}\hbox{$\scriptstyle{\tilde{\phi}_{L}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-23.51598pt}{-24.38832pt}\pgfsys@lineto{-0.61594pt}{-24.38832pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-23.61597pt}{-24.38832pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.41595pt}{-24.38832pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.4625pt}{-31.60217pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\phi_{L}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad,$ (64)

yields a graded Lagrangian submanifold $(L,\tilde{\phi}_{L})$ with graded lift
$\tilde{\phi}_{L}$.

Choose now transversely-intersecting graded Lagrangian
submanifolds$(L_{0},\tilde{\phi}_{L_{0}})$,
$(L_{1},\tilde{\phi}_{L_{1}})\subset M$ with
$[\phi_{L_{0}}]=[\phi_{L_{1}}]=0$, so that there are sections as in diagram
(62). Pick any $p\in L_{0}\cap L_{1}$. Then there exists a favoured (homotopy
class of a) path $l_{p}:[0,1]\rightarrow LGr(T_{p}M)$ such that
$l_{p}(0)=T_{p}L_{0}$ and $l_{p}(1)=T_{p}L_{1}$, namely the one selected by
the covering map $\upsilon_{p}$ after looking at the universal cover
$\widetilde{LGr}(T_{p}M)$.

######  Definition 5.22.

Let $(L_{0},\tilde{\phi}_{L_{0}}),(L_{1},\tilde{\phi}_{L_{1}})\subset M$ be
graded Lagrangian submanifolds with vanishing Maslov class, and $p\in
L_{0}\cap L_{1}$. Let $l_{p}:[0,1]\rightarrow LGr(T_{p}M)$ be the path from
previous paragraph, and $\lambda_{p}:[0,1]\rightarrow LGr(T_{p}M)$ the
canonical short path from $T_{p}L_{0}$ to $T_{p}L_{1}$. Then the degree of $p$
is given by

$\text{deg}(p)\coloneqq w(\gamma_{p})\,,$ (65)

where $\gamma_{p}\coloneqq l_{p}\ast\overline{\lambda_{p}}$ is a loop based at
$T_{p}L_{0}$.

After suitably identifying the tangent spaces $T_{x}(L_{0}\cap L_{1})$, an
immediate consequence of Definitions 5.19 and 5.22 is:

######  Corollary 5.23.

Let $u:\mathbb{R}\times[0,1]\rightarrow M$ be a $J$-holomorphic strip solving
the Cauchy–Riemann problem (57)–(59), where $L_{0}$ and $L_{1}$ are
transversely-intersecting graded Lagrangian submanifolds with
$[\phi_{L_{0}}]=[\phi_{L_{1}}]=0$. Then:

$\textup{ind}([u])=\textup{deg}(q)-\textup{deg}(p).$

In particular, $\textup{ind}([u])$ only depends on the two generators $p$, $q$
connected by $u$, so that $CF(L_{0},L_{1})$ can be equipped with a
$\mathbb{Z}$-grading:

$CF^{n}(L_{0},L_{1})\coloneqq\mkern-18.0mu\bigoplus_{\begin{subarray}{c}p\in\mathcal{X}(L_{0},L_{1}):\\\
\textup{deg}(p)=n\end{subarray}}\mkern-18.0mu\Lambda_{\mathbb{K}}\cdot
p\;,\quad\text{for }n\in\mathbb{Z}.$

Without the assumptions $2c_{1}(TM)=0$ and $[\phi_{L_{0}}]=[\phi_{L_{1}}]=0$,
we merely obtain a $\mathbb{Z}_{2}$-grading (this alternative is briefly
discussed in [Sei99, section 4]).

[frame]Henceforth, we assume that all (oriented spin) Lagrangian submanifolds
are
graded, with vanishing Maslov class.

####  5.5 Transversality

Transversality of the moduli spaces $\widehat{\mathcal{M}}(p,q;[u],J)$ is
somewhat more challenging to address, especially in terms of the underlying
theory.

######  Remark 5.24.

Building on the technical bit of Remark 5.14, transversality is achieved if
all zeros of the linearization $D_{\bar{\partial}_{J},u}$ of
$\bar{\partial}_{J}$ are regular, that is, if $D_{\bar{\partial}_{J},u}$ is
surjective for all $u\in\widehat{\mathcal{M}}(p,q;[u],J)$.

A solution for the case $L_{0}\pitchfork L_{1}$ is provided by allowing our
$\omega$-compatible almost complex structure to be $t$-dependent, that is, by
substituting $J$ in (57) with $J_{t}$, for $t\in[0,1]$ (formally, we just take
$J\in C^{\infty}([0,1],\mathfrak{J}(M,\omega))$ and write $J_{t}\coloneqq
J(t,\,\cdot\,)$).

This strategy comes about also if we try to define $CF(L_{0},L_{1})$ in the
most general case when $L_{0}\,\cancel{\pitchfork}\,L_{1}$ — which we really
need to study if we are to find identity morphisms in $CF(L_{i},L_{i})$!
Indeed, in order to drop transversality of the Lagrangian submanifolds, we
consider a time-dependent Hamiltonian function $H\in C^{\infty}([0,1]\times
M)$ such that $L_{0}\,\pitchfork\,(\phi_{H}^{1})^{-1}(L_{1})$, where
$\phi_{H}^{1}:M\rightarrow M$ is the Hamiltonian diffeomorphism of the
associated $t$-dependent Hamiltonian vector field
$X_{H}\in\mathfrak{X}([0,1]\times M)$. Equation (57) then becomes a perturbed
PDE, with perturbation given by $X_{H}$:

$\bar{\partial}_{J,H}(u)\coloneqq\partial_{s}u+J_{t}(u)(\partial_{t}u-X_{H_{t}}(u))=0\;.$
(66)

(This is sometimes called Floer’s equation, and $\bar{\partial}_{J,H}$ Floer’s
differential.) The boundary conditions must be modified accordingly. The
finite energy constraint (59) becomes:

$E(u)=\frac{1}{2}\int_{-\infty}^{+\infty}\int_{0}^{1}(|\partial_{s}u|^{2}+|\partial_{t}u-X_{H_{t}}(u)|^{2})dsdt\overset{!}{<}\infty\;,$
(67)

and we still want $u(s,i)\in L_{i}$ for $i=0,1$. However, the asymptotes for
$s\rightarrow\pm\infty$ change due to the next observation.

######  Remark 5.25.

The perturbed Cauchy–Riemann equation (66) can be made into a homogeneous
Cauchy–Riemann equation of the form (57) as follows. Let
$\phi_{H}^{t}:M\rightarrow M$ be the flow of $X_{H}$, and consider the
preimage of $u$ under it, $\tilde{u}:\mathbb{R}\times[0,1]\rightarrow
M,\,\tilde{u}(s,t)\coloneqq(\phi_{H}^{t})^{-1}(u(s,t))$, so that the chain
rule yields
$\partial_{t}\tilde{u}=(\phi_{H}^{t})_{*}^{-1}(\partial_{t}u-X_{H})$. Define
also $\tilde{J}_{t}\coloneqq(\phi_{H}^{t})_{*}^{-1}(J_{t})$. Then (66)
acquires the familiar aspect

$\partial_{s}\tilde{u}+\tilde{J}_{t}(\tilde{u})\partial_{t}\tilde{u}=0\;,$

a PDE which we know to have as solutions $\tilde{J}$-holomorphic strips
$\tilde{u}$ fulfilling $\tilde{u}(s,0)=u(s,0)\in L_{0}$ and
$\tilde{u}(s,1)=(\phi_{H}^{1})^{-1}(u(s,1))\in(\phi_{H}^{1})^{-1}(L_{1})$.

By last remark, the Definition 5.11 of Floer complex now requires the
modification
$\mathcal{X}(L_{0},L_{1})\equiv\mathcal{X}(L_{0},L_{1};H)\coloneqq
L_{0}\cap(\phi_{H}^{1})^{-1}(L_{1})$, which is again a discrete finite set due
to $L_{0}\,\pitchfork\,(\phi_{H}^{1})^{-1}(L_{1})$. The generators are
intersection points perturbed according to flow lines $\gamma:[0,1]\rightarrow
M$ (fulfilling $\dot{\gamma}(t)=X_{H_{t}}(t,\gamma(t))$, by definition) such
that $\gamma(0)\in L_{0}$ and $\gamma(1)\in L_{1}$. Accordingly, the Floer
differential counts perturbed $J$-holomorphic strips in moduli spaces of the
form $\mathcal{M}(p,q;[u],J,H)$, now depending on $H$ as well.

######  Remark 5.26.

(Warning) We have overlooked a non-trivial fact when picking a suitable $H\in
C^{\infty}([0,1]\times M)$ such that
$L_{0}\,\pitchfork\,(\phi_{H}^{1})^{-1}(L_{1})$: $H$ should actually be chosen
in $\mathcal{H}_{\text{reg}}\subset C^{\infty}([0,1]\times M)$, the subset of
all Hamiltonian functions whose Hamiltonian diffeomorphisms fulfill such
transversality property for some pair of Lagrangian submanifolds. However,
this does not invalidate our analysis since $\mathcal{H}_{\text{reg}}$ turns
out to be a dense subset of $C^{\infty}([0,1]\times M)$ (more precisely, of
second category in the sense of Baire; cf. [Sal97, Theorem 1.24]).

Similarly, the time-dependent almost complex structure $J$ assuring
surjectivity of the linearization $D_{\bar{\partial}_{J},u}$ (see Remark 5.24)
should actually belong to a dense subset $\mathfrak{J}_{\text{reg}}\subset
C^{\infty}([0,1],\mathfrak{J}(M,\omega))$.

We bear these considerations in mind when selecting the data $(J,H)$.

Let us consider the perturbed problem (66) from a slightly different
perspective, namely that offered by [Oh91], [McS94] and [Mer14, chapter 3].

######  Definition 5.27.

Let $(M,\omega)$ be a symplectic manifold, $L_{0},L_{1}\subset M$ any
Lagrangian submanifolds and $H\in C^{\infty}([0,1]\times M)$ a time-dependent
Hamiltonian. Define the path space

$\mathcal{P}=\mathcal{P}(L_{0},L_{1})\coloneqq\\{\gamma\in
C^{\infty}([0,1],M)\mid\gamma(0)\in L_{0},\gamma(1)\in L_{1}\\}\,.$

This is an infinite-dimensional smooth manifold of countably many connected
components, with tangent spaces

$T_{\gamma}\mathcal{P}=\Gamma(\gamma^{*}TM)\cong\Gamma_{\gamma}(TM)\coloneqq\\{\xi\in
C^{\infty}([0,1],TM)\mid\xi(t)\in T_{\gamma(t)}M\\}\,,$

for every $\gamma\in\mathcal{P}$. We define the closed 1-form
$a_{H}\in\Omega^{1}(\mathcal{P})$ via the $C^{\infty}(\mathcal{P})$-linear map
$\mathfrak{X}(\mathcal{P})\rightarrow C^{\infty}(\mathcal{P})$ it identifies:

$\displaystyle(a_{H})_{\gamma}(\xi):\\!$
$\displaystyle=\int_{0}^{1}\big{(}\omega_{\gamma(t)}(\xi(t),\dot{\gamma}(t))-(dH_{t})_{\gamma(t)}(\xi(t))\big{)}dt$
$\displaystyle=\int_{0}^{1}\omega_{\gamma(t)}\big{(}\xi(t),\dot{\gamma}(t)-X_{H_{t}}(\gamma(t))\big{)}dt\;,$
(68)

for $\gamma\in\mathcal{P}$ (so that
$\dot{\gamma}\coloneqq\frac{d}{dt}|_{t}\gamma\in\Gamma_{\gamma}(TM)$) and
$\xi\in T_{\gamma}\mathcal{P}$, where last equality makes use of
$dH_{t}=\omega(\,\cdot\,,X_{H_{t}}):\mathfrak{X}(M)\rightarrow C^{\infty}(M)$,
by definition of Hamiltonian vector field (see Definition 5.4). Observe that
closedness of $a_{H}$ is an immediate consequence of the closedness of
$\omega$.

It is then straightforward to see:

######  Corollary 5.28.

Let $(M,\omega)$ be a symplectic manifold, $L_{0},L_{1}\subset M$ any
Lagrangian submanifolds and $H\in C^{\infty}([0,1]\times M)$ a time-dependent
Hamiltonian function generating a flow $\phi_{H}^{t}:M\rightarrow M$. Define
$a_{H}\in\Omega^{1}(\mathcal{P})$ as above. Then $\gamma\in\mathcal{P}$ is a
zero of $a_{H}$, $\gamma\in\textup{zeros}(a_{H})$, if and only if $\gamma$ is
a flow line, that is, $\gamma(t)=\phi_{H}^{t}(x)$ for some $x\in L_{0}$. In
particular, there is a bijection given by evaluation at time $0$:

$\\{\gamma\in\textup{zeros}(a_{H})\\}\longleftrightarrow\\{\gamma(0)=x\in
L_{0}\cap(\phi_{H}^{1})^{-1}(L_{1})\\}\;.$ (69)

Corollary 5.28 tells us that we may identify generators of the Floer complex
$CF(L_{0},L_{1})$ with zeros of the closed 1-form $a_{H}$, which we call
time-1 Hamiltonian chords from $L_{0}$ to $L_{1}$.

However, we can still improve our interpretation.

######  Definition 5.29.

Let $(M,\omega)$ be a symplectic manifold, $L_{0},L_{1}\subset M$ any
Lagrangian submanifolds. Define the path space
$\mathcal{P}=\mathcal{P}(L_{0},L_{1})$ as above, and choose fixed reference
paths $\gamma_{k}\in\mathcal{P}_{k}$ for each connected component of
$\mathcal{P}=\bigsqcup_{k\in I}\mathcal{P}_{k}$. We call242424This terminology
is non-standard. a bridge for $\gamma\in\mathcal{P}_{k}$ any $\bar{\gamma}\in
C^{\infty}(\overline{\mathbb{R}}\times[0,1],M)$ such that
$\bar{\gamma}(-\infty,\,\cdot\,)=\gamma_{k}$,
$\bar{\gamma}(+\infty,\,\cdot\,)=\gamma$ and
$\bar{\gamma}(s,\cdot)\in\mathcal{P}_{k}$ for all $s\in\overline{\mathbb{R}}$
(so that we may also write $\bar{\gamma}\in
C^{\infty}(\overline{\mathbb{R}},\mathcal{P}_{k})$).

######  Definition 5.30.

Let $(M,\omega)$ be a symplectic manifold, $L_{0},L_{1}\subset M$ Lagrangian
submanifolds and $H\in C^{\infty}([0,1]\times M)$. Let
$\mathcal{P}=\mathcal{P}(L_{0},L_{1})$ be the corresponding path space, and
fix some $\gamma_{0}\in\mathcal{P}$ such that $\gamma_{0}(0)=q\in
L_{0}\cap(\phi_{H}^{1})^{-1}(L_{1})$ is a perturbed generator of the Floer
complex $CF(L_{0},L_{1})$ (or equivalently by (69), such that
$\gamma_{0}\in\text{zeros}(a_{H})$). We define the Hamiltonian action
functional $A_{H}:\mathcal{P}\rightarrow\mathbb{R}$ by

$A_{H}(\gamma)\coloneqq\int_{\overline{\mathbb{R}}\times[0,1]}\bar{\gamma}^{*}\omega-\int_{0}^{1}H_{t}(\gamma(t))dt\;,$
(70)

where $\bar{\gamma}\in C^{\infty}(\overline{\mathbb{R}}\times[0,1],M)$ is any
bridge for $\gamma\in\mathcal{P}$ such that $\bar{\gamma}(-\infty)=\gamma_{0}$
(for this to make sense, observe that $\gamma$ must belong to the same
connected component of $\gamma_{0}$!).

As it is given, the definition of $A_{H}$ is still incomplete, since we must
prove it to be independent from the specific choice of bridge $\bar{\gamma}$ —
in fact, we did cheat a little: without suitable assumptions on $M$, $L_{0}$
and $L_{1}$, the definition of a Hamiltonian action functional on
$\mathcal{P}=\mathcal{P}(L_{0},L_{1})$ is in general impossible, and one is
forced to work instead on its Novikov covering $\overline{\mathcal{P}}$. This
is the covering space of elements all pairs $[\gamma,\bar{\gamma}]$ such that
$\bar{\gamma}$ is a bridge for $\gamma\in\mathcal{P}$, obtained after
identifying all bridges which are homotopic relative to
$\partial(\overline{\mathbb{R}}\times[0,1])$. With some more work, one can
further adapt $\overline{\mathcal{P}}$ to produce a covering map
$\pi:\overline{\mathcal{P}}\rightarrow\mathcal{P}$ fulfilling
$dA_{H}=\pi^{*}(a_{H})$ when $H=0$ (see for example [Oh91, Proposition
13.5.1]). This is still valid in the perturbed case $H\neq 0$:

######  Lemma 5.31.

Let $A_{H}:\overline{\mathcal{P}}\rightarrow\mathbb{R}$ be the Hamiltonian
action functional (70), $a_{H}\in\Omega^{1}(\mathcal{P})$ the $1$-form (5.27).
Then it holds that $dA_{H}=\pi^{*}(a_{H})$. Consequently, the critical points
of $A_{H}$ coincide with the zeros of $a_{H}$,
$\textup{crit}(A_{H})=\textup{zeros}(a_{H})$.

This, together with Corollary 5.28, helps us refine our interpretation of the
generators of generic Floer complexes $CF(L_{0},L_{1};H)$:

$\\{\gamma(0)=x\in\mathcal{X}(L_{0},L_{1};H)\\}\longleftrightarrow\\{\gamma\in\text{crit}(A_{H})\\}\;.$
(71)

Lemma 5.31 also motivates the following couple of definitions.

######  Definition 5.32.

Let $(M,\omega,J)$ be an almost complex symplectic manifold with
$J\in\mathfrak{J}(M,\omega)$, $L_{0},L_{1}\subset M$ Lagrangian submanifolds
and $H\in C^{\infty}([0,1]\times M)$. Define the $L^{2}$-inner product

$\langle\\!\langle\xi,\zeta\rangle\\!\rangle_{J}\coloneqq\int_{0}^{1}(g_{J})_{\gamma(t)}(\xi(t),\zeta(t))dt=\int_{0}^{1}\omega_{\gamma(t)}(J_{\gamma(t)}(\xi(t)),\zeta(t))dt\;,$

for $\gamma\in\mathcal{P}$ and $\xi,\zeta\in T_{\gamma}\mathcal{P}$. Then we
denote by $\nabla_{J}A_{H}\in\mathfrak{X}(\mathcal{P})$ the vector field

$\nabla_{J}A_{H}(\gamma)\coloneqq
J(\gamma)(\dot{\gamma}-X_{H}(\,\cdot\,,\gamma))\in T_{\gamma}\mathcal{P}$ (72)

(where $J(\gamma)\equiv J_{\gamma(\cdot)}$). Explicitly, we have
$\nabla_{J}A_{H}(\gamma)(t)\coloneqq
J_{\gamma(t)}\big{(}\dot{\gamma}(t)-X_{H_{t}}(\gamma(t))\big{)}$ $\in
T_{\gamma(t)}M$ for any $t\in[0,1]$.

######  Lemma 5.33.

The vector field $\nabla_{J}A_{H}\in\mathfrak{X}(\mathcal{P})$ is the
$L^{2}$-gradient of $A_{H}$ with respect to
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{J}$, that is,
$(dA_{H})_{\gamma}(\xi)=\langle\\!\langle\nabla_{J}A_{H}(\gamma),\xi\rangle\\!\rangle_{J}$
for all $\gamma\in\mathcal{P}$ and $\xi\in T_{\gamma}\mathcal{P}$.

###### Proof.

This is a straightforward computation. Pick any $\gamma\in\mathcal{P}$ and
$\xi\in T_{\gamma}\mathcal{P}$, then:

$\displaystyle(dA_{H})_{\gamma}(\xi)$
$\displaystyle=\int_{0}^{1}\omega_{\gamma(t)}\big{(}\xi(t),\dot{\gamma}(t)-X_{H_{t}}(\gamma(t))\big{)}dt$
$\displaystyle=\int_{0}^{1}\omega_{\gamma(t)}\big{(}J_{\gamma(t)}^{2}\big{(}\dot{\gamma}(t)-X_{H_{t}}(\gamma(t))\big{)},\xi(t)\big{)}dt$
$\displaystyle=\int_{0}^{1}(g_{J})_{\gamma(t)}\big{(}J_{\gamma(t)}\big{(}\dot{\gamma}(t)-X_{H_{t}}(\gamma(t))\big{)},\xi(t)\big{)}dt$
$\displaystyle=\int_{0}^{1}(g_{J})_{\gamma(t)}\big{(}\nabla_{J}A_{H}(\gamma)(t),\xi(t)\big{)}dt$
$\displaystyle=\langle\\!\langle\nabla_{J}A_{H}(\gamma),\xi\rangle\\!\rangle_{J}\;.$

∎

In the language just formulated, the perturbed Cauchy–Riemann equation (66)
reads

$\frac{d}{ds}u(s)+\nabla_{J}A_{H}(u(s))=0\;,$ (73)

an ODE on $\mathcal{P}$ where we reinterpreted $J$-holomorphic strips
$u:\mathbb{R}\times[0,1]\rightarrow M$ as maps
$u:\mathbb{R}\rightarrow\mathcal{P}$, the (bounded) negative gradient flow
lines of $A_{H}$. The energy of a solution $u\in\mathcal{M}(p,q;[u],J,H)$ can
be expressed through the action functional:

$E(u)=A_{H}(\gamma_{-})-A_{H}(\gamma_{+})\,,$

where $\gamma_{\pm}\in\text{crit}(A_{H})$ are such that $\gamma_{-}(0)=q$ and
$\gamma_{+}(0)=p$. Clearly this makes sense, meaning
$\mathcal{M}(p,q;[u],J,H)$ can be nonempty, if and only if
$A_{H}(\gamma_{-})>A_{H}(\gamma_{+})$ or $\gamma_{-}=\gamma_{+}$.

However, this alternative standpoint earns us no further advantage, and we
rather keep our focus252525Indeed, this was Andreas Floer’s brilliant
intuition when developing Floer homology: to regard the ill-posed ODE (73) as
the perturbation of the PDE (57), whose solutions are the well-known
$J$-holomorphic strips. on the perturbed PDE (66) on $M$. The reader
interested in the underlying Floer theoretic machinery is once again referred
to [Oh91].

[frame]Henceforth, we implicitly choose a time-dependent almost complex
structure
$J\in C^{\infty}([0,1],\mathfrak{J}(M,\omega))$ and Hamiltonian $H\in
C^{\infty}([0,1]\times M)$, turning our
analysis to the perturbed Cauchy–Riemann problem (66) for not necessarily
transversely-intersecting Lagrangian submanifolds.

####  5.6 Compactness

Before proceeding to the definition of Floer’s differential, we must address
the issue of compactness of the smooth manifold
$\widehat{\mathcal{M}}(p,q;[u],J,H)$. This is perhaps the most delicate of the
three requirements listed in Section 5.3, due to the following result (see for
example [Sal97, Proposition 3.3]).

######  Proposition 5.34 (Gromov’s Compactness Theorem).

Consider a sequence of $J$-holomorphic strips
$(u_{n})_{n\in\mathbb{N}}\subset\widehat{\mathcal{M}}(p,q;[u],J,H)$ with
uniformly bounded energy, $\sup_{n\in\mathbb{N}}E(u_{n})<\infty$. Then there
exist finitely many points $(s_{j},t_{j})\equiv
s_{j}+it_{j}=:z_{j}\in\mathbb{D}^{2}\setminus\\{\pm 1\\}$, say for
$j=1,...,l$, and a subsequence
$(u_{n_{k}})_{k\in\mathbb{N}}\subset\widehat{\mathcal{M}}(p,q;[u],J,H)$ which
converges uniformly to some solution $u\in\widehat{\mathcal{M}}(p,q;[u],J,H)$
of (66), with derivatives on compact subsets of
$(\mathbb{D}^{2}\setminus\\{\pm 1\\})\setminus\\{z_{1},...,z_{l}\\}$.

######  Definition 5.35.

Consider the subsequence of $J$-holomorphic curves
$(u_{n_{k}})_{k\in\mathbb{N}}$ $\subset\widehat{\mathcal{M}}(p,q;[u],J,H)$ as
in previous proposition. Then the derivatives $(du_{n_{k}})_{k}$ at any among
those $z_{j}=(s_{j},t_{j})\in\mathbb{R}\times[0,1]$ are unbounded, which
causes an “energy concentration” for $k\rightarrow\infty$. We say that the
limit strip $u$ exhibits a bubble. There are three possible such “bubbling”
phenomena (see Figure 3):

  * •

Strip breaking, which occurs when $\lim_{k\to\infty}|(du_{n_{k}})(z)|=\infty$
at $z=(s,t)\in\mathbb{R}\times[0,1]$ for $s\rightarrow\pm\infty$, so that the
limit strip is “broken” (that is, has an intermediate node). Regarding each
component as a $J$-holomorphic disk, this is equivalent to having a bubble at
one of the endpoints $p,q$. Formally, this means there is a sequence
$a_{n}\rightarrow\pm\infty$ such that the translated strips $u_{n}(\cdot-
a_{n},\cdot)$ converge to a non-constant limit strip.

  * •

Disk bubbling, which occurs when $\lim_{k\to\infty}|(du_{n_{k}})(z)|=\infty$
at some $z=(s,t)\in\mathbb{R}\times[0,1]$ for $t\in\\{0,1\\}$, so that the
limit strip has a bubble consisting of a $J$-holomorphic disk whose boundary
intersects — and after suitable rescalings is entirely contained in — $L_{0}$
or $L_{1}$.

  * •

Sphere bubbling, which occurs when $\lim_{k\to\infty}|(du_{n_{k}})(z)|=\infty$
at some $z=(s,t)\in(\mathbb{R}\times[0,1])^{\circ}$, so that the limit strip
exhibits a bubble which — after suitable rescalings — is a $J$-holomorphic
sphere $\mathbb{C}P^{1}\rightarrow M$ intersecting $\text{im}(u)$ transversely
at some interior point of $\text{im}(u)$.

Figure 3: Two possible “bubbling” phenomena: strip breaking (left) and disk
bubbling (right). [Source:[Aur13]]

Strip-breaking is the reason why Floer’s differential squares to zero in
absence of disk and sphere bubbles. The latters are undesirable since they
also obstruct the transversality of $\widehat{\mathcal{M}}(p,q;[u],J,H)$; not
even the perturbations $J\in C^{\infty}([0,1],\mathfrak{J}(M,\omega))$ and
$H\in C^{\infty}([0,1]\times M)$ adopted in Section 5.5 can solve the problem
this time! Therefore, we wish to retain strip-breaking phenomena while
discarding disk and sphere bubbling. This is achieved by simply requiring that
the symplectic area of the associated $J$-holomorphic disks and spheres
vanishes.

######  Lemma 5.36.

Let $(M,\omega)$ be a symplectic manifold, $L_{0},L_{1}\subset M$ Lagrangian
submanifolds such that262626Observe that $\langle\omega\rangle\in
H_{dR}^{2}(M)\cong
H^{2}(M;\mathbb{R})\cong\textup{Hom}(H_{2}(M),\mathbb{R})\oplus\text{Ext}(H_{1}(M),\mathbb{R})=\textup{Hom}(H_{2}(M),\mathbb{R})$,
hence, by identifying $\pi_{2}(M)$ with its image under the Hurewicz map
$h:\pi_{2}(M)\rightarrow H_{2}(M)$, we can restrict $\langle\omega\rangle$ to
$\pi_{2}(M)$. In other words, we impose the pairing
$\langle\langle\omega\rangle,h(\cdot)\rangle\equiv\langle\omega\rangle\cdot\text{im}(h)$
between cohomology and homology to vanish, whence the notation adopted. The
same holds when regarding $\langle\omega\rangle\in H_{dR}^{2}(M,L_{i})$.
$\langle\omega\rangle\cdot\pi_{2}(M,L_{i})=0$. Then $L_{0}$ and $L_{1}$ bound
no $J$-holomorphic disk or sphere.

This is just one of several possible conditions preventing the formation of
disk and sphere bubbles. Later we will see that this comes for free when
working with exact symplectic manifolds.

[frame]Henceforth, we assume that all Lagrangian submanifolds $L\subset M$
satisfy the
“no-bubbling” condition $\langle\omega\rangle\cdot\pi_{2}(M,L)=0$.

####  5.7 The Floer differential and Floer cohomology

We have finally collected all the necessary assumptions which make the ensuing
theory well-posed. We proceed to define the second piece of data for Floer
complexes.

######  Definition 5.37.

Let $(M,\omega)$ be a symplectic manifold, $L_{0},L_{1}\subset M$ Lagrangian
submanifolds with corresponding Floer complex $CF(L_{0},L_{1})\equiv
CF(L_{0},L_{1};J,H)$ as in equation (56) (for
$\mathcal{X}(L_{0},L_{1})=L_{0}\cap(\phi_{H}^{1})^{-1}(L_{1})$). The Floer
differential $\partial\equiv\partial_{J,H}:CF(L_{0},L_{1})\rightarrow
CF(L_{0},L_{1})$ is the $\Lambda_{\mathbb{K}}$-linear extension of

$\partial(p)\coloneqq\mkern-18.0mu\sum_{\begin{subarray}{c}q\in\mathcal{X}(L_{0},L_{1})\\\
[u]:\text{ind}([u])=1\end{subarray}}\mkern-18.0mu(\\#\mathcal{M}(p,q;[u],J,H))T^{\omega([u])}q\;,$
(74)

where $\\#\mathcal{M}(p,q;[u],J,H)\in\mathbb{Z}$ (or $\mathbb{Z}_{2}$) is the
(un)signed cardinality of the moduli space $\mathcal{M}(p,q;[u],J,H)$ coming
from the perturbed Cauchy–Riemann problem (66), and $\omega([u])$ is the
symplectic area (independent from the choice of representative, by the no-
bubbling condition). Due to Corollary 5.23, $\partial$ is of degree 1. (Notice
that the coefficient of $q$ in $\partial(p)$,

$\sum_{[u]:\text{ind}([u])=1}\mkern-18.0mu(\\#\mathcal{M}(p,q;[u],J,H))T^{\omega([u])}\in\Lambda_{\mathbb{K}}\;,$

is an element of the Novikov field.)

To prove that $\partial$ is well defined, we need one last (non-trivial)
result from manifold theory.

######  Lemma 5.38.

Every compact $1$-dimensional smooth manifold $M$ has total signed number of
boundary points equal zero.

###### Proof.

(Sketch) This is a consequence of the Classification Theorem for compact
1-dimensional smooth manifolds: any such must be diffeomorphic to a disjoint,
finite union of closed real intervals and copies of $\mathbb{S}^{1}$. The
formers have two boundary points each, the latters none. Since boundary points
are preserved under diffeomorphism, it follows that $M$ has an even number of
boundary points.

This readily proves the case for the unsigned count (when we work modulo 2).
If we do care about orientations, thus taking into account signs, a careful
analysis reveals that the signed cardinality of $\partial M$ still cancels
out. ∎

######  Theorem 5.39.

Let $(M,\omega)$ be a symplectic manifold, $L_{0},L_{1}\subset M$ Lagrangian
submanifolds such that $\langle\omega\rangle\cdot\pi_{2}(M,L_{i})=0$ for
$i=0,1$. Then $\partial$ is well defined and satisfies $\partial^{2}=0$.
Moreover, the induced cohomology group does not depend (up to isomorphism) on
the choice of time-dependent $J$ and $H$.

###### Proof.

(Sketch) The counting in (74) happens for all homotopy classes
$[u]\in\pi_{2}(M,L_{0}\cup L_{1})$ with
$\text{ind}([u])=\text{dim}(\widehat{\mathcal{M}}(p,q;[u],J,H))=1$, hence the
corresponding moduli spaces
$\mathcal{M}(p,q;[u],J,H)=\widehat{M}(p,q;[u],J,H)/\mathbb{R}$ (which are
smooth manifolds by transversality) are compact, oriented and 0-dimensional,
thus are finite sets which can be counted with signs. Moreover, the sum in
(74) is well defined because, as a consequence of Gromov’s Compactness
Theorem, the finite energy bound (67) is satisfied by only finitely many
classes $[u]$ whose moduli space is nonempty. Recall also that
$\mathcal{X}(L_{0},L_{1})$ is finite.

Fix two generators $p,q\in\mathcal{X}(L_{0},L_{1})$ and some
$[u]\in\pi_{2}(M,L_{0}\cup L_{1})$ with Maslov index 2. Then
$\text{dim}(\widehat{\mathcal{M}}(p,q;[u],J,H))=2$, hence the smooth manifold
$\mathcal{M}(p,q;[u],J,H)=\widehat{\mathcal{M}}(p,q;[u],J,H)/\mathbb{R}$ is
1-dimensional. Again by Proposition 5.34, it can be compactified to a space
$\overline{\mathcal{M}}(p,q;[u],J,H)$ by just adding broken strips connecting
$p$ to $q$ and representing $[u]$ (by assumption, there is no risk to incur
into disk/sphere bubbling). Accordingly,
$\partial\overline{\mathcal{M}}(p,q;[u],J,H)$ is contained in the disjoint
union of products of moduli spaces having the form
$\mathcal{M}(p,r;[u^{\prime}],J,H)\times\mathcal{M}(r,q;[u^{\prime\prime}],J,H)$,
where $r\in CF(L_{0},L_{1})$ is any generator and
$[u^{\prime}]+[u^{\prime\prime}]=[u]\in\pi_{2}(M,L_{0}\cup L_{1})$ (see Figure
3 left). Since the Maslov index is additive under strip-breaking (by
construction) and any non-constant strip has Maslov index at least 1 (by
transversality), $\text{ind}([u])=2$ implies that
$\text{ind}([u^{\prime}])=\text{ind}([u^{\prime\prime}])=1$. Conversely, a
gluing theorem272727Refer to [Mer14, section 8] or [Sal97, section 3.3] for
some aimed formulations, and think about their Morse theoretic analogues.
tells us that $\overline{\mathcal{M}}(p,q;[u],J,H)$ is actually a
1-dimensional manifold with boundary

$\partial\overline{\mathcal{M}}(p,q;[u],J,H)=\mkern-36.0mu\bigsqcup_{\footnotesize\begin{subarray}{c}r\in\mathcal{X}(L_{0},L_{1})\\\
[u^{\prime}]+[u^{\prime\prime}]=[u]\\\
\text{ind}([u^{\prime}])=\text{ind}([u^{\prime\prime}])=1\end{subarray}}\mkern-36.0mu\mathcal{M}(p,r;[u^{\prime}],J,H)\times\mathcal{M}(r,q;[u^{\prime\prime}],J,H)\;,$
(75)

whose orientation is compatible with the natural ones of its components
(coming from the implicit choice of orientations and spin structures for
$L_{0}$ and $L_{1}$). By Lemma 5.38, the boundary of the compact 1-dimensional
smooth manifold $\overline{\mathcal{M}}(p,q;[u],J,H)$ has zero signed
cardinality:

$\mkern-36.0mu\sum_{\footnotesize\begin{subarray}{c}r\in\mathcal{X}(L_{0},L_{1})\\\
[u^{\prime}]+[u^{\prime\prime}]=[u]\\\
\text{ind}([u^{\prime}])=\text{ind}([u^{\prime\prime}])=1\end{subarray}}\mkern-36.0mu(\\#\mathcal{M}(p,r;[u^{\prime}],J,H))\cdot(\\#\mathcal{M}(r,q;[u^{\prime\prime}],J,H))=0\;.$

Multiplying last expression by the formal variable $T$ of
$\Lambda_{\mathbb{K}}$ weighted by the total strip energy,
$T^{\omega([u])}=T^{\omega([u^{\prime}])+\omega([u^{\prime\prime}])}$, and
summing over all classes $[u]\in\pi_{2}(M,L_{0}\cup L_{1})$, we obtain
precisely $\partial^{2}(p)$. Therefore, $\partial^{2}(p)=0$.

We must still prove that the cohomology groups induced by
$\partial=\partial_{J,H}$ do not depend on the choice of perturbation data $J$
and $H$. We just give an overview.282828More details can be found in [Mer14,
chapter 9], for the non-Lagrangian case. Take another pair
$(J^{\prime},H^{\prime})$. Then, since
$C^{\infty}([0,1],\mathfrak{J}(M,\omega))$ and $C^{\infty}([0,1]\times M)$ are
both path-connected, there is an asymptotically constant path
$\chi:\tau\in\mathbb{R}\mapsto(J(\tau),H(\tau))\in
C^{\infty}([0,1],\mathfrak{J}(M,\omega))\times C^{\infty}([0,1]\times M)$ such
that $\chi(-\tau)=(J,H)$ and $\chi(\tau)=(J^{\prime},H^{\prime})$ for all
$\tau\geq N$, given $N\geq 0$ big enough. One proceeds to construct a
continuation map $\Phi_{\chi}:CF(L_{0},L_{1};J,H)\rightarrow
CF(L_{0},L_{1};J^{\prime},H^{\prime})$ by counting solutions to

$\partial_{s}u+J_{t}(\tau,u)(\partial_{t}u-X_{H_{t}}(\tau,u))=0$

as follows: the coefficient of
$p^{\prime}\in\mathcal{X}(L_{0},L_{1};H^{\prime})$ in $F(p)$ is the signed
count of index 0 solutions converging to $p$ respectively $p^{\prime}$ for
$s\rightarrow\pm\infty$. Symmetrically, one can define a continuation map
$\Phi_{\chi}^{\prime}:CF(L_{0},L_{1};J^{\prime},H^{\prime})\rightarrow
CF(L_{0},L_{1};J,H)$. Then it can be shown that
$\partial_{J^{\prime},H^{\prime}}\circ\Phi_{\chi}=\Phi_{\chi}\circ\partial_{J,H}$
and
$\partial_{J,H}\circ\Phi_{\chi}^{\prime}=\Phi_{\chi}^{\prime}\circ\partial_{J^{\prime},H^{\prime}}$,
meaning $\Phi_{\chi}$ and $\Phi_{\chi}^{\prime}$ are inverse chain maps, so
that $\Phi_{\chi}^{\prime}\circ\Phi_{\chi}$ is chain homotopic to
$\textup{id}_{CF(L_{0},L_{1};J,H)}$ respectively
$\Phi_{\chi}\circ\Phi_{\chi}^{\prime}$ to
$\textup{id}_{CF(L_{0},L_{1};J^{\prime},H^{\prime})}$. Passing to cohomology,
it follows that $H(\Phi_{\chi}):H(CF(L_{0},L_{1};J,H))\rightarrow
H(CF(L_{0},L_{1};J^{\prime},H^{\prime}))$ is an isomorphism, as desired. ∎

######  Definition 5.40.

Let $(M,\omega)$ be a symplectic manifold, $L_{0},L_{1}\subset M$ Lagrangian
submanifolds such that $\langle\omega\rangle\cdot\pi_{2}(M,L_{i})=0$ for
$i=0,1$. Then we call

$HF(L_{0},L_{1})\coloneqq
H(CF(L_{0},L_{1}),\partial)=\frac{\text{ker}(\partial)}{\text{im}(\partial)}$
(76)

Lagrangian Floer cohomology (by Theorem 5.39, we can unambiguously choose any
$J\in C^{\infty}([0,1],\mathfrak{J}(M,\omega))$ and $H\in
C^{\infty}([0,1]\times M)$ to define $CF(L_{0},L_{1})$ and $\partial$).
Explicitly, the $n$-th group reads
$HF^{n}(L_{0},L_{1})=\text{ker}(\partial_{n})/\text{im}(\partial_{n-1})$.

From the constructions of Section 5.5 and the proof of Theorem 5.39, it
readily follows that Hamiltonian isotopic Lagrangian submanifolds
$L_{1},L_{1}^{\prime}\subset M$ induce an isomorphism on Floer cohomology:
$HF(L_{0},L_{1})\cong HF(L_{0},L_{1}^{\prime})$. On the other hand, if
$L_{0}\cap L_{1}=\emptyset$, then $HF(L_{0},L_{1})=\\{0\\}$.

######  Remark 5.41.

To obtain Floer homology, we must look at all previous constructions
“backwards”. For example, the strip of Figure 2 goes from $q$ to $p$,
ultimately causing a reversal of all gradings.

Returning to cohomology, with a suitable choice of data $(J,H)$ one can show
that the Floer complexes $CF(L_{0},L_{1})$ and $CF(L_{1},L_{0})$ along with
their differentials are dual to one another (the strip from $q$ to $p$ is then
an element of the latter).

######  Example 5.42.

It is instructive to see what goes wrong when we drop the assumptions
$\langle\omega\rangle\cdot\pi_{2}(M,L_{i})=0$, that is, when we allow
Lagrangian submanifolds to bound $J$-holomorphic disks and spheres. Consider
the 2-dimensional manifold $M\coloneqq\mathbb{R}\times\mathbb{S}^{1}$ with
standard area form $\omega\coloneqq ds\wedge dt$, and Lagrangian submanifolds
$L_{0},L_{1}$ as in Figure 4 left. Then $L_{0}$ and $L_{1}$ bound two
$J$-holomorphic strips $u,v$ of Maslov index 1, which go from $p$ to $q$
respectively from $q$ to $p$. By definition, $\partial(p)=\pm
T^{\omega([u])}q$ and $\partial(q)=\pm T^{\omega([v])}p$, whence
$\partial^{2}(p)=\pm T^{\omega([u])+\omega([v])}p\neq 0$.

Figure 4: If Lagrangian submanifolds bound $J$-holomorphic disks, then Floer’s
differential does not square to 0. [Source:[Aur13]]

What happens algebraically? Look at the local picture of $M$ in Figure 4
right, after identifying
$\mathbb{R}\times[0,1]\simeq\mathbb{D}^{2}\setminus\\{\pm
1\\}\subset\mathbb{C}$ so that $L_{0}\supset[-1,1]$ lies on $\mathbb{R}$ and
$L_{1}$ coincides with $\mathbb{S}^{1}$. Focus on the index 2 moduli space of
$J$-holomorphic strips from $p$ to itself whose image arises by removing a
slit along $L_{0}$ (bold in figure). Any such strip is given by

$w_{\alpha}(z)\coloneqq\frac{z^{2}+\alpha}{1+\alpha z^{2}}\,,\quad\text{with
}dw_{\alpha}(z)=2z\frac{1-\alpha^{2}}{(1+\alpha z^{2})^{2}}\,,$

for $\alpha\in(-1,1)$ regulating the slit length. There are two limit cases:

  * •

If $\alpha\rightarrow-1$, the index two strips $(w_{\alpha})_{\alpha}$
converge to a broken strip $w$ with index 1 components $u$ and $v$, just like
in Figure 3 left (indeed, we can “open up” the disk of Figure 4 along the slit
and diffeomorphically map it to the broken configuration). This means that
energy concentrates at $z=\pm 1$, as confirmed by:

$\lim_{\alpha\to-1}|dw_{\alpha}(\pm
1)|=\lim_{\alpha\to-1}\Big{|}\frac{-4\alpha}{2(1+\alpha)}\Big{|}=\infty\;.$

  * •

If $\alpha\rightarrow 1$, $(w_{\alpha})_{\alpha}$ converges to a constant
strip at $p$ with a disc bubble at $z=i$ mapping to the disk enclosed by
$L_{1}$. Hence, energy concentrates at $i$:

$\lim_{\alpha\to 1}|dw_{\alpha}(i)|=\lim_{\alpha\to
1}\Big{|}\frac{4\alpha}{2(1-\alpha)}\Big{|}=\infty\;.$

The disk bubbling occurring for $\alpha\rightarrow 1$ causes
$\partial^{2}(p)\neq 0$! $\blacklozenge$

Finally, we describe Floer cohomology in the case $L_{0}=L_{1}$. Again, the
proof of the following statement is rather advanced for our level — it
involves Morse theory — hence we just give a sketch and point the reader to
[Aur13, Example 1.12].

######  Proposition 5.43.

Let $(M^{2n},\omega)$ be a symplectic manifold, $L^{n}\subset M$ a Lagrangian
submanifold such that $\langle\omega\rangle\cdot\pi_{2}(M,L)=0$. Then
$HF(L,L)\cong H(L;\Lambda_{\mathbb{K}})$ (singular cohomology with
coefficients in the Novikov field).

###### Proof.

(Sketch) First, let $L$ be any compact real $n$-dimensional manifold. It is
known292929By Weinstein’s Lagrangian Neighbourhood Theorem; see [Sil12,
Theorem 8.4]. that a neighbourhood of $L\subset M$ is symplectomorphic to one
of $L_{0}\coloneqq\mathfrak{o}(L)\subset T^{*}L$, where
$\mathfrak{o}\in\Gamma(L,T^{*}L)$ denotes the zero section and $T^{*}L=$
$(T^{*}L,\theta=\sum_{i}dq_{i}\wedge dp_{i})$ the cotangent bundle of $L$.
Taking a Morse function $f\in C^{\infty}(L)$ and some $\varepsilon>0$, let
$L_{1}\coloneqq\text{graph}(\varepsilon df)=\\{(x,\varepsilon(df)_{x})\in
T_{x}^{*}L\mid x\in L\\}$. Then $L_{0},L_{1}\subset T^{*}L$ are Hamiltonian
isotopic exact Lagrangian submanifolds, intersecting transversely at
$\text{crit}(f)$, the set of critical points of $f$.

Choosing graded lifts, we obtain a graded Floer complex $CF(L_{0},L_{1})$ with
generators any $p\in\text{crit}(f)$, of $\text{deg}(p)=n-i(p)$ (for $i(p)$ the
Morse index of $p$). After suitable perturbations and rescalings, one can show
that $HF(L_{0},L_{1})\cong HM(f;\Lambda_{\mathbb{K}})$, the Morse cohomology
of $L$, in turn isomorphic to the ordinary singular cohomology
$H(L;\Lambda_{\mathbb{K}})$. Together with invariance under Hamiltonian
isotopy, we get $HF(L,L)\cong HF(L_{0},L_{0})\cong HF(L_{0},L_{1})\cong
H(L;\Lambda_{\mathbb{K}})$.

Now, let $L^{n}\subset M$ be a Lagrangian submanifold such that
$\langle\omega\rangle\cdot\pi_{2}(M,L)=0$. For a small enough perturbing
Hamiltonian, any $J$-holomorphic strip determining $HF(L,L)$ must be contained
in a small tubular neighbourhood of $L$ (due to energy constraints).
Therefore, the analysis carried above also applies to this general setting. ∎

Once again, the no-bubbling assumption
$\langle\omega\rangle\cdot\pi_{2}(M,L)=0$ is essential: otherwise, $HF(L,L)$
might be — whenever defined — a smaller subgroup of
$H(L;\Lambda_{\mathbb{K}})$.

####  5.8 The exact manifold setting

Before moving on, let us briefly revisit the theory of last section in
presence of exact Lagrangian submanifolds $L$ in an exact symplectic manifold
$M$. As before, we assume that $2c_{1}(TM)=0$ and all $L$ are compact,
oriented, spin, graded, with vanishing Maslov class.

######  Lemma 5.44.

Let $(M^{2n},\omega)$ be an exact symplectic manifold, $L_{0},L_{1}\subset M$
exact Lagrangian submanifolds. Then the energy of any $J$-holomorphic strip
defining $CF(L_{0},L_{1})$ is constant. Moreover, disk/sphere bubbling
phenomena do not occur (even though we do not require
$\langle\omega\rangle\cdot\pi_{2}(M,L_{i})=0$!).

###### Proof.

Let $\omega=d\nu\in\Omega^{2}(M)$ be the exact form attached to $M$, for some
$\nu\in\Omega^{1}(M)$, and let $f_{i}\in C^{\infty}(L_{i})$ be such that
$\nu|_{L_{i}}=df_{i}\in\Omega^{1}(L_{i})$ for $i=0,1$ (cf. Definition 5.2).
Consider a $J$-holomorphic strip $u:\mathbb{R}\times[0,1]\rightarrow M$ from
$p$ to $q$ with boundary in $L_{0}\cup L_{1}$, as in Figure 2. Its energy is:

$\displaystyle E(u)$
$\displaystyle=\int_{\mathbb{R}\times[0,1]}\mkern-24.0muu^{*}\omega=\int_{\mathbb{R}\times[0,1]}\mkern-24.0mud(u^{*}\nu)$
(since $d$ commutes with pullbacks)
$\displaystyle=\int_{\partial(\mathbb{R}\times[0,1])}\mkern-30.0muu^{*}\nu$
(by Stokes’ Theorem)
$\displaystyle=\int_{\mathbb{R}\times\\{1\\}}\mkern-24.0muu^{*}(\nu|_{L_{1}})\;-\\!\int_{\mathbb{R}\times\\{0\\}}\mkern-24.0muu^{*}(\nu|_{L_{0}})$
(because $\mathbb{R}\times\\{i\\}$ maps to $L_{i}$)
$\displaystyle=\int_{\mathbb{R}\times\\{1\\}}\mkern-24.0mud(u^{*}f_{1})\;-\\!\int_{\mathbb{R}\times\\{0\\}}\mkern-24.0mud(u^{*}f_{0})$
(since $d$ commutes with pullbacks)
$\displaystyle=\int_{\partial(\mathbb{R}\times\\{1\\})}\mkern-30.0muf_{1}\circ
u\;-\\!\int_{\partial(\mathbb{R}\times\\{0\\})}\mkern-30.0muf_{0}\circ u$ (by
Stokes’ Thm and def. of $u^{*}$)
$\displaystyle=f_{1}(u(\,\cdot\,,1))\Big{|}_{+\infty}^{-\infty}-f_{0}(u(\,\cdot\,,0))\Big{|}_{+\infty}^{-\infty}$
(because we integrate $p\rightarrow q$)
$\displaystyle=(f_{1}(q)-f_{1}(p))-(f_{0}(q)-f_{0}(p))\,.$ (77)

Therefore, the energy of $u$ is constant, as claimed.

Consider now possible bubbling phenomena. If $v:\mathbb{D}^{2}\rightarrow M$
is a $J$-holomorphic disk whose image intersects $u$ at some $r\in L_{1}$ (see
Figure 3 right), then its energy is:

$E(v)=\int_{\mathbb{D}^{2}}v^{*}\omega=\int_{\mathbb{S}^{1}}v^{*}\nu=\int_{\partial\mathbb{S}^{1}}v^{*}f_{1}=0$

(using commutativity of $d$ with $v^{*}$ and Stokes’ Theorem twice, and
$\partial\mathbb{S}^{1}=\emptyset$). Similarly, let
$w:\mathbb{C}P^{1}\simeq\mathbb{S}^{2}\rightarrow M$ be a $J$-holomorphic
sphere intersecting $u$ at some interior point. The associated energy is:

$E(w)=\int_{\mathbb{S}^{2}}w^{*}\omega=\int_{\partial\mathbb{S}^{2}}w^{*}\nu=0$

(since $\partial\mathbb{S}^{2}=\emptyset$). It follows that, in both cases,
there are no singular energy concentrations at any region of the strip.
Accordingly, disk and sphere bubbling is excluded. ∎

Therefore, if we work with exact Lagrangian submanifolds, the compactness
property required to make Floer’s differential well-defined is automatically
achieved. However, if $M$ is noncompact, then one needs an additional
assumption on it (specifically, $M$ must be either convex at infinity or
geometrically bounded; consult for example [CGK02]). Also, in light of
equation (5.8) — which tells us that all symplectic areas are equal — we can
rescale generators $p\in\mathcal{X}(L_{0},L_{1})$ by a factor
$T^{f_{1}(p)-f_{0}(p)}$, so that the weights $T^{\omega([u])}$ in (74) can be
ignored, allowing us to directly work over the field $\mathbb{K}$.

Most importantly, Lemma 5.44 tells us that exact Lagrangian submanifolds
cannot be obstructed, so that each one of them is a (non-zero) object of the
Fukaya category. We summarize this scenario.

######  Definition 5.45.

Let $(M,\omega)$ be an exact symplectic manifold equipped with some $J\in
C^{\infty}([0,1],\mathfrak{J}(M,\omega))$ and $H\in C^{\infty}([0,1]\times
M)$. Let $L_{0},L_{1}\subset M$ be exact Lagrangian submanifolds (endowed with
all data listed above). The Floer complex of $(L_{0},L_{1})$ is the free
module

$CF(L_{0},L_{1})\equiv
CF(L_{0},L_{1};H)\coloneqq\mkern-18.0mu\bigoplus_{\footnotesize p\in
L_{0}\cap(\phi_{H}^{1})^{\text{-1}}(L_{1})}\mkern-18.0mu\mathbb{K}\cdot p$

together with the boundary operator
$\partial\equiv\partial_{J,H}:CF(L_{0},L_{1})\rightarrow CF(L_{0},L_{1})$
defined as the $\mathbb{K}$-linear extension of

$\partial(p)\coloneqq\mkern-18.0mu\sum_{\footnotesize\begin{subarray}{c}q\in
L_{0}\cap(\phi_{H}^{1})^{\text{-1}}(L_{1})\\\
[u]:\text{ind}([u])=1\end{subarray}}\mkern-18.0mu(\\#\mathcal{M}(p,q;[u],J,H))q$

(in case one wishes to ignore orientations, it suffices to substitute
$\mathbb{K}\leftarrow\mathbb{Z}_{2}$ and
$\\#\mathcal{M}(p,q;[u],J,H)\leftarrow|\mathcal{M}(p,q;[u],J,H)|\,(\text{mod}\,2)\,$).
We obtain the Lagrangian Floer cohomology $HF(L_{0},L_{1})\coloneqq
H(CF(L_{0},L_{1};H),\partial_{J,H})$, which by Theorem 5.39 we know to be
$(J,H)$-independent.

###  6 The Fukaya category

####  6.1 The Floer product

We bring into play a third Lagrangian submanifold and adapt the geometrical
constructions of the previous chapter so to define a product between two Floer
complexes, the Floer product. Our primary reference is [Aur13, section 2].

Consider as usual an almost complex symplectic manifold $M^{2n}=(M,\omega,J)$
with $2c_{1}(TM)=0$ and $J\in\mathfrak{J}(M,\omega)$, and three Lagrangian
submanifolds $L_{0},L_{1},L_{2}\subset M$, which we assume to be compact,
oriented, spin, graded, with vanishing Maslov classes and enclosing no
bubbles. Furthermore, we take them to be pairwise transversely intersecting,
$L_{i}\pitchfork L_{j}$ for $i\neq j\in\\{0,1,2\\}$. Then transversality,
compactness and orientability are readily achieved (the general case will be
discussed afterwards). We seek a product operation between the associated
Floer complexes of the form:

$CF(L_{1},L_{2})\otimes CF(L_{0},L_{1})\rightarrow CF(L_{0},L_{2})\;.$

With reference to Figure 5, we study the following problem: given
$p_{1}\in\mathcal{X}(L_{0},L_{1})$, $p_{2}\in\mathcal{X}(L_{1},L_{2})$,
$q\in\mathcal{X}(L_{0},L_{2})$ (where $\mathcal{X}(L_{i},L_{j})=L_{i}\cap
L_{j}$ for now), and points $z_{1}=e^{-i\pi/3}$, $z_{2}=e^{i\pi/3}$,
$z_{0}=-1\in\mathbb{S}^{1}$, we want $J$-holomorphic maps
$u:\mathbb{D}^{2}\setminus\\{z_{0},z_{1},z_{2}\\}\rightarrow M$ with boundary
conditions

$\left\\{\begin{array}[]{llll}u(e^{i\alpha})\in L_{0}&\text{if
}\alpha\in\big{(}-\pi,-\frac{\pi}{3}\big{)}&\text{and}&\displaystyle\lim_{z\to
z_{0}}u(z)=q\\\ u(e^{i\alpha})\in L_{1}&\text{if
}\alpha\in\big{(}-\frac{\pi}{3},\frac{\pi}{3}\big{)}&\text{and}&\displaystyle\lim_{z\to
z_{1}}u(z)=p_{1}\\\ u(e^{i\alpha})\in L_{2}&\text{if
}\alpha\in\big{(}\frac{\pi}{3},\pi\big{)}&\text{and}&\displaystyle\lim_{z\to
z_{2}}u(z)=p_{2}\\\ \end{array}\right.\;,$ (78)

and finite energy $E(u)<\infty$. These are solutions of a suitably formulated
Cauchy–Riemann problem (see equation (81) below), and the so obtained homotopy
classes $[u]\in\pi_{2}(M,L_{0}\cup L_{1}\cup L_{2})$ define Moduli spaces
$\mathcal{M}(p_{1},p_{2},q;[u],J)$ which — thanks to our assumptions regarding
compactness and transversality — is a smooth manifold of finite dimension.
Similarly to Floer’s differential, the desired product operation involves a
weighted count of all such $J$-holomorphic maps, which we may also call
pseudoholomorphic triangles, as Figure 5 suggests.

Figure 5: A three-punctured $J$-holomorphic disk $u$ belonging to
$\mathcal{M}(p_{1},p_{2},q;[u],J)$. [Source:[Aur13]]

######  Remark 6.1.

The similarities with (57)–(59) are apparent. Indeed, we observe that any
small enough open neighbourhood of each “puncture” $z_{i}$ in $\mathbb{D}^{2}$
is biholomorphic to
$\mathbb{R}\times[0,1]\subset\mathbb{R}^{2}\cong\mathbb{C}$, as highlighted by
Figure 5, which exhibits a biholomorphism between the three-punctured disk
$\mathbb{D}^{2}\setminus\\{z_{0},z_{1},z_{2}\\}$ and the “three-ends strip” on
the left-hand side.

On the other hand, Figure 2 already alluded to the identification of
$\mathbb{D}^{2}\setminus\\{\pm 1\\}$ with $\mathbb{R}\times[0,1]$. Thus one
could — and should, as we will shortly see — interpret the problem (78) as a
“degree 3 analogue” of what we discussed throughout Chapter 5.

Notice that, in contrast to Definition 5.12, we did not require to quotient
$\mathcal{M}(p_{1},p_{2},q;[u],J)$ by $\mathbb{R}$, as there is no global free
action by translation on the space of solutions. Actually, since the
automorphism group $\text{Aut}(\mathbb{D}^{2})$ acts transitively on
cyclically ordered triples of marked boundary points, even the choice of
$z_{j}$’s is irrelevant. Then the dimension of
$\mathcal{M}(p_{1},p_{2},q;[u],J)$ directly coincides with the Maslov index
$\text{ind}([u])$:

######  Lemma 6.2.

Let $M^{2n}=(M,\omega,J)$ be an almost complex symplectic manifold with
$2c_{1}(TM)=0$, and $L_{0},L_{1},L_{2}\subset M$ Lagrangian submanifolds as
above, in particular graded with $[\phi_{L_{i}}]=0$. Then it holds:

$\textup{dim}(\mathcal{M}(p_{1},p_{2},q;[u],J))=\textup{ind}([u])=\textup{deg}(q)-\textup{deg}(p_{1})-\textup{deg}(p_{2})\,.$

(Again, this is equal to the Fredholm index defined through
$D_{\bar{\partial}_{J},u}$.)

###### Proof.

(Sketch) One needs only adapt the definition of Maslov index given in Section
5.4. Renaming $q\coloneqq p_{0}$ provisionally, we move counterclockwise along
$\partial(\text{im}(u))$: after suitably trivializing $u^{*}TM$, consider
paths $l_{i}$ from $T_{p_{j}}L_{j}$ to $T_{p_{k}}L_{j}$ as described in
Definition 5.17, as well as canonical short paths $\lambda_{p_{i}}$ from
$T_{p_{i}}L_{k}$ to $T_{p_{i}}L_{i}$, where $(i,j,k)$ runs over cyclic
permutations of $(0,1,2)$; then we study the loop $\gamma_{u}\coloneqq
l_{2}\ast\lambda_{p_{1}}\ast l_{0}\ast\lambda_{p_{2}}\ast
l_{1}\ast\lambda_{p_{0}}$ in $LGr(n)$ based at $T_{p_{0}}L_{0}$. Definition
5.22 about the degree of a generator is still valid in this context (and makes
sense thanks to our assumptions). Then the claimed equality comes from
additivity under path concatenation. ∎

The proof of Lemma 6.2 additionally shows that $CF(L_{1},L_{2})\otimes
CF(L_{0},L_{1})$ is equipped with the natural $\mathbb{Z}$-grading induced by
the two individual Floer complexes (hence, in particular,
$\text{deg}(p_{2}\otimes p_{1})=\text{deg}(p_{2})+\text{deg}(p_{1})$).

######  Definition 6.3.

Let $M=(M,\omega,J)$ be an almost complex symplectic manifold with
$2c_{1}(TM)=0$, and $L_{0},L_{1},L_{2}\subset M$ Lagrangian submanifolds with
the usual decorations, in particular such that
$\langle\omega\rangle\cdot\pi_{2}(M,L_{i})=0$. The Floer product
$\boldsymbol{\cdot}:\,CF(L_{1},L_{2})\otimes CF(L_{0},L_{1})\rightarrow
CF(L_{0},L_{2})$ is the $\Lambda_{\mathbb{K}}$-linear extension of

$p_{2}\otimes p_{1}\longmapsto
p_{2}\boldsymbol{\cdot}p_{1}\coloneqq\mkern-18.0mu\sum_{\begin{subarray}{c}q\in\mathcal{X}(L_{0},L_{2})\\\
[u]:\text{ind}([u])=0\end{subarray}}\mkern-18.0mu(\\#\mathcal{M}(p_{1},p_{2},q;[u],J))T^{\omega([u])}q\;,$
(79)

where $\mathcal{M}(p_{1},p_{2},q;[u],J)$ is the moduli space of solutions to
the Cauchy–Riemann problem (78) representing $[u]\in\pi_{2}(M,L_{0}\cup
L_{1}\cup L_{2})$. The sum in (79) is well defined (argue as in the proof of
Theorem 5.39), and Lemma 6.2 tells us that $\boldsymbol{\cdot}$ is of degree
0.

Now we investigate the key feature of the Floer product in relation to Floer’s
differential: it fulfills a graded Leibniz rule. We use the geometric
reasoning behind this to fine-tune the choice of perturbing, time-dependent
almost complex structures and Hamiltonians.

######  Proposition 6.4.

The Floer product satisfies the graded Leibniz rule

$\partial(p_{2}\boldsymbol{\cdot}p_{1})=(-1)^{|p_{1}|}\partial(p_{2})\boldsymbol{\cdot}p_{1}-p_{2}\boldsymbol{\cdot}\partial(p_{1})\,,$
(80)

where $|p_{1}|\coloneqq\textup{deg}(p_{1})$ and each $\partial$ is a suitable
Floer differential as given in Definition 5.37. Therefore, the product
$HF(L_{1},L_{2})\otimes HF(L_{0},L_{1})\rightarrow HF(L_{0},L_{2})$ induced on
cohomology is well defined, and actually independent from the choice of
perturbation data involved (cf. the proof).

###### Proof.

(Sketch) We explain qualitatively the geometric idea involved, keeping an eye
on what we did in the proof of Theorem 5.39. Consider generators
$p_{1}\in\mathcal{X}(L_{0},L_{1})$, $p_{2}\in\mathcal{X}(L_{1},L_{2})$,
$q\in\mathcal{X}(L_{0},L_{2})$ as in Figure 5, and a class
$[u]\in\pi_{2}(M,L_{0}\cup L_{1}\cup L_{2})$ with $\text{ind}([u])=1$.
Assuming transversality holds, $\mathcal{M}(p_{1},p_{2},q;[u],J)$ is a smooth
1-dimensional manifold which, by Gromov’s Compactness Theorem 5.34, can be
compactified to some $\overline{\mathcal{M}}(p_{1},p_{2},q;[u],J)$, by adding
broken strips (disk/sphere bubbling is excluded, by assumption).

But now we have three “strip-like” ends where energy can concentrate. By
additivity of the Maslov index under strip-breaking, its non-negativity for
disks (by transversality), and because non-constant strips have index at least
1, we have only three possible partitions
$\text{ind}([u])=\text{ind}([u^{\prime}])+\text{ind}([u^{\prime\prime}])$: up
to relabeling, $u^{\prime}$ must represent an index 1 strip and
$u^{\prime\prime}$ an index 0 disk with boundary on
$L_{0},L_{1},L_{2}$.303030This is different from the disk bubbles we are
excluding by assumption — the latters have boundary on a single submanifold,
and are the only disks causing energy blow-ups. Bluntly said, we can just
“pinch” the pseudoholomorphic triangle $u$ at one corner; see Figure 6.
Specifically, the pictured configurations contribute to the coefficient of
$T^{\omega([u])}q$ in $\partial(p_{2}\boldsymbol{\cdot}p_{1})$,
$p_{2}\boldsymbol{\cdot}\partial(p_{1})$ and
$\partial(p_{2})\boldsymbol{\cdot}p_{1}$ respectively.

Figure 6: The three possible limit configurations of a three-punctured
$J$-holomorphic disk in $\mathcal{M}(p_{1},p_{2},q;[u],J)$. [Source:[Aur13]]

This means that $\partial\overline{\mathcal{M}}(p_{1},p_{2},q;[u],J)$ is
contained in the disjoint union of products of moduli spaces like
$\mathcal{M}(p_{i},r_{i};[u^{\prime}],J)\times\mathcal{M}(r_{i},p_{j},p_{k};[u^{\prime\prime}],J)$,
where $r_{j}\in\mathcal{X}(L_{i},L_{j})$ are the “pinched” points (and we set
again $q\coloneqq p_{0}$). The converse inclusion is provided by another
gluing theorem, so that $\overline{\mathcal{M}}(p_{1},p_{2},q;[u],J)$ is
indeed a compact 1-dimensional smooth manifold with (coherently oriented)
boundary as just described. Lemma 5.38 tells us that the latter’s (un)signed
cardinality is zero, thus summing over all homotopy classes — and carefully
accounting degrees — we obtain the Leibniz rule (80).

The fact that (79) induces a well-defined product on cohomology is now an easy
argument: if $\partial(p_{1})=\partial(p_{2})=0$, then
$\partial(p_{2}\boldsymbol{\cdot}p_{1})=0$ by (80); if moreover
$p_{1}=\partial(p_{1}^{\prime})$, then
$p_{2}\boldsymbol{\cdot}p_{1}=\partial(p_{2}\boldsymbol{\cdot}p_{1}^{\prime})$.
Thus cocycles and coboundaries are preserved, as desired.

Finally, we discuss the general case when $L_{i}\,\cancel{\pitchfork}\,L_{j}$.
To achieve transversality, we reintroduce time-dependent almost complex
structures and Hamiltonians, specifically a pair $(J_{ij},H_{ij})\in
C^{\infty}([0,1],\mathfrak{J}(M,\omega))\times C^{\infty}([0,1]\times M)$ for
each strip-like end, with labels $0\leq i<j\leq 2$ according to the bounding
Lagrangian submanifolds (see Figure 5 left). Notice this does make sense by
Remark 6.1. This time, however, such data cannot be extended on the whole
punctured disk: we must choose some global $J\in
C^{\infty}(\mathbb{D}^{2},\mathfrak{J}(M,\omega))$ and $H\in
C^{\infty}(\mathbb{D}^{2}\times M)$ (that is, families smoothly varying in
$\mathbb{D}^{2}$) which in a fixed neighbourhood of each $z_{j}$ correspond to
the chosen pair $(J_{ij},H_{ij})$.

So we recover the perturbed PDE (66) only locally about the punctures, where
we can use holomorphic local complex coordinates $z=s+it$ for
$(s,t)\in\mathbb{R}\times[0,1]$. To see this, fix some
$\beta\in\Omega^{1}(\mathbb{D}^{2})$ such that $\beta|_{\mathbb{S}^{1}}=0$ and
$\beta=dt$ about the $z_{j}$’s. Perturbing the Cauchy–Riemann equation (57),
we obtain

$(du-X_{H}\otimes\beta)_{J}^{0,1}=0\;,$ (81)

which about the three punctures coincides exactly with Floer’s equation:
$0=(du-X_{H}\otimes
dt)_{J}^{0,1}=\partial_{s}u+J_{t}(u)(\partial_{t}u-X_{H_{t}}(u))$, where $J$
and $H$ are merely $t$-dependent. Analogously to what found in Section 5.5, it
holds $\mathcal{X}(L_{i},L_{j})=L_{i}\cap(\phi_{H_{ij}}^{1})^{-1}(L_{j})$, and
the proof of a few paragraphs above carries over to the perturbed product map
$\boldsymbol{\cdot}:CF(L_{1},L_{2};J_{12},H_{12})\otimes
CF(L_{0},L_{1};J_{01},H_{01})\rightarrow CF(L_{0},L_{2};J_{02},H_{02})$, then
fulfilling the Leibniz rule. ∎

####  6.2 The higher order composition maps

Let us now consider higher order products involving $(d+1)$-many Lagrangian
submanifolds $L_{0},...,L_{d}\subset M$, with the usual decorations.

Given $d+1$ generators $p_{1}\in\mathcal{X}(L_{0},L_{1})$,
$p_{2}\in\mathcal{X}(L_{1},L_{2})$, …, $p_{d}\in\mathcal{X}(L_{d-1},L_{d})$,
$q\in\mathcal{X}(L_{d},L_{0})$ and boundary punctures
$z_{j}=e^{i\pi(1+2j/(d+1))}$, $z_{0}=-1\in\mathbb{S}^{1}$ for $j=1,...,d$, we
seek $J$-holomorphic maps
$u:\mathbb{D}^{2}\setminus\\{z_{0},z_{1},...,z_{d}\\}\rightarrow M$ satisfying
a degree-$(d+1)$ analogue of the perturbed Cauchy–Riemann equation (81) with
boundary conditions

$\left\\{\begin{array}[]{llll}u(e^{i\alpha})\in L_{0}&\text{if
}\alpha\in\big{(}\pi,\pi+\frac{2\pi}{d+1}\big{)}&\text{and}&\displaystyle\lim_{z\to
z_{0}}u(z)=q\\\ u(e^{i\alpha})\in L_{j}&\text{if
}\alpha\in\big{(}\pi+\frac{2j\pi}{d+1},\pi+\frac{2(j+1)\pi}{d+1}\big{)}&\text{and}&\displaystyle\lim_{z\to
z_{j}}u(z)=p_{j}\\\ \end{array}\right.,$ (82)

and finite energy $E(u)<\infty$. We call this the generalized Cauchy–Riemann
problem. Its solutions representing a fixed class
$[u]\in\pi_{2}(M,\bigcup_{i=0}^{d}L_{i})$ form up the moduli space
$\mathcal{M}(p_{1},...,p_{d},q;[u],J,H)$. The situation is depicted in Figure
7, which also makes clear why the images $\text{im}(u)$ of $J$-holomorphic
solutions $u$ (and by extension, the $u$’s themselves) are called
pseudoholomorphic polygons.

Figure 7: A $(d+1)$-punctured $J$-holomorphic disk $u$ in
$\mathcal{M}(p_{1},...,p_{d},q;[u],J,H)$.

Here we already chose suitable domain-dependent $J\in
C^{\infty}(\mathbb{D}^{2},\mathfrak{J}(M,\omega))$ and $H\in
C^{\infty}(\mathbb{D}^{2}\times M)$, paying attention (see Definition 6.9
below) to the following fact: in contrast to the case $d=2$ of previous
section, here there are many choices of cyclically ordered tuples
$(z_{0},z_{1},...,z_{d})\subset\mathbb{S}^{1}$, differing in the exact
reciprocal position of each puncture along its arc.

######  Definition 6.5.

Let $\text{Punct}_{d+1}$ be the space of all possible ordered configurations
$(z_{0},z_{1},...,z_{d})\subset\mathbb{S}^{1}$ of boundary punctures, for
$d\geq 2$, and $\text{Aut}(\mathbb{D}^{2})$ the group of biholomorphisms of
$\mathbb{D}^{2}$. Then
$\mathcal{M}_{0,d+1}\coloneqq\text{Punct}_{d+1}/\text{Aut}(\mathbb{D}^{2})$ is
the moduli space of conformal structures on $\mathbb{D}^{2}$, a contractible
$(d-2)$-dimensional smooth manifold.

The space $\mathcal{M}_{0,d+1}$ has a natural compactification to a
$(d-2)$-dimensional polytope $\overline{\mathcal{M}}_{0,d+1}$ — which is
(homeomorphically) none other than the associahedron $K_{d}$ we encountered
back in Section 1.1! Its boundary $\partial\overline{\mathcal{M}}_{0,d+1}$
parametrizes all possible degenerations of $\mathbb{D}^{2}$ (nodal
degenerations) into subdisks each carrying at least one puncture $z_{j}$.
Specifically, each $(d-3)$-skeleton component corresponds to a (homotopy class
of a) degeneration into a pair of disks each retaining at least two punctures;
likewise, to the $(d-4)$-skeleton we can assign configurations with three
disks, and so on.

######  Example 6.6.

If $d=3$, we know the compactified 1-dimensional polytope
$\overline{\mathcal{M}}_{0,4}=K_{3}$ to be homeomorphic to
$[0,1]\subset\mathbb{R}$ (see Figure 1, bottom left). The two extremal points
forming its 0-skeleton label the two possible degenerations of
$\mathbb{D}^{2}$ for fixed punctures $z_{1},z_{2},z_{3}\in\mathbb{S}^{1}$
(without loss of generality): either $z_{0}\in\mathbb{S}^{1}$ can “slide”
along $L_{0}$ towards $z_{1}$ and end up with it in a subdisk of the
degenerate configuration corresponding to 0, or it can slide along $L_{3}$
towards $z_{3}$ to produce that associated to 1 (see Figure 8).

Figure 8: The polytope $\overline{\mathcal{M}}_{0,4}=K_{3}$ and its
corresponding degenerate limit configurations (white dots denote newly
generated nodes).

Alternatively, we can fix $z_{0},z_{1},z_{2}$ and let $z_{3}$ vary. This
“sliding operation” — formalized by the homotopy $m_{3}$ of Section 1.1 —
contributes to some higher order composition maps on Floer complexes, as we
will show below. $\blacklozenge$

######  Remark 6.7.

Building on last example, we can justify the definition of
$\mathcal{M}_{0,d+1}$ as follows. The group
$\text{Aut}(\mathbb{D}^{2})\cong\text{PSL}(2,\mathbb{R})$ is a 3-dimensional
Lie group whose action can uniquely fix at most three boundary punctures of
$\mathbb{D}^{2}$ (indeed
$\mathcal{M}_{0,3}=\text{Punct}_{3}/\text{Aut}(\mathbb{D}^{2})$ is
0-dimensional, thus negligible, as discussed right after Remark 6.1). If
$d>2$, that is, if we work with more than three punctures, then these can
slide freely along the arcs mapping to the Lagrangian submanifolds to produce
non-trivial distinct classes in $\mathcal{M}_{0,d+1}$. Modulo the three
punctures which can be fixed, it follows that $\mathcal{M}_{0,d+1}$ has
structure of a smooth manifold of dimension $(d+1)-3=d-2$.

In analogy to Example 6.6, nodal degenerations of $\mathbb{D}^{2}$ arise when
two or more punctures hit each other, causing a bubbling effect which resolves
into subdisks carrying the points involved. The compactification of
$\mathcal{M}_{0,d+1}$ amounts to including these limit configurations, then
labelled by $\partial\overline{\mathcal{M}}_{0,d+1}$.

######  Example 6.8.

The intuition for $\overline{\mathcal{M}}_{0,5}=K_{4}$ is then the following.
Given five punctures $z_{0},z_{1},z_{2},z_{3},z_{4}\in\mathbb{S}^{1}$, fix a
(consecutive!) triple, say $z_{1},z_{2},z_{3}$, and let $z_{0},z_{4}$ vary. We
have ten ways to “crash” this pair. For example, $z_{0}$ can hit $z_{1}$,
which we may suggestively write as $(z_{0}\ast z_{1})$. The other combinations
are (moving counterclockwise in Figure 9): $(z_{4}\ast(z_{0}\ast z_{1}))$,
$(z_{4}\ast z_{0}\ast z_{1})$, $((z_{4}\ast z_{0})\ast z_{1})$, $(z_{4}\ast
z_{0})$, $(z_{3}\ast(z_{4}\ast z_{0}))$, $(z_{3}\ast z_{4}\ast z_{0})$,
$((z_{3}\ast z_{4})\ast z_{0})$, $(z_{3}\ast z_{4})$ and $(z_{0}\ast
z_{1})(z_{3}\ast z_{4})$.

Parentheses matter, as they regulate the collision order! Indeed, $(z_{4}\ast
z_{0}\ast\leavevmode\nobreak\ z_{1})$ corresponds to the nodal degeneration
into two subdisks with one carrying $z_{0},z_{1},z_{4}$, while
$(z_{4}\ast(z_{0}\ast z_{1}))$ encodes a configuration of three disks whose
medial one carries $z_{4}$ and whose terminal one collects $z_{0}$ and
$z_{1}$. Clearly, five among the listed combinations label the edges of the
pentagon $K_{4}$ (that is, its 1-skeleton), while the remaining five its
vertices (the 0-skeleton).

Figure 9: The polytope $\overline{\mathcal{M}}_{0,5}=K_{4}$ and its
corresponding degenerate limit configurations (white dots denote newly
generated nodes).

We purposefully kept the variables in cyclic order to highlight yet another
interpretation: a component of the $(d-r)$-skeleton of $K_{d}$ corresponds to
a suitable insertion of $r-2$ pairs of parentheses in the string
$z_{0}z_{1}...z_{d-1}z_{d}$. For example, $(z_{4}\ast(z_{0}\ast
z_{1}))\equiv(z_{0}z_{1}))z_{2}z_{3}(z_{4}$. This makes the study of higher
dimensional associahedra far easier. The reader may wish to investigate
$K_{5}$ (the rightmost in Figure 1), whose nine faces, twenty-one edges and
fourteen vertices parametrize the possible nodal degenerations into
respectively two, three and four subdisks. $\blacklozenge$

We finally describe how to choose compatible perturbation data for the general
case of $d+1$ Lagrangian submanifolds. The proof of its existence is addressed
in [Sei08, Lemma 9.5].

######  Definition 6.9.

Given a conformal structure $(z_{0},z_{1},...,z_{d})\in\mathcal{M}_{0,d+1}$,
consider the associated generalized Cauchy–Riemann problem of the form (81)
with boundary conditions (82) and finite energy constraint. Consider also the
time-dependent $(J_{ij},H_{ij})\in
C^{\infty}([0,1],\mathfrak{J}(M,\omega))\times C^{\infty}([0,1]\times M)$
determining the $CF(L_{i},L_{j})$’s, for each strip-like end (that is, defined
on fixed neighbourhoods of the $p_{j}$’s and $q$ where holomorphic local
coordinates are available), with $0\leq i<j\leq d$ according to the bounding
Lagrangian submanifolds. To achieve transversality, we choose $J\in
C^{\infty}(\mathbb{D}^{2},\mathfrak{J}(M,\omega))$ and $H\in
C^{\infty}(\mathbb{D}^{2}\times M)$ such that:

  * •

$(J,H)$ agrees with each $(J_{ij},H_{ij})$ in the respective strip-like end,

  * •

upon nodal degeneration of $\mathbb{D}^{2}$ as in Definition 6.5, $J$ and $H$
are translation-invariant in the (strip-like) regions about the nodal points,
and indeed coincide with the local Hamiltonians and almost complex structures
there.

We call such a pair $(J,H)$ Floer datum; by construction, it depends on
$(z_{0},z_{1},...,z_{d})$ as well as all $J_{ij}$’s and $H_{ij}$’s. Then the
arising Floer cohomology is said to be “unobstructed”.

The involvement of $\mathcal{M}_{0,d+1}$ in our discussion also implies we
must adapt Lemma 6.2 and its proof accordingly.

######  Lemma 6.10.

Fix a conformal structure $(z_{0},z_{1},...,z_{d})\in\mathcal{M}_{0,d+1}$ and
a compatible Floer datum $(J,H)\in
C^{\infty}(\mathbb{D}^{2},\mathfrak{J}(M,\omega))\times
C^{\infty}(\mathbb{D}^{2}\times M)$ as specified above. Consider the
associated generalized Cauchy–Riemann problem, yielding moduli spaces
$\mathcal{M}(p_{1},...,p_{d},q;[u],J,H)$ for each solution
$[u]\in\pi_{2}(M,\bigcup_{i=0}^{d}L_{i})$. Then:

$\displaystyle\textup{dim}(\mathcal{M}(p_{1},...,p_{d},q;[u],J,H))$
$\displaystyle=\textup{dim}(\mathcal{M}_{0,d+1})+\textup{ind}([u])$
$\displaystyle=d-2+\textup{deg}(q)-\sum_{i=1}^{d}\textup{deg}(p_{i})\;,$ (83)

where $\textup{dim}(\mathcal{M}_{0,d+1})$ accounts for possible deformations
of $(z_{0},z_{1},...,z_{d})$.

######  Definition 6.11.

Let $(M,\omega,J)$ be an almost complex symplectic manifold with
$2c_{1}(TM)=0$, and $L_{0},...,L_{d}\subset M$ Lagrangian submanifolds with
the usual decorations, in particular such that
$\langle\omega\rangle\cdot\pi_{2}(M,L_{i})=0$. Choose a Floer datum $(J,H)\in
C^{\infty}(\mathbb{D}^{2},\mathfrak{J}(M,\omega))\times
C^{\infty}(\mathbb{D}^{2}\times M)$ to achieve transversality. The $d$-th
Floer composition map $\mu^{d}:\,CF(L_{d-1},L_{d})\otimes...\otimes
CF(L_{0},L_{1})\rightarrow CF(L_{0},L_{d})$ is the
$\Lambda_{\mathbb{K}}$-linear extension of

$\mu^{d}(p_{d},...,p_{1})\coloneqq\mkern-18.0mu\sum_{\begin{subarray}{c}q\in\mathcal{X}(L_{0},L_{d})\\\
[u]:\text{ind}([u])=2-d\end{subarray}}\mkern-18.0mu(\\#\mathcal{M}(p_{1},...,p_{d},q;[u],J,H))T^{\omega([u])}q\;,$
(84)

where $\mathcal{M}(p_{1},...,p_{d},q;[u],J,H)$ is the moduli space of
solutions to the generalized Cauchy–Riemann problem (81) representing
$[u]\in\pi_{2}(M,\bigcup_{i=0}^{d}L_{i})$. The sum is well defined, by the
usual argument, and we note that Lemma 6.10 makes $\mu^{d}$ a graded map of
degree $2-d$.

In particular, $\mu^{1}=\partial_{J,H}$ is the (degree 1) Floer differential
(74) and $\mu^{2}=\boldsymbol{\cdot}$ the (degree 0) Floer product (79).

######  Proposition 6.12.

The Floer composition maps $\mu^{d}$ fulfill the $A_{\infty}$-associativity
equations (6):

$\mkern-30.0mu\sum_{\begin{subarray}{c}1\leq m\leq d\\\ \quad 0\leq n\leq
d-m\end{subarray}}\mkern-24.0mu(-1)^{\dagger_{n}}\mu^{d-m+1}(p_{d},...,p_{n+m+1},\mu^{m}(p_{n+m},...,p_{n+1}),p_{n},...,p_{1})=0\;,$
(85)

in $CF(L_{0},L_{d})$, where $p_{i}\in CF(L_{i-1},L_{i})$ and313131In light of
equation (85), we definitely swap $\text{deg}(p_{i})$ for $|p_{i}|$.
$\dagger_{n}\coloneqq(\sum_{i=1}^{n}|p_{i}|)-n$.

For $d=1,2$, the first two such equations read $\partial(\partial(p_{1}))=0$
and
$\partial(p_{2}\boldsymbol{\cdot}p_{1})=(-1)^{|p_{1}|}\partial(p_{2})\boldsymbol{\cdot}p_{1}-p_{2}\boldsymbol{\cdot}\partial(p_{1})$
(the graded Leibniz rule (80)). In particular, the case $d=3$ implies that
Floer’s product $\mu^{2}$ is associative up to homotopy given by $\mu^{3}$,
hence associative on cohomology.

###### Proof.

(Sketch) The cases $d=1,2$ were addressed in Theorem 5.39 respectively
Proposition 6.4. Proof of the latter also showed associativity of $\mu^{2}$ on
cohomology.

To show that the general identity (85) holds, we follow the “usual” steps: in
the framework of Definition 6.11, we look at classes
$[u]\in\pi_{2}(M,\bigcup_{i=0}^{d}L_{i})$ of $\text{ind}([u])=3-d$ and their
moduli spaces $\mathcal{M}(p_{1},...,p_{d},q;[u],J,H)$, which, by equation
(6.10), transversality and Proposition 5.34, are then 1-dimensional smooth
manifolds compactifying to some
$\overline{\mathcal{M}}(p_{1},...,p_{d},q;[u],J,H)$ whose boundary points
either correspond to

  * •

index 1 $J$-holomorphic strips breaking off at one among the $d+1$ generators,
which will then be separated from the pseudoholomorphic polygon (as in Figure
6), or

  * •

nodal degenerations of $\mathbb{D}^{2}$ described by the $(d-3)$-skeleton of
$\overline{\mathcal{M}}_{0,d+1}$, which are pairs of disks each carrying at
least two punctures (as in Figure 8).

The former case is caused by energy build-ups at one of the generators and
produces all terms involving $\mu^{1}$ in (85), while the second originates
when punctures “hit” each other (as expalined in Remark 6.7), contributing
with all the remaining summands of (85). The desired equation follows by
arguing once again that the space
$\overline{\mathcal{M}}(p_{1},...,p_{d},q;[u],J,H)$ is a compact 1-dimensional
manifold with (coherently oriented) boundary whose signed cardinality must
then vanish, and finally summing over all homotopy classes.323232Consult
[Fuk02a, section 3] for a slightly different approach, from the standpoint of
Morse–Bott Floer theory.

Let us discuss the case $d=3$. Boundary points in
$\overline{\mathcal{M}}(p_{1},p_{2},p_{3},q;[u],J,H)$ of the first type are
index 1 strips breaking off from the index $-1$ pseudoholomorphic square at
one of its four punctures; they are responsible for the terms
$\partial(\mu^{3}(p_{3},p_{2},p_{1}))$, $\mu^{3}(p_{3},p_{2},\partial p_{1})$,
$\mu^{3}(p_{3},\partial p_{2},p_{1})$ and $\mu^{3}(\partial
p_{3},p_{2},p_{1})$. Boundary points of the second type correspond instead to
the two nodal configurations described in Example 6.6, pairs of twice
punctured index 0 disks; they yield
$p_{3}\boldsymbol{\cdot}(p_{2}\boldsymbol{\cdot}p_{1})$ and
$(p_{3}\boldsymbol{\cdot}p_{2})\boldsymbol{\cdot}p_{1}$. With coefficients
dictated by the degrees of the generators, we see that the associated
$A_{\infty}$-identity is indeed fulfilled. ∎

####  6.3 The $A_{\infty}$-structure of the Fukaya category

We are finally ready to start reaping the harvest of Part I.

######  Definition 6.13.

Let $M=(M,\omega)$ be a symplectic manifold with $2c_{1}(TM)=0$. Its (compact)
Fukaya category $\mathscr{F}(M)$ is the $A_{\infty}$-category characterized by
the following data:

  * –

A set of objects $\textup{obj}(\mathscr{F}(M))$ consisting of compact oriented
spin graded Lagrangian submanifolds $L_{i}\subset M$ of vanishing Maslov class
$\mu_{L_{i}}\equiv[\phi_{L_{i}}]=0\in H^{1}(L;\mathbb{Z})$ and fulfilling the
no-bubbling condition $\langle\omega\rangle\cdot\pi_{2}(M,L_{i})=0$, together
with an implicit choice of graded lift
$\tilde{\phi}_{L_{i}}:L_{i}\rightarrow\mathbb{R}$ and spin structure.

  * –

Perturbation data $(J_{L,L^{\prime}},H_{L,L^{\prime}})\in
C^{\infty}([0,1],\mathfrak{J}(M,\omega))\times C^{\infty}([0,1]\times M)$ for
every pair $(L,L^{\prime})\subset\textup{obj}(\mathscr{F}(M))$, and Floer data
$(J,H)\in C^{\infty}(\mathbb{D}^{2},\mathfrak{J}(M,\omega))\times
C^{\infty}(\mathbb{D}^{2}\times M)$ as in Definition 6.9 for every tuple
$(L_{0},...,L_{d})\subset\textup{obj}(\mathscr{F}(M))$, thus compatible with
all $(J_{L_{i},L_{j}},H_{L_{i},L_{j}})$’s in the strip-like regions of the
$J$-holomorphic curves solving the underlying generalized Cauchy–Riemann
problem (82).

  * –

For any two $L_{0},L_{1}\in\textup{obj}(\mathscr{F}(M))$ (possibly equal), the
$\mathbb{Z}$-graded free $\Lambda_{\mathbb{K}}$-module

$\textup{hom}_{\mathscr{F}(M)}(L_{0},L_{1})\coloneqq
CF(L_{0},L_{1};H_{L_{0},L_{1}})\,,$ (86)

resulting from the Hamiltonian perturbation by $H_{L_{0},L_{1}}$ of (56).

  * –

(Multi)linear $\mathbb{Z}$-graded composition maps
$\mu_{\mathscr{F}(M)}^{d}\coloneqq\mu^{d}$ as in (84), for every $d\geq 1$ and
$L_{0},...,L_{d}\in\textup{obj}(\mathscr{F}(M))$ (then fulfilling the
$A_{\infty}$-associativity equations, by Proposition 6.12).

######  Remark 6.14.

  * •

We already hinted that there are many different definitions of Fukaya
category, depending on the precise assumptions for both $M$ and its Lagrangian
submanifolds. For example, dropping the requirements $2c_{1}(TM)$ $=0$ and
$[\phi_{L_{i}}]=0$ for each Lagrangian submanifold reduces the
$\mathbb{Z}$-grading to a $\mathbb{Z}_{2}$-grading, and choosing $\mathbb{K}$
of $\text{char}(\mathbb{K})=2$ allows us to ignore spin structures. Most
prominently, the wrapped Fukaya category considers only exact manifolds,
building on what seen in Section 5.8; we briefly discuss this variant in
Section 6.5 below. The relative and projective versions of Fukaya categories
are also discussed in [Sei13, section 8].

  * •

We can even tinker with the perturbation data; however, all the so obtained
$A_{\infty}$-categories turn out to be quasi-isomorphic (in the sense of
Definition 2.8). This means that, up to quasi-isomorphism, $\mathscr{F}(M)$ is
a “symplectic invariant” of $M$ (more on this in [Sei08, chapter 10]).

  * •

If we drop instead the no-bubbling condition
$\langle\omega\rangle\cdot\pi_{2}(M,L_{i})=0$ and reshape our definitions in
Morse–Bott Floer theoretic language, we ultimately obtain the notion of curved
$A_{\infty}$-category, where disk bubbles are encoded by images of a degree 0
composition map $\mu^{0}:L_{i}\subset M\mapsto\mu_{L_{i}}^{0}\in
CF(L_{i},L_{i})$ which fits into the $A_{\infty}$-associativity equations
(more details in [Aur13, Remark 2.11] and [Fuk02a, section 3]). In this
scenario, one restricts the set of objects to those whose $\mu_{L}^{0}$ is a
multiple of the c-unit $e_{L}$ (see equation (87) below), which in mirror
symmetry corresponds to the so-called “superpotential” or “central charge”.

  * •

In low dimension, the existence and well-definiteness of Fukaya categories
might be guaranteed by topological constraints: for example, any 4-dimen-
sional symplectic manifold $M^{4}$ (such as classical Calabi-Yau manifolds)
with $2c_{1}(TM)=0$ admits no bubbling phenomena (see [AS09, Lemma 3.3]),
hence its associated Fukaya category is properly defined (even though
invariance under Hamiltonian isotopies of its objects — Lagrangian surfaces
$L^{2}\subset M$ of vanishing Maslov class — is not a priori achieved; cf.
[AS09, Lemma 3.4]).

######  Definition 6.15.

Let $M$ be a symplectic manifold with $2c_{1}(TM)=0$, and $\mathscr{F}(M)$ its
Fukaya category. The induced cohomological category $H(\mathscr{F}(M))$ (as
described in Definition 2.6) is called the Donaldson–Fukaya category, or
quantum category. Specifically, its objects are still Lagrangian submanifolds
and its Hom-sets are the total Floer cohomology rings (76) induced by the
Floer differentials.

The subcategory $H^{0}(\mathscr{F}(M))\subset H(\mathscr{F}(M))$ only
considers the zeroth cohomology of each morphism space,
$\textup{Hom}_{H^{0}(\mathscr{F}(M))}(L_{0},L_{1})\coloneqq
HF^{0}(L_{0},L_{1})$.

Observe that, since Hamiltonian isotopic Lagrangian submanifolds
$L_{1},L_{1}^{\prime}\subset M$ induce isomorphisms
$\textup{Hom}_{H(\mathscr{F}(M))}(L_{0},L_{1})=HF(L_{0},L_{1})\cong
HF(L_{0},L_{1}^{\prime})=\textup{Hom}_{H(\mathscr{F}(M))}(L_{0},L_{1}^{\prime})$
for any $L_{0}\subset M$ (cf. Definition 5.5 and Remark 5.25), setting
$L_{0}\coloneqq L_{1}$ proves that they are isomorphic as objects of
$H(\mathscr{F}(M))$. This ticks off the only restriction we imposed back in
Section 5.2.

Now that Fukaya categories are ready to go, we may wonder about their
unitality as $A_{\infty}$-categories. The next few observations are discussed,
for example, in [AS09, section 3].

######  Lemma 6.16.

Given any Fukaya category $\mathscr{F}(M)$, each Lagrangian submanifold
$L\in\textup{obj}(\mathscr{F}(M))$ admits a cohomological unit
$e_{L}\in\textup{hom}_{\mathscr{F}(M)}^{0}(L,L)$, which we recall is a degree
$0$ cocycle (thus $\mu_{\mathscr{F}(M)}^{1}(e_{L})=0$) such that $\langle
e_{L}\rangle=\textup{id}_{L}\in HF(L,L)$ in the Donaldson–Fukaya category.
Therefore, $H(\mathscr{F}(M))$ is an actual category.

###### Proof.

(Sketch) Recall that

$CF^{0}(L,L)=\mkern-18.0mu\bigoplus_{p\in\mathcal{X}(L,L):|p|=0}\mkern-18.0mu\Lambda_{\mathbb{K}}\cdot
p\;,$

where $\mathcal{X}(L,L)=L\cap(\phi_{H}^{1})^{-1}(L)$ is the transverse
intersection of $L$ with a copy perturbed by a suitable Hamiltonian function
$H\in C^{\infty}(M\times[0,1])$. Among all these perturbed degree 0 generators
$p$, the cohomological unit $e_{L}$ is obtained by considering the degree 1
(so $d=0$) analogue of the Cauchy–Riemann problem (66), where we choose
$L_{0}=L_{1}\coloneqq L$ and start with a single boundary puncture of
$\mathbb{D}^{2}$ (then mapping to $e_{L}$). The cocycle we seek turns out to
be

$e_{L}\coloneqq\mkern-18.0mu\sum_{q\in\mathcal{X}(L,L):|q|=0}\mkern-18.0mu(\\#\mathcal{M}(q;[u],J,H))T^{\omega([u])}q\;.$
(87)

One can show with help of Proposition 5.43 that this does indeed represent the
identity morphism in $HF(L,L)\cong H(L;\Lambda_{\mathbb{K}})$. The precise
proof involves a gluing argument from TQFT — far exceeding our scope. ∎

######  Lemma 6.17.

The perturbation data defining $\mathscr{F}(M)$ can be chosen in such a way
that the cohomological unit $e_{L}\in\textup{hom}_{\mathscr{F}(M)}^{0}(L,L)$
of any Lagrangian submanifold $L\subset M$ fulfills
$\mu_{\mathscr{F}(M)}^{2}(e_{L},e_{L})=e_{L}$ and
$\mu_{\mathscr{F}(M)}^{d}(e_{L},...,e_{L})=0$ if $d>2$.

###### Proof.

(Sketch) Consider the Morse function $f\in C^{\infty}(L)$ we used in the proof
of Proposition 5.43, obtaining $HF(L,L)\cong HM(f;\Lambda_{\mathbb{K}})$.
Morse homology is isomorphic to singular homology, which, in the case of a
cell complex (such as a smooth manifold), is in turn isomorphic to cellular
homology (see [Hat09, Theorem 2.35]). Since $L^{n}\subset M^{2n}$ is
$n$-dimensional, it follows that $CF(L,L)$ is concentrated in the degrees
between 0 and $n$. Therefore, as $\mu_{\mathscr{F}(M)}^{d}(e_{L},...,e_{L})\in
CF^{2-d}(L,L)$ ($\mu_{\mathscr{F}(M)}^{d}$ being of degree $2-d$), $d>2$
forces it to vanish.

To prove $\mu_{\mathscr{F}(M)}^{2}(e_{L},e_{L})=e_{L}$, we just take $f$ to
have a unique maximum and minimum on each connected component of $L$. ∎

The last lemma tells us that $\mathscr{F}(M)$ is not strictly unital, since
back in equation (• ‣ 2.23) we imposed the stronger requirement that the two
identities should hold whenever at least one of the inputs is a unit $e_{L}$.

However, we can resort to some formal diffeomorphism
$\Phi:\mathscr{F}(M)\rightarrow\widetilde{\mathscr{F}}(M)$ with target a
strictly unital $A_{\infty}$-category $\widetilde{\mathscr{F}}(M)$, even
though this will be geometrically meaningless: the composition maps
$\mu_{\widetilde{\mathscr{F}}(M)}^{d}$ will no longer be counts of
pseudoholomorphic polygons!

Therefore, when dealing with Fukaya categories, we can work with cohomological
units at best — this is the reason why we focused so intensely on c-unital
$A_{\infty}$-categories, after all! With the fundamental structure of
$\mathscr{F}(M)$ clarified, let us now summarize the main results we know from
Part I:

  * •

First, we can study $\mathscr{F}(M)$ through the $A_{\infty}$-functors mapping
to it: for any non-unital $A_{\infty}$-category $\mathcal{C}$,
$\mathcal{Q}\coloneqq nu\text{-}\\!fun(\mathcal{C},\mathscr{F}(M))$ is
c-unital, and if
$E_{\mathcal{G}}\in\textup{hom}_{\mathcal{Q}}^{0}(\mathcal{G},\mathcal{G})$ is
a c-unit, then $E_{\mathcal{G}}^{0}(X)\in
CF^{0}(\mathcal{G}(X),\mathcal{G}(X))$ is the c-unit (87) for any
$X\in\textup{obj}(\mathcal{C})$ (by Proposition 2.26). Moreover, homotopic
$A_{\infty}$-functors $\mathcal{G}_{0}\simeq\mathcal{G}_{1}\in
nu\text{-}\\!fun(\mathcal{C},\mathscr{F}(M))$ are isomorphic as objects of
$H^{0}\big{(}nu\text{-}\\!fun(\mathcal{C},\mathscr{F}(M))\big{)}$ (by Lemma
2.28).
On the other hand, for any c-unital $A_{\infty}$-category $\mathcal{B}$ and
quasi-equivalence $\mathcal{G}\in fun(\mathcal{B},\mathscr{F}(M))$, there
exists an inverse quasi-equivalence $\mathcal{F}\in
fun(\mathscr{F}(M),\mathcal{B})$ so that the compositions are identity
functors in the respective zeroth cohomological categories of c-unital
$A_{\infty}$-functors (by Theorem 2.31, particularly useful if one wishes to
relate Fukaya categories of two different manifolds).

  * •

We can consider the $A_{\infty}$-category

$\mathscr{Q}(M)\\!\coloneqq\\!mod(\mathscr{F}(M))$

of c-unital $A_{\infty}$-modules over $\mathscr{F}(M)$, whose objects are
c-unital $A_{\infty}$-functors
$\mathcal{M}\\!:\\!\mathscr{F}(M)^{opp}\\!\\!\rightarrow\mathsf{Ch}$, yielding
$\\{\mathcal{M}(L)\\!=\\!\bigoplus_{n\in\mathbb{Z}}\mathcal{M}(L)^{n}\mid
L\in\textup{obj}(\mathscr{F}(M))\\}$, with composition maps

$\mu_{\mathcal{M}}^{d}:\mathcal{M}(L_{d-1})\otimes
CF(L_{d-2},L_{d-1})\otimes...\otimes
CF(L_{0},L_{1})\rightarrow\mathcal{M}(L_{0})[2-d]$

for all $d\geq 1$ and $L_{i}\in\textup{obj}(\mathscr{F}(M))$, and pre-module
homomorphisms as in (25) (cf. Definition 2.20).
Then we can define a Yoneda embedding

$\Upsilon\equiv\Upsilon_{\mathscr{F}(M)}:\mathscr{F}(M)\rightarrow\mathscr{Q}(M)\,,$

cohomologically full and faithful, given by $\Upsilon(L)(\tilde{L})\coloneqq
CF(\tilde{L},L)$ and
$\mu_{\Upsilon(L)}^{d}\coloneqq\mu^{d}:CF(L_{d-1},L)\otimes...\otimes
CF(L_{0},L_{1})\rightarrow CF(L_{0},L)$ as in (84) (by Definition 2.35). For
any $\mathcal{M}\in\textup{obj}(\mathscr{Q}(M))$ and
$L\in\textup{obj}(\mathscr{F}(M))$, the Yoneda Lemma 2.36 makes the extension
$\lambda_{\mathcal{M}}:\mathcal{M}(L)\rightarrow\textup{hom}_{\mathscr{Q}(M)}(\Upsilon(L),\mathcal{M}(L))$
of

$\displaystyle\Upsilon^{1}(p)^{d}:\;$ $\displaystyle
CF(L_{d-1},\tilde{L})\otimes CF(L_{d-2},L_{d-1})\otimes...\otimes
CF(L_{0},L_{1})\rightarrow CF(L_{0},L),$
$\displaystyle(\tilde{p},p_{d-1},...,p_{1})\mapsto\mu^{d+1}(p,\tilde{p},p_{d-1},...,p_{1})$

a quasi-isomorphism (where $p\in CF(\tilde{L},L)$).

  * •

The map $\lambda$ comes into play with Lemma 3.11, which produces the
following identifications between Floer cohomology rings:333333The same
warning we gave at the beginning of Section 3.2 applies: direct sums, tensor
products and shifts of Lagrangian submanifolds are to be understood as formal
algebraic operations between objects of $\mathscr{F}(M)$, not geometrically
meaningful — and in any case, only when corresponding quasi-representatives
exist!

$\displaystyle HF(L,L_{0}\oplus L_{1})$ $\displaystyle\cong HF(L,L_{0})\oplus
HF(L,L_{1})\;,$ $\displaystyle HF(L_{0},Z\otimes L_{1})$ $\displaystyle\cong
H(Z)\otimes HF(L_{0},L_{1})\;,$ $\displaystyle HF(L_{0},S^{\sigma}L_{1})$
$\displaystyle\cong HF(L_{0},L_{1})[\sigma]\;,$ $\displaystyle
HF(S^{\sigma}L_{0},L_{1})$ $\displaystyle\cong HF(L_{0},L_{1})[-\sigma]\;,$
$\displaystyle HF(S^{\sigma}L_{0},S^{\sigma}L_{1})$ $\displaystyle\cong
HF(L_{0},L_{1})\;,$

for all $L,L_{0},L_{1}\in\textup{obj}(\mathscr{F}(M))$ and
$Z\in\textup{obj}(\mathsf{Ch})$.
Most significantly, any cocycle $p\in CF^{0}(L_{0},L_{1})$ identifies an
abstract mapping cone
$\mathcal{C}one(p)=\mathcal{C}\in\textup{obj}(\mathscr{Q}(M))$ (as in
Definition 3.16), which assigns to any $L\in\textup{obj}(\mathscr{F}(M))$ the
cochain complex

$\mathcal{C}one(p)(L)=(\mathcal{C}(L),\mu_{\mathcal{C}}^{1})\coloneqq\Big{(}CF(L,L_{0})[1]\oplus
CF(L,L_{1}),\begin{pmatrix}\mu^{1}&0\\\
\mu^{2}(p,\cdot)&\mu^{1}\end{pmatrix}\Big{)}\;,$

and fits into the diagram (36) formed by the “abtract” inclusion and
projection, which in turn detects exact triangles (37) in the Donaldson–Fukaya
category $H(\mathscr{F}(M))$ (all those whose Yoneda image is isomorphic to
(36)).

  * •

Constructing an additive enlargement $\varSigma\mathscr{F}(M)$ for the Fukaya
category as in Definition 3.24, we can ultimately define the
$A_{\infty}$-category $Tw\mathscr{F}(M)$ of twisted complexes over
$\mathscr{F}(M)$: objects are pairs $(L,\delta_{L})$ where
$L=(I,\\{L^{i}\\},\\{V^{i}\\})\in\textup{obj}(\varSigma\mathscr{F}(M))$ and
$\delta_{L}=(\delta_{L}^{j,i})\in\textup{hom}_{\varSigma\mathscr{F}(M)}^{1}(L,L)$
is strictly lower-triangular, morphism spaces are as in equation (40), and
composition maps are given by (46).
Our usual Lagrangian submanifolds can be seen as trivial twisted complexes
thanks to the full inclusion $\mathscr{F}(M)\hookrightarrow Tw\mathscr{F}(M)$.
In particular, $Tw\mathscr{F}(M)$ is c-unital (by Lemma 3.31), and Corollary
3.32 relates it through quasi-equivalences to other $A_{\infty}$-categories of
twisted complexes.

  * •

In $Tw\mathscr{F}(M)$ the direct sum, tensor product and shift operations
listed above are valid also at the level of Floer complexes (Definition 3.33).
Similarly, any $p\in\textup{hom}_{Tw\mathscr{F}(M)}(L_{0},L_{1})$ yields a
twisted mapping cone

$Cone(p)=(C,\delta_{C})\coloneqq\Big{(}SL_{0}\oplus
L_{1},\begin{pmatrix}\delta_{L_{0}}&0\\\
p&\delta_{L_{1}}\end{pmatrix}\Big{)}\in\textup{obj}(Tw\mathscr{F}(M))$

such that $\Upsilon(Cone(p))\cong\mathcal{C}one(p)$ (cf. Definition 3.34,
which involves the $A_{\infty}$-shift functor $S:Tw\mathscr{F}(M)\rightarrow
Tw\mathscr{F}(M)$), and fitting into diagram (50) in
$H^{0}(Tw\mathscr{F}(M))$.
As discussed, the advantage of twisted complexes is that $Tw\mathscr{F}(M)$ is
a triangulated $A_{\infty}$-category (by Corollary 3.38): we can associate to
any cocycle $p\in CF^{0}(L_{0},L_{1})$ the aforementioned canonical exact
triangle

${L_{0}}$${L_{1}}$${Cone(p)}$$\scriptstyle{\langle
p\rangle}$$\scriptstyle{\langle i\rangle}$$\scriptstyle{\langle\pi\rangle[1]}$
(88)

in $H^{0}(Tw\mathscr{F}(M))$ (where we denoted the canonical projection by
$\pi$, so to avoid confusion). Proposition 3.39 shows that
$(H^{0}(Tw\mathscr{F}(M)),H^{0}(S))$ is a standard triangulated category, then
fulfilling the axioms of Section 3.1. For example, we can rotate diagram (88)
to produce the two exact triangles of (51), or form long exact sequences of
vector spaces in $H(Tw\mathscr{F}(M))$:

$\displaystyle...\rightarrow HF^{n}(L,L_{0})\rightarrow
HF^{n}(L,L_{1})\rightarrow HF^{n}(L,C)\rightarrow
HF^{n+1}(L,L_{0})\rightarrow...\;,$ $\displaystyle...\rightarrow
HF^{n}(L_{1},L)\rightarrow HF^{n}(L_{0},L)\rightarrow HF^{n}(C,L)\rightarrow
HF^{n+1}(L_{1},L)\rightarrow...\;,$

for any $L\in\textup{obj}(Tw\mathscr{F}(M))$.
Then we can use these results to show the criteria of Proposition 3.43 testing
exactness of triangles in $H(\mathscr{F}(M))$ and of Lemma 3.46 for
triangularity of $\mathscr{F}(M)$.

  * •

Moreover, $Tw\mathscr{F}(M)$ is generated by $\mathscr{F}(M)$ (by Lemma 3.45),
and vice versa $(Tw\mathscr{F}(M),\imath)$ is a triangulated envelope for
$\mathscr{F}(M)$ (by Proposition 3.50), so that we can associate to the Fukaya
category its bounded derived category $\mathsf{D^{b}}(\mathscr{F}(M))\coloneqq
H^{0}(Tw\mathscr{F}(M))$.
We can include split objects by taking the split-closure
$(\Pi\mathscr{F}(M),\\!\Upsilon_{\\!\mathscr{F}(M)})$ of $\mathscr{F}(M)$
(Proposition 4.13). For twisted complexes, $\Pi(Tw\mathscr{F}(M))$ has the
advantage of being still $A_{\infty}$-triangulated (by Lemma 4.14), so that
the split-closed derived category

$\mathsf{D}^{\pi}(\mathscr{F}(M))\coloneqq
H^{0}\big{(}\Pi(Tw\mathscr{F}(M))\big{)}$

is a standard triangulated category. Specifically, $\mathscr{F}(M)$ split-
generates$\Pi(Tw\mathscr{F}(M))$, meaning that any object of the latter can be
constructed from Lagrangian submanifolds by forming all possible mapping cones
and shifts, iterating and finally taking images of all the resulting
idempotent endomorphisms.
A useful result is that split-closed derived categories of quasi-equivalent
Fukaya categories are themselves equivalent, which can be refined to quasi-
equivalent split-generating full subcategories (this is Corollary 4.19).

  * •

About twisting: as per Definition 4.20, we can define an abstract twist
functor $\mathcal{T}_{L_{0}}:\mathscr{Q}(M)\rightarrow\mathscr{Q}(M)$ along
any $L_{0}\in\textup{obj}(\mathscr{F}(M))$, which on Yoneda modules
$\Upsilon(L_{1})\in\textup{obj}(\mathscr{Q}(M))$ produces graded vector spaces

$\mathcal{T}_{L_{0}}(\Upsilon(L_{1}))(L)\coloneqq(CF(L_{0},L_{1})\otimes
CF(L,L_{0}))[1]\oplus CF(L,L_{1}),$

for each $L\in\mathscr{F}(M)$. Any quasi-representative
$T_{L_{0}}(L_{1})\in\textup{obj}(\mathscr{F}(M))$ thereof is called twist of
$L_{1}$ along $L_{0}$, and sits into the exact triangle

${HF(L_{0},L_{1})\otimes
L_{0}}$${L_{1}}$${T_{L_{0}}(L_{1})}$$\scriptstyle{\langle\textup{ev}\rangle}$$\scriptstyle{\langle
i\rangle}$$\scriptstyle{\langle\pi\rangle[1]}$ (89)

formed by the morphism $\textup{ev}:CF(L_{0},L_{1})\otimes L_{0}\rightarrow
L_{1}$ mapping under $\Upsilon$ to the abstract evaluation $\epsilon\in
CF^{0}(CF(L_{0},L_{1})\otimes\Upsilon(L_{0}),\Upsilon(L_{1}))$ of Definition
4.21. If we work with twisted complexes, then we obtain
$T_{L_{0}}(L_{1})=S(HF(L_{0},L_{1})\otimes L_{0})\oplus
L_{1}=Cone(\textup{ev})\in\textup{obj}(Tw\mathscr{F}(M))$, allowing us to
legitimately define a twist functor $T_{L_{0}}:Tw\mathscr{F}(M)\rightarrow
Tw\mathscr{F}(M)$.

Roughly speaking, $\mathsf{D}^{\pi}(\mathscr{F}(M))$ maximizes the load of
nice “algebraic features” we can ever hope to extract from $\mathscr{F}(M)$,
whereas taking cohomology immediately, thus dealing with the Donaldson–Fukaya
category, would for example result in the loss of the triangulation
properties.

But why do we care so much about exact triangles? And what is even the point
of studying Fukaya categories if all we can infer about them is just a matter
of notational reformulation of Part I? Well, it turns out that the notion of
mapping cones is geometrically meaningful in this particular context. We
discuss this in the coming section.

####  6.4 Geometric features of Fukaya categories

We begin with a brief aside regarding the geometrical relevance of Hochschild
cohomology (discussed in [AS09, section 3]).

######  Remark 6.18.

Consider a Hamiltonian function $H\in C^{\infty}(\mathbb{R}/\mathbb{Z}\times
M)$ and the Hamiltonian Floer complex it defines,

$CF(H)\coloneqq\mkern-12.0mu\bigoplus_{\gamma\in\mathcal{P}_{1}^{\circ}(H)}\mkern-12.0mu\mathbb{Z}_{2}\cdot\langle\gamma\rangle\;,$

where $\mathcal{P}_{1}^{\circ}(H)$ is the set of contractible 1-periodic
orbits of the associated Hamiltonian vector field
$X_{H}\in\mathfrak{X}(\mathbb{R}/\mathbb{Z}\times M)$ (see for example [Mer14,
chapter 2]). Pick one such generator
$\gamma_{x}\coloneqq\phi_{H}^{1}(x)\in\mathcal{P}_{1}^{\circ}(H)$ ($x\in M$
must be a fixed point of the flow $\phi_{H}^{t}$ of $H$). Then we can define a
map

$\Psi:CF(H)\rightarrow CC(\mathscr{F}(M))$

with target the Hochschild complex of Definition 2.14 which sends
$\langle\gamma_{x}\rangle$ to a sequence of (multi)linear graded maps
$\Psi(\langle\gamma_{x}\rangle)=(h^{0},h^{1},...)$ whose $d$-th term $h^{d}$
counts pseudoholomorphic disks with an interior puncture converging to
$\gamma_{x}$ and having $d$-many boundary punctures (the inputs of $h^{d}$, as
usual).

The map $\Psi$ is actually a chain map, then inducing a homomorphism $H(\Psi)$
from the quantum cohomology $QH(M)\coloneqq H(M)\otimes\Lambda_{\mathbb{K}}$
of $M$ to the Hochschild cohomology $HH(\mathscr{F}(M))$ of its Fukaya
category. On the other hand, the assignment $\Xi:CC(\mathscr{F}(M))\rightarrow
CF(L,L),\,(h^{d})_{d\geq 0}\mapsto h_{L}^{0}$ is itself a chain map, thus
inducing a homomorphism $H(\Xi):HH(\mathscr{F}(M))\rightarrow HF(L,L)$, for
any $L\in\mathscr{F}(M)$. In particular, as $L$ bounds no pseudoholomorphic
disk (by definition of objects of $\mathscr{F}(M)$), the composition

$H(\Xi)\circ H(\Psi):QH(M)\rightarrow HH(\mathscr{F}(M))\rightarrow HF(L,L)$

is just the restriction on cohomology ([AS09, Lemma 3.11]).

Let us discuss something of more significance to our journey: Dehn twists. The
relevant theory can be found in [Sei02, chapter 1].

######  Definition 6.19.

Consider the cotangent bundle $T\coloneqq
T^{*}\mathbb{S}^{n}\cong\mathbb{S}^{n}\times\mathbb{R}^{n+1}$ equipped with
standard symplectic form $\omega_{T}\in\Omega^{2}(T)$ which, in global
coordinates $(q,p)\in\mathbb{S}^{n}\times\mathbb{R}^{n+1}$ such that $\langle
q,p\rangle_{\text{std}}=0$, reads $\omega_{T}=\sum_{i=1}^{n}dp_{i}\wedge
dq_{i}$. Then for each $\lambda\geq 0$ we define the subbundle

$T(\lambda)\coloneqq\\{(q,p)\in T\mid|\\!|p|\\!|\leq\lambda\\}\subset T\,,$

a tubular neighbourhood of the zero section $\mathbb{S}^{n}\equiv T(0)\subset
T$, and the fibre-length functional
$\varrho:T\rightarrow\mathbb{R},\,\varrho(q,p)\coloneqq|\\!|p|\\!|$, which
generates the normalized geodesic flow on $\mathbb{S}^{n}$:

$\Theta:\;(t,(q,p))\in(0,\pi)\times
T\longmapsto\Big{(}\cos(t)p-\sin(t)|\\!|p|\\!|q,\,\cos(t)q+\sin(t)\frac{p}{|\\!|p|\\!|}\Big{)}\in
T\;.$

(recall that $T\cong T\mathbb{S}^{n}$). Note $\Theta_{\pi}$ would be the
antipodal map $a(q,p)\coloneqq(-q,-p)$, which continuously extends over
$T(0)$.

The following is the key technical step, proved in [Sei02, Lemma 1.8].

######  Lemma 6.20.

Let $h\in C^{\infty}(\mathbb{R})$ be a smooth function such that $h(t)=0$ if
$t\gg 0$, and $h(-t)=h(t)-kt$ for $|t|$ small and some $k\in\mathbb{Z}$.
Define the Hamiltonian function $H\coloneqq h\circ\varrho|_{T\setminus
T(0)}\in C^{\infty}(T\setminus T(0))$. Then the associated Hamiltonin
diffeomorphism $\phi_{H}^{2\pi}:T\setminus T(0)\rightarrow T\setminus T(0)$
extends smoothly over $T(0)$ to a compactly supported
$\phi\in\textup{Symp}(T,\omega_{T})$.

Moreover, if $\textup{supp}(h)\subset(-\infty,\lambda)$ for some $\lambda>0$,
then $\phi|_{T(\lambda)}\\!\in\\!\textup{Symp}(T(\lambda),\omega_{T})$ and is
the identity near the boundary.

######  Definition 6.21.

If $k=1$ and $\text{supp}(h)\subset(-\infty,\lambda)$ in the last lemma, then
we call the resulting symplectomorphism
$\tau_{\,\mathbb{S}^{n}}\coloneqq\phi\in\text{Symp}(T,\omega_{T})$ a (model)
Dehn twist. Explicitly:

$\tau_{\,\mathbb{S}^{n}}(q,p)\coloneqq\begin{cases}\Theta_{2\pi
h^{\prime}(|\\!|p|\\!|)}(q,p)&$if $(q,p)\in T(\lambda)\setminus T(0)\\\
a(q,p)&$if $(q,p)\in T(0)\end{cases}\;.$ (90)

An easy manipulation, along with the assumptions on $h$, shows that the angle
of rotation $2\pi h^{\prime}(|\\!|p|\\!|)$ continuously decreases from $2\pi
h^{\prime}(0)=\pi$, where $\Theta_{\pi}\equiv a$ is the antipodal map sending
$\mathbb{S}^{n}$ onto itself, to $2\pi h^{\prime}(\lambda)=0$, where
$\Theta_{0}\equiv\textup{id}$. Any other Dehn twist in $T(\lambda)$ is
symplectically isotopic to (90).

Figure 10: The Dehn twist $\tau_{S}$ along the sphere
$S\coloneqq\mathbb{S}^{n}$ at work: the chosen fibre $L_{1}$ spirals around
$S$ once and is asymptotically preserved. [Source:[Aur13]]

The action of the Dehn twist $\tau_{\,\mathbb{S}^{n}}$ on a fibre of
$T(\lambda)$ is depicted in Figure 10. It is sometimes convenient to control
the “wobbliness” of $\tau_{\,\mathbb{S}^{n}}$ by imposing that
$h^{\prime}|_{\mathbb{R}_{\geq 0}}\geq 0$ and $h^{\prime\prime}<0$ whenever
the “twisting speed” $h^{\prime}$ exceeds some tolerance $0<\varepsilon<1/2$.
Then one can deduce where the twisted fibre will intersect (transversely)
another non-twisted one; this will happen exactly at the zero section,
precisely as shown in Figure 10, if the two fibers have antipodal footpoints
(see [Sei02, Lemma 1.9]).

We transport the theory to the manifold world.

######  Definition 6.22.

Let $M^{2n}=(M,\omega)$ be a symplectic manifold. A compact Lagrangian
submanifold $S^{n}\subset M$ is called a Lagrangian sphere if there exists a
diffeomorphism $f:\mathbb{S}^{n}\rightarrow S$. It is framed if there is a
class $[f]$ of such diffeomorphisms, where $f_{1}\sim f_{2}$ if and only if
$f_{2}^{-1}\circ f_{1}$ can be continuously deformed inside
$\text{Diff}(\mathbb{S}^{n})$ to some orthogonal matrix of $\text{O}(n+1)$.

We can associate a Dehn twist $\tau_{S}\in\text{Symp}(M,\omega)$ to any framed
Lagrangian sphere $S=(S,[f])$. Given some $\lambda>0$, choose any
representative map$f:\mathbb{S}^{n}\equiv T(0)\rightarrow S$ and extend it to
$T(\lambda)$, which embeds symplectically into $M$ via
$\iota:T(\lambda)\hookrightarrow M$, and construct the Dehn twist (90)
supported inside $T(\lambda)$. The Dehn twist along $S$ is343434The reader
should not be fooled by the notation adopted: it is currently unknown whether
$[\tau_{S}]\in\pi_{0}(\text{Symp}(M,\omega))$ depends on the specific class
$[f]$ or not, although all other choices made in the construction are in fact
irrelevant. For practicity, we avoid mention of the specific framing $[f]$,
and implicitly consider all Lagrangian spheres to be framed. then:

$\tau_{S}(x)\coloneqq\begin{cases}\iota\circ\tau_{\,\mathbb{S}^{n}}\circ\iota^{-1}(x)&$if
$x\in\text{im}(\iota)\\\ x&$else $\end{cases}\;.$ (91)

(Concretely, $\tau_{S}$ is the unique possible extension of
$\phi_{H}^{2\pi}:M\setminus S\rightarrow M\setminus S$ over $S$, where
$H\coloneqq h\circ\varrho\circ\iota^{-1}|_{\text{im}(\iota)\setminus S}$ is
zero on $M\setminus\text{im}(\iota)$.) In particular, $\tau_{S}$ maps $S$ to
itself antipodally.

The last definition made implicit use of the Weinstein’s Lagrangian
Neighbourhood Theorem ([Sil12, Theorem 8.4]): any neighbourhood of $S\subset
M$ is symplectomorphic to a neighbourhood $T(\lambda)$ of
$\mathbb{S}^{n}\equiv T(0)\subset T^{*}\mathbb{S}^{n}$ (we exploited this also
back in Proposition 5.43).

Being a symplectomorphism, $\tau_{S}$ maps Lagrangian submanifolds to
Lagrangian submanifolds, preserving all their decorations (non-trivial!). In
other words, $L\in\textup{obj}(\mathscr{F}(M))$ implies
$\tau_{S}(L)\in\textup{obj}(\mathscr{F}(M))$. The main result we are
interested in is:

######  Theorem 6.23.

Let $M$ be a symplectic manifold with associated Fukaya category
$\mathscr{F}(M)$. Let $S\in\textup{obj}(\mathscr{F}(M))$ be a Lagrangian
sphere and $L\in\textup{obj}(\mathscr{F}(M))$ any Lagrangian submanifold. Then

$\tau_{S}(L)\cong
T_{S}(L)=Cone(\textup{ev})\in\textup{obj}\big{(}\mathsf{D^{b}}(\mathscr{F}(M))\big{)}\,,$
(92)

where $T_{S}(L)$ is the algebraic twist of $L$ along $S$, coinciding with the
mapping cone of $\textup{ev}:CF(S,L)\otimes S\rightarrow L$ as in Definition
4.21. Therefore, there is an exact triangle

${HF(S,L)\otimes
S}$${L}$${\tau_{S}(L)}$$\scriptstyle{\langle\textup{ev}\rangle}$$\scriptstyle{\langle
c\rangle}$$\scriptstyle{\langle\pi\rangle[1]}$ (93)

in $\mathsf{D^{b}}(\mathscr{F}(M))$. Consequently, application of
$HF(\tilde{L},\square)$ respectively $HF(\square,\tilde{L})$ yields long exact
sequences

$\displaystyle...\rightarrow(HF(S,L)\otimes
HF(\tilde{L},S))^{n}\xrightarrow{\langle\mu^{2}\rangle}HF^{n}(\tilde{L},L)\rightarrow
HF^{n}(\tilde{L},\tau_{S}(L))\xrightarrow{[1]}...$
$\displaystyle...\rightarrow HF^{n}(L,\tilde{L})\rightarrow(HF(S,L)\otimes
HF(S,\tilde{L}))^{n}\xrightarrow{\langle
d\rangle[1]}HF^{n+1}(\tau_{S}(L),\tilde{L})\rightarrow...$

for any $\tilde{L}\in\textup{obj}(\mathscr{F}(M))$, where $\mu^{2}$ is the
Floer product and $\langle d\rangle$ is a composition involving the canonical
isomorphism $HF(L,S)\cong HF(\tau_{S}(L),\tau_{S}(S))=HF(\tau_{S}(L),S)$ and
the Donaldson “pair-of-pants” product.

###### Proof.

(Sketch) We just give a very vague outline of how to proceed — a precise proof
is the scope of the whole paper [Sei02] by Seidel. Goal is to work first at
the $A_{\infty}$-module level in order to apply Lemma 4.23. Relabeling
$L_{0}\coloneqq S$, $L_{1}\coloneqq L$ for consistency and choosing
$L_{2}\in\textup{obj}(\mathscr{F}(M))$ to be isotopic to
$\tau_{L_{0}}(L_{1})$, one defines a Floer cocycle $c\in CF(L_{1},L_{2})$
(thus $\mu^{1}(c)=0$) and a chain homotopy $k:CF(L_{0},L_{1})\rightarrow
CF(L_{0},L_{2})[-1]$ between $\mu^{2}(c,\cdot):CF(L_{0},L_{1})\rightarrow
CF(L_{0},L_{2})$ and the zero map: $\mu^{1}\circ
k+k\circ\mu^{1}+\mu^{2}(c,\cdot)=0$.

After producing the suitable additional data required by [Sei08, Lemma 5.3]
(this is done throughout sections (17d)–(17f) of [Sei08]), Lemma 4.23 yields a
well-defined module isomorphism
$t:\mathcal{T}_{L_{0}}(\Upsilon(L_{1}))\rightarrow\Upsilon(L_{2})$. One can
improve this result to take into account signs, gradings and brane structures
[Sei08, Theorem 17.16]. Therefore,
$\mathcal{T}_{L_{0}}(\Upsilon(L_{1}))\cong\Upsilon(L_{2})\in\textup{obj}\big{(}H^{0}(\mathscr{Q}(M))\big{)}$.

Now, we move to twisted complexes by applying the isomorphisms
$\mathcal{T}_{Y_{0}}(\tilde{\Upsilon}(Y_{1}))$
$\cong\tilde{\Upsilon}(T_{Y_{0}}(Y_{1}))$ of the second bullet point in Remark
4.22. Letting without loss of generality
$L_{2}=\tau_{L_{0}}(L_{1})\in\textup{obj}(Tw\mathscr{F}(M))$, we obtain:

$\tilde{\Upsilon}(T_{S}(L))=\tilde{\Upsilon}(T_{L_{0}}(L_{1}))\cong\mathcal{T}_{L_{0}}(\tilde{\Upsilon}(L_{1}))\cong\tilde{\Upsilon}(\tau_{L_{0}}(L_{1}))=\tilde{\Upsilon}(\tau_{S}(L))\,,$

whence
$T_{S}(L)\cong\tau_{S}(L)\in\textup{obj}\big{(}\mathsf{D^{b}}(\mathscr{F}(M))\big{)}$
(recall that $\tilde{\Upsilon}$ is a quasi-equivalence).

It follows immediately from diagram (89) that $\tau_{S}(L)$ fits into the
exact triangle (93), where we recognize $c:L\rightarrow\tau_{S}(L)$ to be
precisely the Floer cocycle constructed above, while
$\langle\pi\rangle\in\textup{Hom}_{\mathsf{D^{b}}(\mathscr{F}(M))}\big{(}\tau_{S}(L),(HF(S,L)\otimes
S)[1]\big{)}\cong\big{(}HF(S,L)\otimes
HF(\tau_{S}(L),S)\big{)}^{1}\cong\big{(}HF(S,L)\otimes HF(L,S)\big{)}^{n}$.
Moreover, one easily checks that indeed
$HF(\tilde{L},\langle\textup{ev}\rangle)=\langle\mu^{2}\rangle$. For the exact
definition of $d$, we point again the reader to [Sei02]. ∎

Recalling that twisted mapping cones quasi-represent their abstract
counterpart through the Yoneda embedding, from the proof of Theorem 6.23 we
also deduce:

######  Corollary 6.24.

Let $M$ be a symplectic manifold with associated Fukaya category
$\mathscr{F}(M)$. Let $S\in\textup{obj}(\mathscr{F}(M))$ be a Lagrangian
sphere and $L\in\textup{obj}(\mathscr{F}(M))$ any Lagrangian submanifold. Then
the Dehn twist
$\tau_{S}(L)\in\textup{obj}\big{(}\mathsf{D^{b}}(\mathscr{F}(M))\big{)}$ of
$L$ along $S$ quasi-represents the abstract algebraic twist
$\mathcal{T}_{S}(\Upsilon(L))\in\textup{obj}(\mathscr{Q}(M))$ of the Yoneda
module $\Upsilon(L)\in\textup{obj}(\mathscr{Q}(M))$ along $S$.

As the reader may have suspected by now, the underlying theory is way deeper
than what we touched upon — it revolves around Lefschetz fibrations, which are
among the fundamental objects of interest of Picard–Lefschetz theory.
Unfortunately, a thorough study falls out of our reach; we refer the hungry
reader to the third part of [Sei08].

A second geometrically meaningful aspect of Fukaya categories is the so-called
Lagrangian surgery, whose construction is outlined in [FOOO07, section 10.54].
First, another piece of technicality.

######  Definition 6.25.

For $\varepsilon\in\mathbb{R}$ with $|\varepsilon|>0$ small enough, consider
the smooth function
$f_{\varepsilon}:\mathbb{R}^{n}\setminus\\{0\\}\rightarrow\mathbb{R},\,f_{\varepsilon}(x)\coloneqq\varepsilon\log(|\\!|x|\\!|)$
and the 1-form it defines:

$df_{\varepsilon}(x)=\varepsilon\frac{x\cdot
dx}{|\\!|x|\\!|^{2}}=\sum_{j=1}^{n}y_{j}dx_{j}\in
T_{x}^{*}(\mathbb{R}^{n}\setminus\\{0\\})\,,\quad\text{with
}y_{j}\coloneqq\varepsilon\frac{x_{j}}{|\\!|x|\\!|^{2}}\in\mathbb{R}\;.$

Then
$H_{\varepsilon}\coloneqq\text{graph}(df_{\varepsilon})=\\{(x_{1},y_{1},...,x_{n},y_{n})\mid
x\in\mathbb{R}^{n}\setminus\\{0\\}\\}\subset
T^{*}(\mathbb{R}^{n}\setminus\\{0\\})\cong\mathbb{C}^{n}\setminus(i\mathbb{R})^{n}$
is a Lagrangian submanifold which tends asymptotically to $(i\mathbb{R})^{n}$
(the cotangent fibre over $0\in\mathbb{R}^{n}$) as $|\\!|x|\\!|\rightarrow 0$,
and to the zero section $\mathbb{R}^{n}$ as $|\\!|x|\\!|\rightarrow\infty$,
where the identification with the complex space follows by setting
$z_{j}\coloneqq x_{j}+iy_{j}\in\mathbb{C}$.

Observe that, denoting by $r:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$ the
reflection along the diagonal $(x_{j}+iy_{j}\mapsto y_{j}+ix_{j})$ for all
$j=1,...,n$, it holds $r(H_{\varepsilon})=H_{\varepsilon}$ (indeed, symmetry
of the construction implies that $H_{\varepsilon}$ is also the mirrored graph
consisting of all $(y_{1},x_{1},...,y_{n},x_{n})\in
T^{*}(\mathbb{R}^{n}\setminus\\{0\\})$ such that
$y\in\mathbb{R}^{n}\setminus\\{0\\}$ and $x_{j}=\varepsilon
y_{j}/|\\!|y|\\!|^{2}$).

Now take a corrective smooth function $g\in C^{\infty}(\mathbb{R}_{>0})$ such
that $g(x)=\log(x)-|\varepsilon|$ if $0<x\leq x_{0}$, where $x_{0}\equiv
x_{0}(|\varepsilon|)$ is small enough, and is constant if $x\geq 2x_{0}$, with
$g^{\prime}\geq 0$ and $g^{\prime\prime}\leq 0$ (compare this with the map $h$
of Lemma 6.20). We define the adjusted smooth function

$\tilde{f}_{\varepsilon}:\mathbb{R}^{n}\setminus\\{0\\}\rightarrow\mathbb{R},\,\tilde{f}_{\varepsilon}(x)\coloneqq\varepsilon
g(|\\!|x|\\!|)=\begin{cases}f_{\varepsilon}(x)-\varepsilon|\varepsilon|&\text{if
}0<x\leq x_{0}\\\ \text{constant}&\text{else}\end{cases}\;,$

and look at its graph
$\tilde{H}_{\varepsilon}\coloneqq\text{graph}(d\tilde{f}_{\varepsilon})\subset
T^{*}(\mathbb{R}^{n}\setminus\\{0\\})$, again a Lagrangian submanifold
invariant under reflection by $r$. In particular,
$\tilde{H}_{\varepsilon}=\mathbb{R}^{n}\cup(i\mathbb{R})^{n}$ outside the
$n$-dimensional complex ball $B_{2x_{0}}^{n}(0)\subset\mathbb{C}^{n}$ at 0 of
radius $2x_{0}$.

######  Definition 6.26.

Let $M^{2n}=(M,\omega)$ be a symplectic manifold with associated Fukaya
category $\mathscr{F}(M)$. Let $L_{1},L_{2}\in\textup{obj}(\mathscr{F}(M))$ be
transversely intersecting at some $p\in M$. With a suitable Darboux
chart353535This is a chart of local coordinates $(x^{j},y^{j})_{j=1}^{n}$ such
that $\omega|_{U}=\sum_{j=1}^{n}dx^{j}\wedge dy^{j}$, always existing by
[Sil12, Theorem 8.1]. $\psi:U\subset M\rightarrow
V\subset\mathbb{R}^{2n}\cong\mathbb{C}^{n}$ on a neighbourhood $U$ of $p$,
centered at $p$ (so $\psi(p)=0$), we can arrange the two Lagrangian
submanifolds in such a way that $\psi(L_{1}\cap U)=\mathbb{R}^{n}\cap V$ and
$\psi(L_{2}\cap U)=(i\mathbb{R})^{n}\cap V$.

Now construct $\tilde{H}_{\varepsilon}$ as in Definition 6.25, adapting
$\varepsilon$ so that $B_{2x_{0}}^{n}(0)\subset\psi(U)$. Then

$L_{1}\\#_{\varepsilon}L_{2}\coloneqq\begin{cases}(L_{1}\cup L_{2})\setminus
U&\text{on }M\setminus U\\\ \psi^{-1}(\tilde{H}_{\varepsilon})\cap U&\text{on
}U\end{cases}\;,$ (94)

is the Lagrangian connected sum (or Lagrangian surgery) of $L_{1}$ and $L_{2}$
at $p$ (sometimes just shortened $L_{1}\\#L_{2}$). It is a Lagrangian
submanifold of $M$ inheriting all the desired decorations (non-trivial!),
meaning that $L_{1}\\#_{\varepsilon}L_{2}\in\textup{obj}(\mathscr{F}(M))$. The
region $(L_{1}\\#_{\varepsilon}L_{2})\cap
U=\psi^{-1}(\tilde{H}_{\varepsilon})\cap U$ is called Lagrangian handle.

The 2-dimensional case is shown in Figure 11, while the 4-dimensional is
discussed for example in [AS09, section 3].

From the above constructions, it follows that
$L_{1}\\#_{\varepsilon}L_{2}=L_{2}\\#_{-\varepsilon}L_{1}$ (not
$L_{2}\\#_{\varepsilon}L_{1}$, which is not even isotopic to
$L_{1}\\#_{\varepsilon}L_{2}$!). Moreover, $L_{1}\\#_{\varepsilon}L_{2}$ is
Hamiltonian isotopic to some $L_{1}\\#_{\delta}L_{2}$ as soon as
$\text{sgn}(\varepsilon)=\text{sgn}(\delta)$ (this can be seen by working in a
suitable tubular neighbourhood of the Lagrangian handle). However, if we
perform surgeries at multiple distinct points, the end result loses isotopy
invariance.

Figure 11: The Lagrangian connected sum $L_{1}\\#L_{2}$ of two 1-dimensional
Lagrangian submanifolds $L_{1},L_{2}\subset M^{2}$ transversely intersecting
at $p\in M$, which are mapped by a suitable Darboux chart to the real
respectively imaginary axis in a neighbourhood of 0 in $\mathbb{C}$. Here the
“curved region” of $L_{1}\\#L_{2}$ is what we call Lagrangian handle.
[Source:[Aur13]]

######  Remark 6.27.

Take $L_{2}=S\in\textup{obj}(\mathscr{F}(M))$ to be a Lagrangian sphere
transversely intersecting $L_{1}$ at a single point $p$. Then $L_{1}\\#S$ is
Hamiltonian isotopic to the Dehn twist $\tau_{S}(L_{1})$,

$L_{1}\\#S\simeq\tau_{S}(L_{1})$ (95)

(compare to Figure 10). Therefore, $S$ regulates the twisting, while $L_{1}$
governs the asymptotical behaviour. In the 2-dimensional case, the local
Darboux picture of $L_{1}$ corresponds to the imaginary line $i\mathbb{R}$ and
that of $S$ to $\mathbb{S}^{1}$ centered at $1\in\mathbb{C}$. Then the
Lagrangian surgery amounts to smoothing the angular regions squeezed between
$\mathbb{S}^{1}$ and $i\mathbb{R}$ about 0. This can be taken informally as
the general recipe to produce Lagrangian connected sums.

Similarly to Dehn twists, Lagrangian connected sums are closely related to
twisted mapping cones:

######  Theorem 6.28.

Let $L_{1},L_{2}\in\textup{obj}(\mathscr{F}(M))$ intersect transversely at a
single $p\in M$, thus forcedly363636This is immediate by definition of Floer’s
complex and differential: $\mu^{1}$ is of degree 1, and therefore cannot
preserve the only generator $p\in\mathcal{X}(L_{2},L_{1})$. a Floer cocycle of
$CF(L_{2},L_{1})$. Consider a third $\tilde{L}\in\textup{obj}(\mathscr{F}(M))$
with intersections $\tilde{L}\cap(L_{1}\\#_{\varepsilon}L_{2})=(\tilde{L}\cap
L_{1})\cup(\tilde{L}\cap L_{2})$ (possible for $|\varepsilon|$ small enough;
cf. Figure 11). Write $p_{i}$ for the transverse intersection point in
$\tilde{L}\cap L_{i}$. Then for a suitable almost complex structure
$J\in\mathfrak{J}(M,\omega)$ on the region bounded by $\tilde{L}$ and
$L_{1}\\#_{\varepsilon}L_{2}$, we have the bijective correspondence

$\left\\{\begin{aligned} &J\text{-holomorphic strips from}\\\ &\text{$p_{2}$
to $p_{1}$ with boundary}\\\ &\text{on $\tilde{L}$ and
$L_{1}\\#_{\varepsilon}L_{2}$}\end{aligned}\right\\}\longleftrightarrow\left\\{\begin{aligned}
&J\text{-holomorphic triangles from}\\\ &\text{$p_{2}$ to $p_{1}$ to $p$ with
boundary}\\\ &\text{on $\tilde{L}$, $L_{1}$ and
$L_{2}$}\end{aligned}\right\\}\;.$

As a consequence, $CF(\tilde{L},L_{1}\\#_{\varepsilon}L_{2})$ is the standard
mapping cone of $\mu^{2}(p,\cdot):CF(\tilde{L},L_{2})\rightarrow
CF(\tilde{L},L_{1})$ (in the chain complex sense), thus fitting into the short
exact sequence

$\\{0\\}\rightarrow CF(\tilde{L},L_{1})\rightarrow
CF(\tilde{L},L_{1}\\#_{\varepsilon}L_{2})\rightarrow
CF(\tilde{L},L_{2})\rightarrow\\{0\\}$

inducing the long exact sequence

$...\rightarrow HF^{n}(\tilde{L},L_{1})\rightarrow
HF^{n}(\tilde{L},L_{1}\\#_{\varepsilon}L_{2})\rightarrow
HF^{n}(\tilde{L},L_{2})\xrightarrow{\langle\mu^{2}(p,\cdot)\rangle[1]}...\quad.$
(96)

In fact, there is an exact triangle

${L_{2}}$${L_{1}}$${L_{1}\\#_{\varepsilon}L_{2}}$$\scriptstyle{\langle
p\rangle}$$\scriptstyle{\langle i\rangle}$$\scriptstyle{\langle\pi\rangle[1]}$
(97)

in $\mathsf{D^{b}}(\mathscr{F}(M))$, meaning that

$L_{1}\\#_{\varepsilon}L_{2}\cong
Cone(p)\in\textup{obj}\big{(}\mathsf{D^{b}}(\mathscr{F}(M))\big{)}\,.$ (98)

A proof of this result is developed throughout [FOOO07, sections 10.59–10.62]
(where it is referred to as “Theorem Z”). In order to study the moduli space
$\mathcal{M}(p_{2},p_{1};[u],J,H)$ associated to $J$-holomorphic strips
bounded by $L_{1}\\#_{\varepsilon}L_{2}$ and $\tilde{L}$, this proof heavily
relies on the analytical tools we occasionally mentioned in Chapter 5, as well
as gluing techniques, exploiting the local construction in the complex space
$\mathbb{C}^{n}$.

######  Remark 6.29.

Take again $L_{2}=S\in\textup{obj}(\mathscr{F}(M))$ to be a Lagrangian sphere
so that $L_{1}\\#S\simeq\tau_{S}(L_{1})$. The claims of Theorem 6.28 are then
compatible with those of Theorem 6.23. Indeed, the long exact sequence induced
by (93) is just (96) after fixing the input $\langle p\rangle\in HF(S,L_{1})$
for the connecting homomorphism $\langle\mu^{2}\rangle:HF(S,L_{1})\otimes
HF(\tilde{L},S)\rightarrow HF(\tilde{L},L_{1})$ (and renaming $L\coloneqq
L_{1}$). Similarly, diagram (93) takes the shape of (97).

To conclude, we warn that the identifiability of Dehn twists and Lagrangian
connected sums with twisted mapping cones is two-edged: the advantage of such
a dictionary between algebraic and geometrical objects is partially
compensated by the necessity to deal with higher order Floer composition maps
$\mu^{d}$ — think about the local picture of Figure 11 when several surgeries
are performed. Consequently, studying the Lagrangian constituents unavoidably
requires us to enhance the analytical tools at disposal. This is beyond our
scope; we venture no further.

Finally, some heuristics about the importance of the split-closure
$\mathsf{D}^{\pi}(\mathscr{F}(M))$. Given finitely many Lagrangian
submanifolds $L_{i}\in\textup{obj}(\mathscr{F}(M))$ which generate an
$A_{\infty}$-subcategory $\mathcal{A}\subset\mathscr{F}(M)$, the
$A_{\infty}$-category $Tw\mathcal{A}$ contains all possible mapping cones
which can be iteratively constructed from them, in particular Dehn twists and
Lagrangian connected sums. Surgery of all starting objects may produce a
Lagrangian submanifold which is disconnected. Though an element of
$\mathcal{A}$, its components may no longer be as much. Therefore, it is
conjectured that taking the idempotent completion of $Tw\mathcal{A}$ could
include the individual constituents as objects of
$\mathsf{D}^{\pi}(\mathcal{A})$, solving this deficiency. This is particularly
true in the case of the 2-torus:

######  Example 6.30.

Consider the symplectic 2-torus
$\mathbb{T}^{2}=\mathbb{S}^{1}\times\mathbb{S}^{1}$ equipped with the standard
area form $\omega\coloneqq d\theta_{1}\wedge d\theta_{2}$, where $\theta_{1}$
and $\theta_{2}$ are the angle coordinates373737Observe that
$\theta_{j}(e^{2\pi it})\coloneqq t\in(0,1)$ is well defined only on
$\mathbb{S}^{1}\setminus\\{1\\}$, not globally, so that $d\theta_{j}$ does not
denote the 1-form obtained as exterior derivative of $\theta_{j}$! on the
respective factors $\mathbb{S}^{1}\subset\mathbb{C}$. Objects of
$\mathscr{F}(\mathbb{T}^{2})$, our decorated Lagrangian submanifolds, are all
simple loops (that is, non-self-intersecting closed paths) on
$\mathbb{T}^{2}$, identifiable as class representatives of
$\mathbb{Z}\oplus\mathbb{Z}=\pi_{1}(\mathbb{T}^{2})$ and determined (up to
homotopy) by their rational slope in the quotient space representation of
$\mathbb{T}^{2}$.

Fix a preferred longitude
$L_{1}\coloneqq\alpha\in\textup{obj}(\mathscr{F}(\mathbb{T}^{2}))$ and
meridian $L_{2}\coloneqq\beta\in\textup{obj}(\mathscr{F}(\mathbb{T}^{2}))$
along the two factors of the identification polygon of $\mathbb{T}^{2}$
(Figure 12 top left); these belong to the classes $(1,0)$ respectively $(0,1)$
of $\pi_{1}(\mathbb{T}^{2})$. Call $p\in CF(L_{2},L_{1})$ the unique
intersection point of $\alpha$ and $\beta$. Since $\beta$ is clearly
diffeomorphic to $\mathbb{S}^{1}$, it is a Lagrangian sphere, so that we can
construct the Dehn twist of $\alpha$ along it. According to Remark 6.27 and
Theorem 6.28, this yields a new simple loop
$\tau_{\beta}(\alpha)\simeq\alpha\\#\beta\cong
Cone(p)\in\textup{obj}\big{(}\mathsf{D^{b}}(\mathscr{F}(\mathbb{T}^{2}))\big{)}$,
depicted in Figure 12 top right. The Dehn twisting procedure can be iterated
indefinitely (this amounts to repeatedly take algebraic twists as discussed in
Section 4.4), applying also $\tau_{\alpha}$.

Figure 12: The Lagrangian connected sum of two simple loops possibly yields a
direct sum of unbalanced simple loops (bottom row). These are genuine objects
of the split-closed derived category
$\mathsf{D}^{\pi}(\mathscr{F}(\mathbb{T}^{2}))$. [Source: [Aur13]
(identification polygons); special thanks to S. Bone for creating these
gorgeous tori!]

Since we will come across all homotopy classes of $\pi_{1}(\mathbb{T}^{2})$ —
indeed, $\tau_{\alpha}$, $\tau_{\beta}\in\text{Diff}(\mathbb{T}^{2})$ generate
the mapping class group of $\mathbb{T}^{2}$, consisting of ambient isotopy
classes of its diffeomorphisms — one would legitimately expect to obtain all
objects of $\mathscr{F}(\mathbb{T}^{2})$; however, this is far from true: let
$\nu\in\Omega^{1}(\mathbb{T}^{2}\setminus\\{p\\})$ be such that $d\nu=\omega$
(this is possible on the punctured torus!) and
$\int_{\alpha}\nu=\int_{\beta}\nu=0$, which means
$\alpha,\beta\subset\mathbb{T}^{2}$ are balanced (or exact) Lagrangian
submanifolds with $\nu|_{\alpha}=d\theta_{\alpha}$ and
$\nu|_{\beta}=d\theta_{\beta}$ (so that indeed
$\int_{\alpha}\nu=\int_{\alpha}d\theta_{\alpha}=\theta_{\alpha}(e^{2\pi
i})-\theta_{\alpha}(e^{0})=0$, and similarly on $\beta$). Then also all the
successive mapping cones constructed above will be exact: looking at
$\tau_{\beta}(\alpha)$ in Figure 12 top left, we perceive that its bijectivity
on $[0,1]\times[0,1]$ causes the Dehn twisted Lagrangian submanifold it
describes to be exact — put differently, if we follow along the corresponding
simple loop, we see it revolves around each factor exactly once (indeed,
$\tau_{\beta}(\alpha)$ represents $(1,1)\in\pi_{1}(\mathbb{T}^{2})$). This is
manifestly a constraint we need to remove, as not all simple loops are exact!

For example, consider a third simple loop
$L_{3}\coloneqq\gamma\in\textup{obj}(\mathscr{F}(\mathbb{T}^{2}))$ of rational
slope $-1/2$ (Figure 12 bottom left), representing the class
$(2,1)\in\pi_{1}(\mathbb{T}^{2})$ (that of
$\tau_{\alpha}(\tau_{\alpha}(\beta))$), which intersects $\beta$ twice, say at
$q_{1},q_{2}\in CF(L_{2},L_{3})$. Constructing the (double) connected sum of
$\gamma$ with $\beta$ and carefully applying Theorem 6.28, it turns out that
$\tau_{\beta}(\gamma)\simeq\gamma\\#\beta\cong
Cone(T^{w_{1}}q_{1}+T^{w_{2}}q_{2})$ in
$\mathsf{D^{b}}(\mathscr{F}(\mathbb{T}^{2}))$, for suitable weights
$w_{1},w_{2}\in\mathbb{R}$. The geometrical outcome is precisely the direct
sum of the two purple, non-exact simple loops of Figure 12 bottom right, which
both lie in the homotopy class $(1,0)$ of $\alpha$. The mindful reader already
knows the next move: we take the split-closure of
$\mathscr{F}(\mathbb{T}^{2})$, which enables us to include both summands of
the mapping cone as objects of
$\mathsf{D}^{\pi}(\mathscr{F}(\mathbb{T}^{2}))$.

To summarize, we can obtain all simple loops of $\mathbb{T}^{2}$ (up to
Hamiltonian isotopy) by iterated twist along $\alpha$ and $\beta$ and eventual
inclusion of the arising direct summands. A far more systematic discussion is
given in [LP11], with special focus on Lagrangian surgery and the arising
exact triangles. Our take aims to provide a geometrical intuition behind the
next result. $\blacklozenge$

######  Proposition 6.31.

Any pair of longitude and meridian of the symplectic$2$-torus
$\mathbb{T}^{2}=\mathbb{S}^{1}\times\mathbb{S}^{1}$ split-generates its Fukaya
category $\mathscr{F}(\mathbb{T}^{2})$. Consequently, any compact
$2$-dimensional symplectic manifold $M^{2}$ diffeomorphic to $\mathbb{T}^{2}$
has Fukaya category $\mathscr{F}(M)$ split-generated isomorphically by any
pair of longitude and meridian.

###### Proof.

(Sketch) We work directly on $M$. Let
$S_{1},S_{2}\in\textup{obj}(\mathscr{F}(M))$ be diffeomorphic to some fixed
longitude respectively meridian as in previous example. Choose any
homotopically essential closed curve (the manifold analogue of simple loops in
$\mathbb{T}^{2}$). We want to show it lies in the split-closure of
$\\{S_{1},S_{2}\\}$. Then so will the whole Fukaya category $\mathscr{F}(M)$.

Firstly, one can show that for all $k\geq 1$ holds
$(\tau_{S_{1}}\circ\tau_{S_{2}})^{6k}=\mathds{1}\in SL(2,\mathbb{Z})$. We thus
have a canonical morphism
$i:L\rightarrow(\tau_{S_{1}}\circ\tau_{S_{2}})^{6k}(L)\cong L$ in
$\mathsf{D^{b}}(\mathscr{F}(M))$ fitting into the diagram

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where by axiom (T1) for triangulated categories we can find some $Y\cong
Cone(i)$ $\in\textup{obj}\big{(}\mathsf{D^{b}}(\mathscr{F}(M))\big{)}$
completing the exact triangle. Note that $\langle i\rangle\in HF^{0}(L,L)\cong
H^{0}(L)$ is just a singular cohomology class, by Proposition 5.43. If it
vanishes, it follows immediately from the discussion of Remark 3.48 that
$Cone(i)\cong L\oplus L$, making $L$ a direct summand which will therefore be
included in the split-closure of $\\{S_{1},S_{2}\\}$, as claimed.

One can show that if $k=2$, then $\langle i\rangle=0$. However the argument
behind this is too convoluted for the limited set of tools we developed. We
refer the reader to [AS09, Proposition 6.1] for an outline of the main
strategy. ∎

####  6.5 The wrapped Fukaya category

To conclude this chapter, we discuss briefly, and quite vaguely, the wrapped
version of Fukaya categories, which deals with exact symplectic manifolds
(recall the setting of Section 5.8). The main advantage is that we no longer
need to impose compactness of the exact Lagrangian submanifolds, only a
certain behaviour at infinity. We follow yet again the outline of [Aur13,
section 4.1], with an eye on [Abo10, section 2]. First, some more working
tools (see [Sil12]).

######  Definition 6.32.

Let $M^{n}$ be a smooth manifold. A point $p\in M$ is called a contact point
if it has a tangent hyperplane $H_{p}\subset T_{p}M$, that is, a subspace with
$\text{dim}(H_{p})=n-1$.

A smooth section $H\in\Gamma(TM)$ assigning $p\mapsto H_{p}$ (a contact
distribution) can be locally written as $H=\text{ker}(\alpha)$ for some not
unique $\alpha\in\Omega^{1}(M)$ such that $d\alpha|_{H}$ is a symplectic form.
Then each $H_{p}$ is the kernel of a linear map
$0\neq\alpha_{p}:T_{p}\rightarrow\mathbb{R}$. We call $\alpha$ a contact form
on $M$. Its non-degeneracy forces the dimension $n$ of $M$ to be odd.

######  Definition 6.33.

An exact symplectic manifold $M^{2n}=(M,\omega)$ with
$\omega=d\lambda\in\Omega^{2}(M)$ is called a Liouville manifold if equipped
with some complete $Z\in\mathfrak{X}(M)$ which is outward-pointing at infinity
(see next paragraph) and fulfills $\iota_{Z}(\omega)=\lambda$, or
equivalently, $\mathcal{L}_{Z}(\omega)=\omega$. Then $Z$ is the Liouville
vector field associated to the Liouville form $\lambda\in\Omega^{1}(M)$.

We are particularly interested in Liouville manifolds
$(M^{2n},\omega=d\lambda)$ possessing some compact domain $M^{in}\subset M$
with the following properties (see Figure 13):

  * •

the $(2n-1)$-dimensional boundary $\partial M^{in}$ is a smooth hypersurface
endowed with contact form $\alpha\coloneqq\lambda|_{\partial
M^{in}}\in\Omega^{1}(\partial M^{in})$,

  * •

$\text{zeros}(Z)\subset M^{in}$ and $Z$ is positively transverse to $\partial
M^{in}$, which means $Z(x)\notin T_{x}(\partial M^{in})$ and $Z(x)(x^{1})>0$
(on a suitable half-space chart about $x$) for all $x\in\partial M^{in}$.

The flow $\phi_{Z}^{t}:M\rightarrow M$ of $Z$ allows us to identify
$M\setminus M^{in}$ with the symplectization $[1,\infty)\times\partial
M^{in}$, where, given the radial coordinate function $r\in
C^{\infty}([1,\infty))$, we have $Z=r\frac{\partial}{\partial r}$ and
$\lambda=r\alpha$. By abuse of notation, we can informally write our Liouville
manifold as the adjunction space

$M=M^{in}\sqcup_{\partial M^{in}}([1,\infty)\times\partial M^{in})\,.$ (99)

For later use, we let $\psi_{Z}^{\rho}\coloneqq\phi_{Z}^{\log\rho}$ denote the
time-$\log\rho$ flow of $Z$ for $\rho>1$ (observe that $\phi_{Z}^{t}$ is
conformally symplectic, implying that
$\psi_{Z}^{\rho}\in\text{Symp}(M,\omega)$).

Figure 13: Exact Lagrangian submanifolds
$L_{0},L_{1}\in\textup{obj}(\mathscr{W}\\!(M))$ of a Liouville 2-dimensional
manifold $M$. The flow of the “wrapping Hamiltonian” $H\in C^{\infty}(M)$,
quadratic outside of some compact subset of $M$, causes $L_{0}$ to spiral