proof is by induction on $\displaystyle j$. Let us define $\displaystyle
x_{i,0},\ldots,\bar{z}_{i,0}$ by (12).

Assuming that we have defined a sequence
$\displaystyle(x_{i,j},\ldots,\bar{z}_{i,j})_{i\geq I}$ which satisfies the
required conditions, we are going to define a sequence
$\displaystyle(x_{i,j+1},\ldots,\bar{z}_{i,j+1})_{i\geq I}$.

Let us set

$\displaystyle\displaystyle\Delta^{i}_{x}$
$\displaystyle\displaystyle:=x_{i+1,j}-e_{+}(\bar{x}_{i,j})\in F^{i+j}C_{-1},$
$\displaystyle\displaystyle\Delta^{i}_{y}$
$\displaystyle\displaystyle:=y_{i+1,j}-e_{+}(\bar{y}_{i,j})\in
F^{i+j-U-1}C_{2},$ $\displaystyle\displaystyle\Delta^{i}_{z}$
$\displaystyle\displaystyle:=z_{i+1,j}-e_{+}(\bar{z}_{i,j})\in
F^{i+j-2}C_{1}.$

Since $\displaystyle e_{-}$ preserves $\displaystyle\partial$,
$\displaystyle[\,,\,]$ and filtrations, (14) and (15) show that

$\displaystyle\displaystyle\partial
x_{i+1,j}-\frac{1}{2}[x_{i+1,j},x_{i+1,j}]\in F^{i+j+1}C_{-2},$
$\displaystyle\displaystyle\partial
y_{i+1,j}-[x_{i+1,j},y_{i+1,j}]-z_{i+1,j}\in F^{i+j-U}C_{1},$
$\displaystyle\displaystyle\partial z_{i+1,j}-[x_{i+1,j},z_{i+1,j}]\in
F^{i+j-1}C_{-2}.$

Then we obtain

$\displaystyle\displaystyle\partial\Delta^{i}_{x}+e_{+}(\partial\bar{x}_{i,j}-\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}])\in
F^{i+j+1}C_{-2},$
$\displaystyle\displaystyle\partial\Delta^{i}_{y}+e_{+}(\partial\bar{y}_{i,j}-[\bar{x}_{i,j},\bar{y}_{i,j}]-\bar{z}_{i,j})\in
F^{i+j-U}C_{1},$
$\displaystyle\displaystyle\partial\Delta^{i}_{z}+e_{+}(\partial\bar{z}_{i,j}-[\bar{x}_{i,j},\bar{z}_{i,j}])\in
F^{i+j-1}C_{0}.$

On the other hand, we obtain

$\displaystyle\displaystyle\partial\bigg{(}\partial\bar{x}_{i,j}-\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}]\bigg{)}$
$\displaystyle\displaystyle=-\frac{1}{2}\bigg{(}\bigg{[}\partial\bar{x}_{i,j}-\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}],\bar{x}_{i,j}\bigg{]}-\bigg{[}\bar{x}_{i,j},\partial\bar{x}_{i,j}-\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}]\bigg{]}\bigg{)}\in
F^{i+j+1}\bar{C}_{-3},$
$\displaystyle\displaystyle\partial(\partial\bar{y}_{i,j}-[\bar{x}_{i,j},\bar{y}_{i,j}]-\bar{z}_{i,j})$
$\displaystyle\displaystyle=-\bigg{[}\partial\bar{x}_{i,j}-\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}],\bar{y}_{i,j}\bigg{]}+[\bar{x}_{i,j},\partial\bar{y}_{i,j}-[\bar{x}_{i,j},\bar{y}_{i,j}]-\bar{z}_{i,j}]$
$\displaystyle\displaystyle\qquad-(\partial\bar{z}_{i,j}-[\bar{x}_{i,j},\bar{z}_{i,j}])\in
F^{i+j-U}\bar{C}_{0},$
$\displaystyle\displaystyle\partial(\partial\bar{z}_{i,j}-[\bar{x}_{i,j},\bar{z}_{i,j}])$
$\displaystyle\displaystyle=-\bigg{[}\partial\bar{x}_{i,j}-\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}],\bar{z}_{i,j}\bigg{]}+[\bar{x}_{i,j},\partial\bar{z}_{i,j}-[\bar{x}_{i,j},\bar{z}_{i,j}]]\in
F^{i+j-1}\bar{C}_{-1}.$

Applying Lemma 6.3 for

$\displaystyle e_{+}:F^{D}\bar{C}/F^{D+1}\bar{C}\to F^{D}C/F^{D+1}C$

where $\displaystyle D=i+j,i+j-U-1,i+j-2$ (recall that $\displaystyle e_{+}$
is a surjective quasi-isomorphism, as we have seen in Section 4.5), we show
that there exist

$\displaystyle\bar{\Delta}^{i}_{x}\in
F^{i+j}\bar{C}_{-1},\quad\bar{\Delta}^{i}_{y}\in
F^{i+j-U-1}\bar{C}_{2},\quad\bar{\Delta}^{i}_{z}\in F^{i+j-2}\bar{C}_{1}$

such that

$\displaystyle e_{+}(\bar{\Delta}^{i}_{x})-\Delta^{i}_{x}\in
F^{i+j+1}C_{-1},\quad e_{+}(\bar{\Delta}^{i}_{y})-\Delta^{i}_{y}\in
F^{i+j-U}C_{2},\quad e_{+}(\bar{\Delta}^{i}_{z})-\Delta^{i}_{z}\in
F^{i+j-1}C_{1}$

and

$\displaystyle\displaystyle\partial\bar{\Delta}^{i}_{x}+(\partial\bar{x}_{i,j}-\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}])\in
F^{i+j+1}\bar{C}_{-2},$
$\displaystyle\displaystyle\partial\bar{\Delta}^{i}_{y}+(\partial\bar{y}_{i,j}-[\bar{x}_{i,j},\bar{y}_{i,j}]-\bar{z}_{i,j})\in
F^{i+j-U}\bar{C}_{1},$
$\displaystyle\displaystyle\partial\bar{\Delta}^{i}_{z}+(\partial\bar{z}_{i,j}-[\bar{x}_{i,j},\bar{z}_{i,j}])\in
F^{i+j-1}\bar{C}_{0}.$

######  Remark 6.5.

  * •

The $\displaystyle(a,k)$-components of $\displaystyle\Delta^{i}_{x}$ and
$\displaystyle\partial\bar{x}_{i,j}+\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}]$
are nonzero only if $\displaystyle\omega(\bar{a})\geq 2\varepsilon$ or
$\displaystyle a=0$. Then we may take $\displaystyle\bar{\Delta}^{i}_{x}$ so
that its $\displaystyle(a,k)$-component is nonzero only if
$\displaystyle\omega(\bar{a})\geq 2\varepsilon$ or $\displaystyle a=0$.

  * •

Similarly, we can take $\displaystyle\bar{\Delta}^{i}_{z}$ so that
$\displaystyle\bar{\Delta}^{i}_{z}(a,k)\neq 0$ only if
$\displaystyle\omega(\bar{a})\geq 2\varepsilon$ or $\displaystyle a=0$.

  * •

Since $\displaystyle\bar{\Delta}^{i}_{x}\in F^{2}\bar{C}_{-1}$ and
$\displaystyle\bar{\Delta}^{i}_{z}\in F^{0}\bar{C}_{1}$, it follows that
$\displaystyle\bar{\Delta}^{i}_{x}(0,k)=0$ if $\displaystyle k=0,1,2$ and
$\displaystyle\bar{\Delta}^{i}_{z}(0,k)=0$ if $\displaystyle k=0$.

Finally, let us set

$\displaystyle\bar{x}_{i,j+1}:=\bar{x}_{i,j}+\bar{\Delta}^{i}_{x},\quad\bar{y}_{i,j+1}:=\bar{y}_{i,j}+\bar{\Delta}^{i}_{y},\quad\bar{z}_{i,j+1}:=\bar{z}_{i,j}+\bar{\Delta}^{i}_{z}$

and

$\displaystyle x_{i,j+1}:=e_{-}(\bar{x}_{i,j+1}),\quad
y_{i,j+1}:=e_{-}(\bar{y}_{i,j+1}),\quad z_{i,j+1}:=e_{-}(\bar{z}_{i,j+1}).$

Now we have to check that, for every $\displaystyle i\geq I$ there holds

$\displaystyle\displaystyle\partial\bar{x}_{i,j+1}-\frac{1}{2}[\bar{x}_{i,j+1},\bar{x}_{i,j+1}]\in
F^{i+j+1}\bar{C}_{-2},$
$\displaystyle\displaystyle\partial\bar{y}_{i,j+1}-[\bar{x}_{i,j+1},\bar{y}_{i,j+1}]-\bar{z}_{i,j+1}\in
F^{i+j-U}\bar{C}_{1},$
$\displaystyle\displaystyle\partial\bar{z}_{i,j+1}-[\bar{x}_{i,j+1},\bar{z}_{i,j+1}]\in
F^{i+j-1}\bar{C}_{0},$ $\displaystyle\displaystyle
x_{i+1,j+1}-e_{+}(\bar{x}_{i,j+1})\in F^{i+j+1}C_{-1},$
$\displaystyle\displaystyle y_{i+1,j+1}-e_{+}(\bar{y}_{i,j+1})\in
F^{i+j-U}C_{-2},$ $\displaystyle\displaystyle
z_{i+1,j+1}-e_{+}(\bar{z}_{i,j+1})\in F^{i+j-1}C_{1}.$

The first formula holds since

$\displaystyle\partial\bar{x}_{i,j+1}-\frac{1}{2}[\bar{x}_{i,j+1},\bar{x}_{i,j+1}]=\biggl{(}\partial\bar{x}_{i,j}+\partial\bar{\Delta}^{i}_{x}-\frac{1}{2}[\bar{x}_{i,j},\bar{x}_{i,j}]\biggr{)}-\frac{1}{2}[\bar{\Delta}^{i}_{x},\bar{\Delta}^{i}_{x}]-[\bar{x}_{i,j},\bar{\Delta}^{i}_{x}]$

and all three terms in the RHS are in $\displaystyle F^{i+j+1}\bar{C}_{-2}$.
Similarly, the second and the third formulas follow from

$\displaystyle\displaystyle\partial\bar{y}_{i,j+1}-[\bar{x}_{i,j+1},\bar{y}_{i,j+1}]-\bar{z}_{i,j+1}=$
$\displaystyle\displaystyle\quad(\partial\bar{\Delta}^{i}_{y}+\partial\bar{y}_{i,j}-[\bar{x}_{i,j},\bar{y}_{i,j}]-\bar{z}_{i,j})-[\bar{\Delta}^{i}_{x},\bar{y}_{i,j}]-[\bar{x}_{i,j},\bar{\Delta}^{i}_{y}]-[\bar{\Delta}^{i}_{x},\bar{\Delta}^{i}_{y}]-\bar{\Delta}^{i}_{z},$
$\displaystyle\displaystyle\partial\bar{z}_{i,j+1}-[\bar{x}_{i,j+1},\bar{z}_{i,j+1}]=$
$\displaystyle\displaystyle\quad(\partial\bar{z}_{i,j}+\partial\bar{\Delta}^{i}_{z}-[\bar{x}_{i,j},\bar{z}_{i,j}])-[\bar{\Delta}^{i}_{x},\bar{z}_{i,j}]-[\bar{x}_{i,j},\bar{\Delta}^{i}_{z}]-[\bar{\Delta}^{i}_{x},\bar{\Delta}^{i}_{z}].$

The fourth formula holds since

$\displaystyle\displaystyle x_{i+1,j+1}-e_{+}(\bar{x}_{i,j+1})$
$\displaystyle\displaystyle=(x_{i+1,j+1}-x_{i+1,j})+(x_{i+1,j}-e_{+}(\bar{x}_{i,j}))+e_{+}(\bar{x}_{i,j}-\bar{x}_{i,j+1})$
$\displaystyle\displaystyle=e_{-}(\bar{\Delta}^{i+1}_{x})+(\Delta^{i}_{x}-e_{+}(\bar{\Delta}^{i}_{x})).$

Similarly, the fifth and the sixth formulas follow from

$\displaystyle\displaystyle y_{i+1,j+1}-e_{+}(\bar{y}_{i,j+1})$
$\displaystyle\displaystyle=(y_{i+1,j+1}-y_{i+1,j})+(y_{i+1,j}-e_{+}(\bar{y}_{i,j}))+e_{+}(\bar{y}_{i,j}-\bar{y}_{i,j+1})$
$\displaystyle\displaystyle=e_{-}(\bar{\Delta}^{i+1}_{y})+\Delta^{i}_{y}-e_{+}(\bar{\Delta}^{i}_{y}),$
$\displaystyle\displaystyle z_{i+1,j+1}-e_{+}(\bar{z}_{i,j+1})$
$\displaystyle\displaystyle=(z_{i+1,j+1}-z_{i+1,j})+(z_{i+1,j}-e_{+}(\bar{z}_{i,j}))+e_{+}(\bar{z}_{i,j}-\bar{z}_{i,j+1})$
$\displaystyle\displaystyle=e_{-}(\bar{\Delta}^{i+1}_{z})+\Delta^{i}_{z}-e_{+}(\bar{\Delta}^{i}_{z}).$

Finally, by the induction hypothesis, $\displaystyle\bar{x}_{i,j}$ and
$\displaystyle\bar{z}_{i,j}$ satisfy the last two conditions in the statement.
Then Remark 6.5 shows that $\displaystyle\bar{x}_{i,j+1}$ and
$\displaystyle\bar{z}_{i,j+1}$ also satisfy these conditions. ∎

###### Proof of Theorem 5.1 assuming Theorem 6.1.

Let us fix an integer $\displaystyle i\geq I$. Then for every $\displaystyle
j\geq 0$, there holds

$\displaystyle x_{i,j+1}-x_{i,j}\in F^{i+j}C_{-1},\quad y_{i,j+1}-y_{i,j}\in
F^{i+j-U-1}C_{2},\quad z_{i,j+1}-z_{i,j}\in F^{i+j-2}C_{1}.$

Then, the limits

$\displaystyle x:=\lim_{j\to\infty}x_{i,j}\in\widehat{C}_{-1},\quad
y:=\lim_{j\to\infty}y_{i,j}\in\widehat{C}_{2},\quad
z:=\lim_{j\to\infty}z_{i,j}\in\widehat{C}_{1}$

exist, and satisfy

$\displaystyle\partial x-\frac{1}{2}[x,x]=0,\qquad\partial y-[x,y]=z.$

Conditions (iii), (iv) in Theorem 5.1 follow from the last two conditions in
Lemma 6.4. ∎

##  7\. Proof of Theorem 6.1 modulo technical results

Our arguments in the rest of this paper are based on the theory of Kuranishi
structures (abbreviated as K-structures), in particular we heavily rely on
[10] by Fukaya-Oh-Ohta-Ono. Section 10 very briefly explains some notions in
the theory of Kuranishi structures.

The goal of this section is to prove Theorem 6.1 modulo technical results,
which are proved in Sections 8 and 9. Our proof of Theorem 6.1 is based on the
following principle: given a compact K-space
$\displaystyle(X,\widehat{{\mathcal{U}}})$ with a CF-perturbation
$\displaystyle\widehat{{\mathfrak{S}}}=(\widehat{{\mathfrak{S}}}^{\varepsilon})_{0<\varepsilon\leq
1}$ and a strongly smooth map
$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to{\mathcal{L}}_{k+1}$,
one can define a de Rham chain
$\displaystyle\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{{\mathfrak{S}}}^{\varepsilon})\in
C^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1})$ for sufficiently small
$\displaystyle\varepsilon$. In Section 7.1, we state this principle and its
variants in rigorous forms. Proofs of these results will be carried out in
Section 8. In Sections 7.2–7.6, we apply this principle to prove Theorem 6.1.

In Section 7.2, we introduce compactified moduli spaces of (perturbed) pseudo-
holomorphic disks, and equip K-structures (with boundaries and corners) on
these spaces so that the (normalized) boundary of each moduli space is
naturally isomorphic to a disjoint union of fiber products of lower (in terms
of symplectic area) moduli spaces. We call this relation “boundary
$\displaystyle\cong$ fiber product” relation.

To apply the principle in Section 7.1 to moduli spaces defined in Section 7.2,
we have to define strongly smooth maps from these moduli spaces to spaces of
smooth loops with marked points, so that these maps are compatible with the
“boundary $\displaystyle\cong$ fiber product” relation. The idea is to assign
the boundary loop to each pseudo-holomorphic disk, however we have to be
careful to achieve smoothness, which can be very subtle on boundary of moduli
spaces. In this paper, we achieve smoothness in the following two steps. The
first step is to define strongly continuous maps from these moduli spaces to
spaces of continuous loops with marked points. This is much easier and we
explain details in Sections 7.3 and 7.4. The second step is to approximate
(with respect to the $\displaystyle C^{0}$-topology) these continuous maps by
smooth maps. In this step we use an abstract approximation result
($\displaystyle C^{0}$-approximation lemma) which we state in Section 7.5 and
prove in Section 9. In Section 7.5, we also introduce CF-perturbations of
K-structures on these moduli spaces. Finally in Section 7.6 we complete the
proof assuming technical results presented in Sections 7.1 and 7.5, which are
proved in Sections 8 and 9, respectively.

###  7.1. Strongly smooth map from a K-space with a CF-perturbation gives a de
Rham chain

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a K-space equipped with a
differential form $\displaystyle\widehat{\omega}$ and a CF-perturbation
$\displaystyle\widehat{{\mathfrak{S}}}=(\widehat{{\mathfrak{S}}}^{\varepsilon})_{0<\varepsilon\leq
1}$. (The notion of CF-perturbation is explained in Section 7 in [10]). Given
a strongly smooth map (see Definition 7.1 below)
$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to{\mathcal{L}}_{k+1}$,
we define

$\displaystyle\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon})\in
C^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1})$

for sufficiently small $\displaystyle\varepsilon>0$, and state Stokes’ formula
and fiber product formula. We also consider the version that $\displaystyle X$
is an admissible K-space with boundaries and corners. The goal of this
subsection is to state these results in a formal manner so that we can use
them to complete the proof of Theorem 6.1. Proofs of these results are
explained in Section 8.

####  7.1.1. K-space without boundary

Let us start from the following definition:

######  Definition 7.1.

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a K-space without boundary.
A strongly smooth map from $\displaystyle(X,\widehat{{\mathcal{U}}})$ to
$\displaystyle{\mathcal{L}}_{k+1}$ is a family
$\displaystyle\widehat{f}=(f_{p})_{p\in X}$ such that the following conditions
hold:

  * •

$\displaystyle f_{p}$ is a smooth map (in the sense of Definition 4.1) from
$\displaystyle U_{p}$ to $\displaystyle{\mathcal{L}}_{k+1}$ for every
$\displaystyle p\in X$.

  * •

For every $\displaystyle p\in X$ and $\displaystyle q\in\text{\rm
Im}\,\psi_{p}$, there holds $\displaystyle
f_{p}\circ\varphi_{pq}=f_{q}|_{U_{pq}}$.

The underlying set-theoretic map $\displaystyle X\to{\mathcal{L}}_{k+1}$ is
denoted by $\displaystyle f$.

######  Theorem 7.2.

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a compact, oriented K-space
of dimension $\displaystyle d$ with a strongly smooth map
$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to{\mathcal{L}}_{k+1}$,
a differential form $\displaystyle\widehat{\omega}$, and a CF-perturbation
$\displaystyle\widehat{{\mathfrak{S}}}=(\widehat{{\mathfrak{S}}}^{\varepsilon})_{0<\varepsilon\leq
1}$. We assume that $\displaystyle\widehat{{\mathfrak{S}}}$ is transversal to
$\displaystyle 0$, and $\displaystyle\text{\rm
ev}\,_{0}\circ\widehat{f}:(X,\widehat{{\mathcal{U}}})\to L$ is strongly
submersive with respect to $\displaystyle\widehat{{\mathfrak{S}}}$ (see
Definition 9.2 in [10]). Under these assumptions, one can define a de Rham
chain

(18)
$\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon})\in
C^{\text{\rm dR}}_{d-|\hat{\omega}|}({\mathcal{L}}_{k+1})$

for sufficiently small $\displaystyle\varepsilon$, so that Stokes’ formula
(Theorem 7.5) and the fiber product formula (Theorem 7.6) hold.

######  Remark 7.3.

When $\displaystyle\widehat{\omega}\equiv 1$, we abbreviate the LHS of (18) as
$\displaystyle\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{{\mathfrak{S}}}^{\varepsilon})$.

######  Remark 7.4.

In the statement of Theorem 7.2, “for sufficiently small
$\displaystyle\varepsilon$” is used slightly loosely. Strictly speaking, it
means the following: the definition of
$\displaystyle\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon})$
involves some auxiliary choices (good coordinate system and partition of
unity), however it is well-defined in the sense of $\displaystyle\spadesuit$
(see Definition 8.12). Namely, for any choices $\displaystyle c_{1}$ and
$\displaystyle c_{2}$, there exists $\displaystyle\varepsilon(c_{1},c_{2})>0$
such that, the definition with $\displaystyle c_{1}$ coincides with the
definition with $\displaystyle c_{2}$ when
$\displaystyle\varepsilon\in(0,\varepsilon(c_{1},c_{2}))$. A similar remark
applies to all places in Section 7.1 where we use “for sufficiently small
$\displaystyle\varepsilon$”.

Stokes’ formula is easy to state:

######  Theorem 7.5.

For sufficiently small $\displaystyle\varepsilon>0$, there holds

$\displaystyle\partial(\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon}))=(-1)^{|\widehat{\omega}|+1}\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},d\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon}).$

To state the fiber product formula, we need some notations. Suppose, for
$\displaystyle i=1,2$, we have the following data:

  * •

$\displaystyle(X_{i},\widehat{{\mathcal{U}}}_{i})$: a compact oriented K-space
of dimension $\displaystyle d_{i}$,

  * •

a strongly smooth map
$\displaystyle\widehat{f}_{i}:(X_{i},\widehat{{\mathcal{U}}_{i}})\to{\mathcal{L}}_{k_{i}+1}$,

  * •

a differential form $\displaystyle\widehat{\omega}_{i}$ on
$\displaystyle(X_{i},\widehat{{\mathcal{U}}}_{i})$,

  * •

a CF-perturbation $\displaystyle\widehat{{\mathfrak{S}}}_{i}$ on
$\displaystyle(X_{i},\widehat{{\mathcal{U}}}_{i})$ such that
$\displaystyle\text{\rm
ev}\,_{0}\circ\widehat{f}_{i}:(X_{i},\widehat{{\mathcal{U}}}_{i})\to L$ is
strongly submersive with respect to
$\displaystyle\widehat{{\mathfrak{S}}}_{i}$.

Due to Theorem 7.2, one can define
$\displaystyle(\widehat{f}_{i})_{*}(X_{i},\widehat{{\mathcal{U}}}_{i},\widehat{\omega}_{i},\widehat{{\mathfrak{S}}}^{\varepsilon}_{i})\in
C^{\text{\rm dR}}_{d_{i}-|\widehat{\omega}_{i}|}({\mathcal{L}}_{k_{i}+1})$ for
sufficiently small $\displaystyle\varepsilon$. On the other hand, for every
$\displaystyle j\in\\{1,\ldots,k_{1}\\}$, one can take a fiber product of
K-spaces and define

$\displaystyle(X_{12},\widehat{{\mathcal{U}}}_{12}):=(X_{1},\widehat{{\mathcal{U}}}_{1})\,_{\text{\rm
ev}\,_{j}\circ\widehat{f}_{1}}\\!\times_{\text{\rm
ev}\,_{0}\circ\widehat{f}_{2}}(X_{2},\widehat{{\mathcal{U}}}_{2}).$

For the definition of fiber product of K-spaces, see Section 4.1 in [10]. Our
sign convention for the fiber product is explained in Section 4.2. One can
also define fiber product of CF-perturbations
$\displaystyle\widehat{{\mathfrak{S}}}_{1}\times\widehat{{\mathfrak{S}}}_{2}$
on $\displaystyle(X_{12},\widehat{{\mathcal{U}}}_{12})$ (see Definition 10.13
in [10]). Finally we define a differential form
$\displaystyle\widehat{\omega}_{12}$ on
$\displaystyle(X_{12},\widehat{{\mathcal{U}}}_{12})$ by

$\displaystyle\widehat{\omega}_{12}:=(-1)^{(d_{1}-|\hat{\omega}_{1}|-n)|\hat{\omega}_{2}|}\cdot\widehat{\omega}_{1}\times\widehat{\omega}_{2}$

and a strongly smooth map
$\displaystyle\widehat{f}_{12}:(X_{12},\widehat{{\mathcal{U}}}_{12})\to{\mathcal{L}}_{k_{1}+k_{2}}$
by

(19)
$\displaystyle\displaystyle(f_{12})_{(p_{1},p_{2})}(x_{1},x_{2}):=\text{\rm
con}\,_{j}((f_{1})_{p_{1}}(x_{1}),(f_{2})_{p_{2}}(x_{2})),$
$\displaystyle\displaystyle(x_{1}\in U_{p_{1}},\,x_{2}\in
U_{p_{2}},\,\text{\rm ev}\,_{j}\circ f_{p_{1}}(x_{1})=\text{\rm ev}\,_{0}\circ
f_{p_{2}}(x_{2})).$

Then one can state the fiber product formula (the proof will be sketched in
Section 8):

######  Theorem 7.6.

In the situation described above, there holds

$\displaystyle(\widehat{f}_{12})_{*}(X_{12},\widehat{{\mathcal{U}}}_{12},\widehat{\omega}_{12},\widehat{{\mathfrak{S}}}_{12}^{\varepsilon})=(\widehat{f}_{1})_{*}(X_{1},\widehat{{\mathcal{U}}}_{1},\widehat{\omega}_{1},\widehat{{\mathfrak{S}}}_{1}^{\varepsilon})\circ_{j}(\widehat{f}_{2})_{*}(X_{2},\widehat{{\mathcal{U}}}_{2},\widehat{\omega}_{2},\widehat{{\mathfrak{S}}}_{2}^{\varepsilon})$

for sufficiently small $\displaystyle\varepsilon>0$, where the RHS is the
fiber product of de Rham chains (see Section 4.3).

####  7.1.2. Admissible K-space

Next we consider the case that $\displaystyle X$ is an admissible K-space,
which roughly means that $\displaystyle X$ is a K-space with boundaries and
corners, and all coordinate change data decay exponentially near boundaries.
For rigorous definitions of admissible manifolds, vector bundles, and
K-structures etc., see Section 25 in [10].

######  Definition 7.7.

  1. (i):

Let $\displaystyle U$ be an admissible manifold. A $\displaystyle
C^{\infty}$-map

$\displaystyle f:U\to{\mathcal{L}}_{k+1};\quad
u\mapsto(T(u),\gamma(u),t_{1}(u),\ldots,t_{k}(u))$

is called admissible if the following conditions hold:

     * •

$\displaystyle T,t_{1},\ldots,t_{k}:U\to{\mathbb{R}}$ are admissible (see
Definition 25.3 in [10]).

     * •

$\displaystyle U\times S^{1}\to L;\,(u,\theta)\mapsto\gamma(u)(T(u)\theta)$ is
admissible (see Remark 7.8 below).

     * •

$\displaystyle\text{\rm ev}\,_{0}\circ f:U\to L$ is strata-wise submersive
(i.e. the restriction of $\displaystyle\text{\rm ev}\,_{0}\circ f$ to each
open strata of $\displaystyle U$ is a submersion).

  2. (ii):

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be an admissible K-space. An
admissible map from $\displaystyle(X,\widehat{{\mathcal{U}}})$ to
$\displaystyle{\mathcal{L}}_{k+1}$ is a family
$\displaystyle\widehat{f}=(f_{p})_{p\in X}$ such that

     * •

$\displaystyle f_{p}$ is an admissible map from $\displaystyle U_{p}$ to
$\displaystyle{\mathcal{L}}_{k+1}$ for every $\displaystyle p\in X$.

     * •

For every $\displaystyle p\in X$ and $\displaystyle q\in\text{\rm
Im}\,\psi_{p}$, there holds $\displaystyle
f_{p}\circ\varphi_{pq}=f_{q}|_{U_{pq}}$.

The underlying set-theoretic map $\displaystyle X\to{\mathcal{L}}_{k+1}$ is
denoted by $\displaystyle f$.

######  Remark 7.8.

In the second bullet of the definition (i) above, the admissible structure on
$\displaystyle U\times S^{1}$ is defined as follows: let
$\displaystyle(x_{1},\ldots,x_{l},y_{1},\ldots,y_{k})\,(x_{i}\in{\mathbb{R}},y_{i}\in{\mathbb{R}}_{\geq
0})$ be an admissible chart on $\displaystyle U$ near a point on the
codimension $\displaystyle k$ corner, and $\displaystyle z$ be any chart on
$\displaystyle S^{1}$. Then we define the admissible structure on
$\displaystyle U\times S^{1}$ so that
$\displaystyle(x_{1},\ldots,x_{l},y_{1},\ldots,y_{k},z)$ is an admissible
chart on $\displaystyle U\times S^{1}$.

The next result is a version of Theorem 7.2 for admissible K-spaces.

######  Theorem 7.9.

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a compact, oriented,
admissible K-space of dimension $\displaystyle d$, and
$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to{\mathcal{L}}_{k+1}$
be an admissible map, $\displaystyle\widehat{\omega}$ be an admissible
differential form on $\displaystyle(X,\widehat{{\mathcal{U}}})$, and
$\displaystyle\widehat{{\mathfrak{S}}}$ be an admissible CF-perturbation of
$\displaystyle(X,\widehat{{\mathcal{U}}})$. We assume that
$\displaystyle\widehat{{\mathfrak{S}}}$ is transversal to $\displaystyle 0$,
and $\displaystyle\text{\rm
ev}\,_{0}\circ\widehat{f}:(X,\widehat{{\mathcal{U}}})\to L$ is strata-wise
strongly submersive with respect to $\displaystyle\widehat{{\mathfrak{S}}}$.
Then one can define

$\displaystyle\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon})\in
C^{\text{\rm dR}}_{d-|\hat{\omega}|}({\mathcal{L}}_{k+1})$

for sufficiently small $\displaystyle\varepsilon>0$, so that Stokes’ formula
(Theorem 7.11) and the fiber product formula (Theorem 7.12) hold.

######  Remark 7.10.

To define the de Rham chain
$\displaystyle\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon})$,
we put a collar on $\displaystyle X$ and take an auxiliary cut-off function on
the collar. It seems that the chain
$\displaystyle\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon})$
depends on choice of the cut-off function. See Section 8.1.2 for more details.

Now we state Stokes’ formula:

######  Theorem 7.11.

In the situation of Theorem 7.9, for sufficiently small
$\displaystyle\varepsilon>0$ there holds

$\displaystyle\partial(\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon}))=(-1)^{|\widehat{\omega}|}\widehat{f}_{*}(\partial
X,\widehat{{\mathcal{U}}}|_{\partial X},\widehat{\omega}|_{\partial
X},\widehat{{\mathfrak{S}}}^{\varepsilon}|_{\partial
X})+(-1)^{|\widehat{\omega}|+1}\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},d\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon}),$

where $\displaystyle\partial X$ denotes the normalized boundary of
$\displaystyle X$, which is again an admissible K-space.

Next we state the fiber product formula. Suppose, for $\displaystyle i=1,2$,
we have the following data:

  * •

An admissible K-space $\displaystyle(X_{i},\widehat{{\mathcal{U}}}_{i})$.

  * •

An admissible map
$\displaystyle\widehat{f}_{i}:(X_{i},\widehat{{\mathcal{U}}}_{i})\to{\mathcal{L}}_{k_{i}+1}$.

  * •

An admissible differential form $\displaystyle\widehat{\omega}_{i}$ on
$\displaystyle(X_{i},\widehat{{\mathcal{U}}}_{i})$.

  * •

An admissible CF-perturbation $\displaystyle\widehat{{\mathfrak{S}}}_{i}$ on
$\displaystyle(X_{i},\widehat{{\mathcal{U}}}_{i})$ such that
$\displaystyle\text{\rm
ev}\,_{0}\circ\widehat{f}_{i}:(X_{i},\widehat{{\mathcal{U}}}_{i})\to L$ is
strata-wise strongly submersive with respect to
$\displaystyle\widehat{{\mathfrak{S}}}_{i}$.

Under these assumptions, for every $\displaystyle j\in\\{1,\ldots,k_{1}\\}$,
the fiber product

$\displaystyle(X_{12},\widehat{{\mathcal{U}}}_{12}):=(X_{1},\widehat{{\mathcal{U}}}_{1})\,_{\text{\rm
ev}\,_{j}\circ f_{1}}\\!\times_{\text{\rm ev}\,_{0}\circ
f_{2}}(X_{2},\widehat{{\mathcal{U}}}_{2})$

has an admissible K-structure. Moreover, both
$\displaystyle\widehat{\omega}_{1}\times\widehat{\omega}_{2}$ and
$\displaystyle\widehat{{\mathfrak{S}}}_{1}\times\widehat{{\mathfrak{S}}}_{2}$
are admissible. Finally, a strongly smooth map
$\displaystyle\widehat{f}_{12}:(X_{12},\widehat{{\mathcal{U}}}_{12})\to{\mathcal{L}}_{k_{1}+k_{2}}$
which is defined by the same formula as (19) is also admissible.

######  Theorem 7.12.

In the situation described above, there holds

$\displaystyle(\widehat{f}_{12})_{*}(X_{12},\widehat{{\mathcal{U}}}_{12},\widehat{\omega}_{12},\widehat{{\mathfrak{S}}}_{12}^{\varepsilon})=(\widehat{f}_{1})_{*}(X_{1},\widehat{{\mathcal{U}}}_{1},\widehat{\omega}_{1},\widehat{{\mathfrak{S}}}_{1}^{\varepsilon})\circ_{j}(\widehat{f}_{2})_{*}(X_{2},\widehat{{\mathcal{U}}}_{2},\widehat{\omega}_{2},\widehat{{\mathfrak{S}}}_{2}^{\varepsilon})$

for sufficiently small $\displaystyle\varepsilon>0$.

####  7.1.3. Admissible K-space over an interval

Finally we consider the case that $\displaystyle X$ is an admissible K-space
and the target of the map $\displaystyle f$ is
$\displaystyle[a,b]\times{\mathcal{L}}_{k+1}$.

######  Definition 7.13.

  1. (i):

Let $\displaystyle U$ be an admissible manifold. A $\displaystyle
C^{\infty}$-map $\displaystyle f:U\to[a,b]\times{\mathcal{L}}_{k+1}$ is called
admissible if the following conditions hold:

     * •

$\displaystyle\text{\rm pr}_{{\mathcal{L}}_{k+1}}\circ
f:U\to{\mathcal{L}}_{k+1}$ is admissible in the sense of Definition 7.7.

     * •

Let us denote $\displaystyle\tau:=\text{\rm pr}_{[a,b]}\circ f$, and suppose
that $\displaystyle\tau(p)=a$ (resp. $\displaystyle\tau(p)=b$) for
$\displaystyle p\in U$. Then $\displaystyle p$ is on codimension
$\displaystyle k\geq 1$ corner of $\displaystyle U$, and there exists an
admissible chart
$\displaystyle(y,t_{1},\ldots,t_{k})\,(t_{i}\in{\mathbb{R}}_{\geq 0})$ defined
near $\displaystyle p$, such that $\displaystyle p$ corresponds to
$\displaystyle(y_{0},0,\ldots,0)$ and
$\displaystyle\tau(y,t_{1},\ldots,t_{k})=t_{1}+a$ (resp. $\displaystyle
b-t_{1}$).

     * •

$\displaystyle(\tau,\text{\rm ev}\,_{0}\circ\text{\rm
pr}_{{\mathcal{L}}_{k+1}}\circ f):U\to[a,b]\times L$ is a corner stratified
submersion (see Definition 26.3 in [10]).

  2. (ii):

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be an admissible K-space. An
admissible map from $\displaystyle(X,\widehat{{\mathcal{U}}})$ to
$\displaystyle[a,b]\times{\mathcal{L}}_{k+1}$ is a family
$\displaystyle\widehat{f}=(f_{p})_{p\in X}$ such that

     * •

$\displaystyle f_{p}$ is an admissible map from $\displaystyle U_{p}$ to
$\displaystyle[a,b]\times{\mathcal{L}}_{k+1}$ for every $\displaystyle p\in
X$.

     * •

For any $\displaystyle p\in X$ and $\displaystyle q\in\text{\rm
Im}\,\psi_{p}$, there holds $\displaystyle
f_{p}\circ\varphi_{pq}=f_{q}|_{U_{pq}}$.

The underlying set-theoretic map $\displaystyle
X\to[a,b]\times{\mathcal{L}}_{k+1}$ is denoted by $\displaystyle f$.

In the situation of Definition 7.13 (ii), the normalized boundary
$\displaystyle\partial X$ is decomposed as $\displaystyle\partial
X=\partial_{v}X\sqcup\partial_{h}X$, where $\displaystyle\partial_{v}X$ (resp.
$\displaystyle\partial_{h}X$) denotes the vertical (resp. horizontal) boundary
(see Definition 26.10 in [10]). Moreover, $\displaystyle\partial_{v}X$ is
decomposed into
$\displaystyle\partial_{-}X:=f^{-1}(\\{a\\}\times{\mathcal{L}}_{k+1})$ and
$\displaystyle\partial_{+}X:=f^{-1}(\\{b\\}\times{\mathcal{L}}_{k+1})$.

######  Theorem 7.14.

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a compact, oriented,
admissible K-space of dimension $\displaystyle d$, $\displaystyle\widehat{f}$
be an admissible map (in the sense of Definition 7.13) from
$\displaystyle(X,\widehat{{\mathcal{U}}})$ to
$\displaystyle[a,b]\times{\mathcal{L}}_{k+1}$, $\displaystyle\widehat{\omega}$
be an admissible differential form on
$\displaystyle(X,\widehat{{\mathcal{U}}})$,
$\displaystyle\widehat{{\mathfrak{S}}}$ be an admissible CF-perturbation of
$\displaystyle(X,\widehat{{\mathcal{U}}})$. Assume that
$\displaystyle\widehat{{\mathfrak{S}}}$ is transversal to $\displaystyle 0$,
and

$\displaystyle(\text{\rm pr}_{[a,b]}\circ\widehat{f},\text{\rm
ev}\,_{0}\circ\text{\rm
pr}_{{\mathcal{L}}_{k+1}}\circ\widehat{f}):(X,\widehat{{\mathcal{U}}})\to[a,b]\times
L$

is a stratified submersion with respect to
$\displaystyle\widehat{{\mathfrak{S}}}$. Then one can define

$\displaystyle\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon})\in\bar{C}^{\text{\rm
dR}}_{d-|\widehat{\omega}|-1}({\mathcal{L}}_{k+1})$

for sufficiently small $\displaystyle\varepsilon>0$, so that there holds

(20)
$e_{+}(\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon}))=(\widehat{f}|_{\partial_{+}X})_{*}(\partial_{+}X,\widehat{{\mathcal{U}}}|_{\partial_{+}X},\widehat{\omega}|_{\partial_{+}X},\widehat{{\mathfrak{S}}}^{\varepsilon}|_{\partial_{+}X}),$
(21)
$e_{-}(\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon}))=(\widehat{f}|_{\partial_{-}X})_{*}(\partial_{-}X,\widehat{{\mathcal{U}}}|_{\partial_{-}X},\widehat{\omega}|_{\partial_{-}X},\widehat{{\mathfrak{S}}}^{\varepsilon}|_{\partial_{-}X}).$

Moreover Stokes’ formula and the fiber product formula hold.

Let us state Stokes’ formula:

######  Proposition 7.15.

For sufficiently small $\displaystyle\varepsilon>0$, there holds

$\displaystyle\partial\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon})=(-1)^{|\hat{\omega}|}(\widehat{f}|_{\partial_{h}X})_{*}(\partial_{h}X,\widehat{{\mathcal{U}}}|_{\partial_{h}X},\widehat{\omega}|_{\partial_{h}X},\widehat{{\mathfrak{S}}}^{\varepsilon}|_{\partial_{h}X})+(-1)^{|\hat{\omega}|+1}\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},d\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon}).$

Finally we state the fiber product formula. Suppose, for $\displaystyle
i\in\\{1,2\\}$, we have $\displaystyle(X_{i},\widehat{{\mathcal{U}}}_{i})$,
$\displaystyle\widehat{\omega}_{i}$,
$\displaystyle\widehat{{\mathfrak{S}}}_{i}$, $\displaystyle
k_{i}\in{\mathbb{Z}}_{\geq 0}$ and
$\displaystyle\widehat{f}_{i}:(X_{i},\widehat{{\mathcal{U}}}_{i})\to[a,b]\times{\mathcal{L}}_{k_{i}+1}$
satisfying the assumptions in Theorem 7.14. For every $\displaystyle
j\in\\{1,\ldots,k_{1}\\}$, the fiber product

$\displaystyle(X_{12},\widehat{{\mathcal{U}}}_{12}):=(X_{1},\widehat{{\mathcal{U}}}_{1})\,_{\text{\rm
ev}\,_{j}\circ\widehat{f}_{1}}\\!\times_{\text{\rm
ev}\,_{0}\circ\widehat{f}_{2}}(X_{2},\widehat{{\mathcal{U}}}_{2})$

has an admissible K-structure. Moreover, both
$\displaystyle\widehat{\omega}_{1}\times\widehat{\omega}_{2}$ and
$\displaystyle\widehat{{\mathfrak{S}}}_{1}\times\widehat{{\mathfrak{S}}}_{2}$
are admissible. Finally, a smooth map
$\displaystyle\widehat{f}_{12}:(X_{12},\widehat{{\mathcal{U}}}_{12})\to[a,b]\times{\mathcal{L}}_{k_{1}+k_{2}}$
which is defined by the same formula as (19) is also admissible.

######  Theorem 7.16.

In the situation described above, there holds

$\displaystyle(\widehat{f}_{12})_{*}(X_{12},\widehat{{\mathcal{U}}}_{12},\widehat{\omega}_{12},\widehat{{\mathfrak{S}}}_{12}^{\varepsilon})=(\widehat{f}_{1})_{*}(X_{1},\widehat{{\mathcal{U}}}_{1},\widehat{\omega}_{1},\widehat{{\mathfrak{S}}}_{1}^{\varepsilon})\circ_{j}(\widehat{f}_{2})_{*}(X_{2},\widehat{{\mathcal{U}}}_{2},\widehat{\omega}_{2},\widehat{{\mathfrak{S}}}_{2}^{\varepsilon})$

for sufficiently small $\displaystyle\varepsilon>0$.

###  7.2. Moduli spaces of (perturbed) pseudo-holomorphic disks with boundary
marked points

####  7.2.1. Uncompactified moduli spaces

Let $\displaystyle D$ denote the unit holomorphic disk, namely $\displaystyle
D:=\\{z\in{\mathbb{C}}\mid|z|\leq 1\\}$. For every $\displaystyle
k\in{\mathbb{Z}}_{\geq 0}$ and $\displaystyle\beta\in
H_{2}({\mathbb{C}}^{n},L)$, we define a set
$\displaystyle{\mathring{{\mathcal{M}}}}_{k+1}(\beta)$ in the following way.
When $\displaystyle\beta=0$ and $\displaystyle k=0$ or $\displaystyle k=1$, we
define both $\displaystyle{\mathring{{\mathcal{M}}}}_{1}(0)$ and
$\displaystyle{\mathring{{\mathcal{M}}}}_{2}(0)$ to be empty set. In the other
cases, namely $\displaystyle\beta\neq 0$ or $\displaystyle k\neq 0,1$, we
define

$\displaystyle{\mathring{{\mathcal{M}}}}_{k+1}(\beta):=\\{(u,z_{0},\ldots,z_{k})\\}/\text{\rm
Aut}\,(D)$

where $\displaystyle u:(D,\partial D)\to({\mathbb{C}}^{n},L)$ satisfies
$\displaystyle\bar{\partial}u=0$, $\displaystyle[u]=\beta$, and $\displaystyle
z_{0},\ldots,z_{k}$ are distinct points on $\displaystyle\partial D$ aligned
in anti-clockwise order. The right action of $\displaystyle\text{\rm
Aut}\,(D)$ is defined so that

$\displaystyle(u,z_{0},\ldots,z_{k})^{\rho}:=(u\circ\rho,\rho^{-1}(z_{0}),\ldots,\rho^{-1}(z_{k})).$

For each $\displaystyle j\in\\{0,\ldots,k\\}$, we define an evaluation map
$\displaystyle\text{\rm ev}\,_{j}:{\mathring{{\mathcal{M}}}}_{k+1}(\beta)\to
L$ by

(22) $\text{\rm ev}\,_{j}[(u,z_{0},\ldots,z_{k})]:=u(z_{j}).$

Next we take a Hamiltonian $\displaystyle H\in
C^{\infty}_{c}([0,1]\times{\mathbb{C}}^{n})$ which displaces $\displaystyle
L$. For every $\displaystyle t\in[0,1]$, let us set $\displaystyle
H_{t}(z):=H(t,z)$, define a Hamiltonian vector field $\displaystyle X_{H_{t}}$
by $\displaystyle\omega_{n}(\,\cdot\,,X_{H_{t}})=dH_{t}(\,\cdot\,)$, and
define an isotopy $\displaystyle(\varphi^{t}_{H})_{t\in[0,1]}$ on
$\displaystyle{\mathbb{C}}^{n}$ by

$\displaystyle\varphi^{0}_{H}=\text{\rm
id}_{{\mathbb{C}}^{n}},\qquad\partial_{t}\varphi^{t}_{H}=X_{H_{t}}(\varphi^{t}_{H})\quad(\forall
t\in[0,1]).$

Then “$\displaystyle H$ displaces $\displaystyle L$” means that
$\displaystyle\varphi^{1}_{H}(L)\cap L=\emptyset$. We also assume that
$\displaystyle H_{t}\equiv 0$ when $\displaystyle t\in[0,1/3]\cup[2/3,1]$.

Let us define a family of perturbed Cauchy-Riemann operators
$\displaystyle(\bar{\partial}_{r})_{r\geq 0}$ in the following way. We first
take a $\displaystyle C^{\infty}$-function
$\displaystyle\chi:{\mathbb{R}}\to[0,1]$ such that $\displaystyle\chi\equiv 0$
on $\displaystyle{\mathbb{R}}_{\leq 0}$ and $\displaystyle\chi\equiv 1$ on
$\displaystyle{\mathbb{R}}_{\geq 1}$. For every $\displaystyle
r\in{\mathbb{R}}_{\geq 0}$, we define
$\displaystyle\chi_{r}(s):=\chi(r+s)\chi(-r-s)$. In particular
$\displaystyle\chi_{0}\equiv 0$.

Let us take a complex structure $\displaystyle J$ on
$\displaystyle{\mathbb{R}}\times[0,1]$ so that $\displaystyle
J(\partial_{s})=\partial_{t}$, where $\displaystyle s$ denotes the coordinate
on $\displaystyle{\mathbb{R}}$ and $\displaystyle t$ denotes the coordinate on
$\displaystyle[0,1]$. Next we take a biholomorphic map from $\displaystyle
D\setminus\\{-1,1\\}$ to $\displaystyle{\mathbb{R}}\times[0,1]$. This defines
$\displaystyle C^{\infty}$-functions $\displaystyle
s:D\setminus\\{-1,1\\}\to{\mathbb{R}}$ and $\displaystyle
t:D\setminus\\{-1,1\\}\to[0,1]$. For every $\displaystyle
r\in{\mathbb{R}}_{\geq 0}$ and $\displaystyle u:(D,\partial
D)\to({\mathbb{C}}^{n},L)$, we define

$\displaystyle\bar{\partial}_{r}u:=(du-X_{\chi_{r}(s)H_{t}}(u)\otimes
dt)^{0,1}.$

Obviously $\displaystyle\bar{\partial}_{0}=\bar{\partial}$. Now let us define

$\displaystyle{\mathring{{\mathcal{N}}}}^{r}_{k+1}(\beta):=\\{(u,z_{0}=1,z_{1},\ldots,z_{k})\\}$

where $\displaystyle u:(D,\partial D)\to({\mathbb{C}}^{n},L)$ satisfies
$\displaystyle\bar{\partial}_{r}u=0$, $\displaystyle[u]=\beta$, and
$\displaystyle 1,z_{1},\ldots,z_{k}$ are distinct points on
$\displaystyle\partial D$ aligned in anti-clockwise order. For each
$\displaystyle j\in\\{0,\ldots,k\\}$, the evaluation map
$\displaystyle\text{\rm
ev}\,_{j}:{\mathring{{\mathcal{N}}}}^{r}_{k+1}(\beta)\to L$ is defined in the
same formula as (22). Finally we define

$\displaystyle{\mathring{{\mathcal{N}}}}^{\geq 0}_{k+1}(\beta):=\bigcup_{r\geq
0}{\mathring{{\mathcal{N}}}}^{r}_{k+1}(\beta).$

Let us summarize basic properties of these moduli spaces:

######  Lemma 7.17.

  1. (i):

If $\displaystyle\omega(\beta)<0$ or $\displaystyle\omega(\beta)=0$ and
$\displaystyle\beta\neq 0$, then
$\displaystyle{\mathring{{\mathcal{M}}}}_{k+1}(\beta)={\mathring{{\mathcal{N}}}}^{0}_{k+1}(\beta)=\emptyset$.

  2. (ii):

Consider the case $\displaystyle\beta=0$:

     * •

$\displaystyle{\mathring{{\mathcal{M}}}}_{k+1}(0)$ consists of constant maps
for every $\displaystyle k\geq 2$.

     * •

$\displaystyle{\mathring{{\mathcal{N}}}}^{0}_{k+1}(0)$ consists of constant
maps for every $\displaystyle k\geq 0$.

  3. (iii):

Let $\displaystyle\|\,\cdot\,\|$ denote the Hofer’s norm, namely

$\displaystyle\|H\|:=\int_{0}^{1}\,(\max H_{t}-\min H_{t})\,dt.$

If $\displaystyle\omega(\beta)+2\|H\|<0$, then
$\displaystyle{\mathring{{\mathcal{N}}}}^{\geq 0}_{k+1}(\beta)=\emptyset$.

  4. (iv):

First notice that, for every $\displaystyle k\in{\mathbb{Z}}_{\geq 0}$ and
$\displaystyle r\in{\mathbb{R}}_{\geq 0}$,

$\displaystyle{\mathring{{\mathcal{M}}}}_{k+1}(\beta)=\emptyset\iff{\mathring{{\mathcal{M}}}}_{1}(\beta)=\emptyset,\qquad{\mathring{{\mathcal{N}}}}^{r}_{k+1}(\beta)=\emptyset\iff{\mathring{{\mathcal{N}}}}^{r}_{1}(\beta)=\emptyset.$

Now for every $\displaystyle c\in{\mathbb{R}}$, sets

$\displaystyle\displaystyle\\{\beta\in
H_{2}({\mathbb{C}}^{n},L)\mid{\mathring{{\mathcal{M}}}}_{1}(\beta)\neq\emptyset,\,\omega(\beta)<c\\},$
$\displaystyle\displaystyle\\{\beta\in
H_{2}({\mathbb{C}}^{n},L)\mid{\mathring{{\mathcal{N}}}}^{r}_{1}(\beta)\neq\emptyset,\,\omega(\beta)<c\\}$

are both finite.

  5. (v):

For every $\displaystyle\beta\in H_{2}({\mathbb{C}}^{n},L)$, there exists
$\displaystyle r(\beta)>0$ such that $\displaystyle\bigcup_{r\geq
r(\beta)}{\mathring{{\mathcal{N}}}}^{r}_{1}(\beta)=\emptyset$.

###### Proof.

(i), (ii) are straightforward from definitions. (iii) follows from standard
computations. (iv) follows from Gromov compactness theorem. If (v) is not the
case, there exists $\displaystyle
v:({\mathbb{R}}\times[0,1],{\mathbb{R}}\times\\{0,1\\})\to({\mathbb{C}}^{n},L)$
satisfying

$\displaystyle\int_{{\mathbb{R}}\times[0,1]}|\partial_{s}v|^{2}\,dsdt<\infty,\qquad\partial_{s}v-J(\partial_{t}v-X_{H_{t}}(v))=0,$

where $\displaystyle J$ denotes the standard complex structure on
$\displaystyle{\mathbb{C}}^{n}$. Then
$\displaystyle\gamma(t):=\lim_{s\to\infty}v(s,t)$ satisfies
$\displaystyle\gamma(0),\gamma(1)\in L$ and
$\displaystyle\partial_{t}\gamma(t)=X_{H_{t}}(\gamma(t))$, contradicting the
assumption that $\displaystyle H$ displaces $\displaystyle L$. ∎

####  7.2.2. Compactified moduli spaces

In this subsubsection, we define compactified moduli spaces
$\displaystyle{\mathcal{M}}_{k+1}(\beta)$,
$\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta)$ and
$\displaystyle{\mathcal{N}}^{\geq 0}_{k+1}(\beta)$, taking fiber products of
uncompactified moduli spaces along decorated rooted ribbon trees (see
Definition 7.18 below). K-structures on these spaces are defined in the next
subsubsection.

The next definition is a modified version of Definition 21.2 in [10].

######  Definition 7.18.

A decorated rooted ribbon tree is a pair $\displaystyle(T,B)$ such that:

  * •

$\displaystyle T$ is a connected tree. Let $\displaystyle C_{0}(T)$ and
$\displaystyle C_{1}(T)$ be the set of all vertices and edges of
$\displaystyle T$, respectively.

  * •

For each $\displaystyle v\in C_{0}(T)$ we fix a cyclic order of the set of
edges containing $\displaystyle v$ (ribbon structure).

  * •

$\displaystyle C_{0}(T)$ is divided into the set of exterior vertices
$\displaystyle C_{0,\text{\rm ext}\,}(T)$ and the set of interior vertices
$\displaystyle C_{0,\text{\rm int}\,}(T)$. For every $\displaystyle v\in
C_{0,\text{\rm int}\,}(T)$, we define $\displaystyle k_{v}$ to be the valency
of $\displaystyle v$ minus $\displaystyle 1$.

  * •

We fix one element of $\displaystyle C_{0,\text{\rm ext}\,}(T)$, which we call
the root.

  * •

The valency of every exterior vertex is $\displaystyle 1$.

  * •

$\displaystyle B$ is a map from $\displaystyle C_{0,\text{\rm int}\,}(T)$ to
$\displaystyle H_{2}({\mathbb{C}}^{n},L)$. For every vertex $\displaystyle v$,
either $\displaystyle\omega_{n}(B(v))>0$ or $\displaystyle B(v)=0$ holds.

  * •

Every vertex $\displaystyle v$ with $\displaystyle B(v)=0$ has valency at
least $\displaystyle 3$.

For every $\displaystyle k\in{\mathbb{Z}}_{\geq 0}$ and $\displaystyle\beta\in
H_{2}({\mathbb{C}}^{n},L)$, we denote by
$\displaystyle{\mathcal{G}}(k+1,\beta)$ the set of decorated rooted ribbon
trees $\displaystyle(T,B)$ such that:

  * (I):

$\displaystyle\\#C_{0,\text{\rm ext}\,}(T)=k+1$.

  * (II):

$\displaystyle\sum_{v\in C_{0,\text{\rm int}\,}(T)}B(v)=\beta$.

An edge is called exterior if it contains an exterior vertex. Otherwise it is
called interior. $\displaystyle C_{1,\text{\rm ext}\,}(T)$ (resp.
$\displaystyle C_{1,\text{\rm int}\,}(T)$) denotes the set of exterior (resp.
interior) edges.

We also define a natural notion of reduction on
$\displaystyle{\mathcal{G}}(k+1:\beta)$.

######  Definition 7.19.

Let $\displaystyle(T,B)\in{\mathcal{G}}(k+1:\beta)$ and $\displaystyle e\in
C_{1,\text{\rm int}\,}(T)$, and $\displaystyle v_{0}$, $\displaystyle v_{1}$
be vertices of $\displaystyle e$. By collapsing $\displaystyle e$ to a new
vertex $\displaystyle v_{01}$, we get
$\displaystyle(T^{\prime},B^{\prime})\in{\mathcal{G}}(k+1:\beta)$ such that

$\displaystyle\displaystyle C_{0}(T^{\prime})$
$\displaystyle\displaystyle:=(C_{0}(T)\setminus\\{v_{0},v_{1}\\})\cup\\{v_{01}\\},$
$\displaystyle\displaystyle C_{1}(T^{\prime})$
$\displaystyle\displaystyle:=C_{1}(T)\setminus\\{e\\},$
$\displaystyle\displaystyle B^{\prime}(v)$
$\displaystyle\displaystyle:=\begin{cases}B(v)&(v\neq v_{01})\\\
B(v_{0})+B(v_{1})&(v=v_{01}).\end{cases}$

An element of $\displaystyle{\mathcal{G}}(k+1:\beta)$ which can be obtained
from $\displaystyle(T,B)$ by repeating this process is called a reduction of
$\displaystyle(T,B)$.