# Nearby Special Lagrangians

Mohammed Abouzaid and Yohsuke Imagi

###### Abstract

Let $X$ be a Calabi–Yau manifold and $Q\subset X$ a closed connected embedded
special Lagrangian. We use a combination of methods from Floer theory and from
geometric analysis, to study those closed immersed special Lagrangians $L$
which are $C^{0}$-close to $Q$ and unobstructed in the sense of Floer theory.
Our results are easiest to state under the assumption that the fundamental
group $\pi_{1}Q$ has no nonabelian free subgroup: we prove that every such
nearby special Lagrangian $L$ as above is unbranched (that is, the projection
$L\to Q$ is a covering map); moreover, if $\pi_{1}Q$ is abelian then $L$ is a
union of smooth perturbations of $Q\subset X.$ In comparison with Tsai–Wang’s
results, their Riemannian geometry assumptions on $Q\subset X$ are replaced by
the purely topological ones on $\pi_{1}Q,$ at the cost of introducing the
Floer theory assumption on $L.$

The $\pi_{1}Q$ conditions, the Floer theory condition and the special
Lagrangian condition are all essential to the statement above, as we show by
constructing counterexamples; they are branched $C^{0}$-nearby Lagranigans of
Maslov class zero, which fail to satisfy the three kinds of conditions
respectively.

Our results have also applications to Morse–Novikov theory.

###### Contents

  1. 1 Introduction
  2. 2 Short Proofs
  3. 3 General Floer Theory
  4. 4 Remaining Proofs
  5. A Novikov Fields in Arbitrary Characteristic
  6. B Proof of Theorem 1.31

##  1 Introduction

We work with the following definitions:

  * •

By a Calabi–Yau manifold we shall mean a Kähler manifold $X$ with a given
holomorphic volume form $\Omega,$ where we do not require that the Kähler
metric be Ricci-flat. This is different from the convention in the literature
on special Lagrangians, in which $(X,\Omega)$ is called an almost Calabi–Yau
manifold; but we shall not follow this because it would mislead to the idea
that the complex structure of $X$ is nonintegrable, especially in the
symplectic context.

  * •

By a Lagrangian in a symplectic manifold $X$ we shall mean, unless otherwise
mentioned, an immersed Lagrangian submanifold of $X.$ We shall suppose that
$X$ is compact or convex at infinity wherever Floer theory is concerned.

  * •

By a special Lagrangian in a Calabi–Yau manifold $X$ we shall mean, unless
otherwise mentioned, a Lagrangian of phase $0$ modulo
$2\pi{\mathbin{\mathbb{Z}}}.$

Let $X$ be a Calabi–Yau manifold and $Q\subset X$ a closed connected embedded
special Lagrangian. The $C^{1}$-nearby problem for $Q\subset X$ is solved by
McLean [42]; that is, the germ of $C^{1}$-nearby special Lagrangians about
$Q\subset X$ is parametrised smoothly by $H^{1}(Q,{\mathbin{\mathbb{R}}}).$ We
study the $C^{0}$-nearby problem, which is to classify those closed special
Lagrangians $L$ contained in a neighbourhood of $Q\subset X.$

Our main result is Theorem 1.12 which we state in §1.3; it essentially solves
the $C^{0}$-nearby problem under additional assumptions on the fundamental
group $\pi_{1}Q$ and the Floer theory of $L$. To place this result in the
appropriate context, we recall in §1.1 a simple analytical method by Tsai and
Wang [57, 58]. In §1.2 we describe the class of Lagrangians for which Theorem
1.12 holds; briefly speaking, we treat only those Lagrangians which underlie
the Fukaya category ${\mathbin{\cal F}}(X).$ Except this constraint from
${\mathbin{\cal F}}(X),$ Theorem 1.12 is stronger than the results of §1.1.

In §1.4 we construct examples which show that the hypotheses of Theorem 1.12
are essential. These examples are all Lagrangian surgeries obtained from Morse
$1$-forms on $Q$ and our results have applications to Morse–Novikov theory. In
comparison, Donaldson and He [16, 29, 30] have a new method of constructing
branched special Lagrangians from ${\mathbin{\mathbb{Z}}}_{2}$ harmonic
$1$-forms.

In §1.5 and §1.6 we state the key facts which we use to prove Theorem 1.12. In
§1.5 we recall the relevant facts about Thomas–Yau’s work [55]. In §1.6 we
show what we need from Floer theory in $T^{*}Q.$

Section 1.1 is entirely analytical; Section 1.6 is mostly topological; and the
other sections are mixed with both analysis and topology.

###  1.1 Tsai–Wang’s Method

Tsai–Wang’s result [57, 58] may be summarised as follows:

######  Proposition 1.1.

Let $X$ be a Ricci-flat Calabi–Yau manifold of complex dimension $n,$ and
$Q\subset X$ a closed embedded special Lagrangian which has positive Ricci
curvature with respect to the induced metric from $X.$ Then there exists a
neighbourhood of $Q\subset X$ in which $Q$ is up to multiplicity a unique
compactly-supported stationary integral $n$-varifold in the sense of geometric
measure theory. In particular, it is a unique closed special Lagrangian
including any phase and any singularities.

######  Remark 1.2.

Tsai and Wang [57, 58] treat only smooth submanifolds but their result extends
immediately to varifolds as we show in §2.1. The idea of the proof is the same
in either case, which is that the fibrewise dilation of the normal bundle over
$Q\subset X$ increases the area of any nontrivial $C^{0}$-nearby varifold.

######  Example 1.3.

Let $X$ be either $T^{*}S^{n},$ for $n>1,$ or
$T^{*}{\mathbin{\mathbb{C}}}P^{n}$ for arbitrary $n$, and let $Q$ denote its
zero-section; then $X$ has a Ricci-flat metric constructed respectively by
Stenzel [54] and Calabi [14]. In these two cases the hypothesis of Proposition
1.1 holds and moreover its conclusion holds in the whole $X$ rather than in a
neighbourhood of $Q.$

By the same method we can prove the following version of Proposition 1.1:

######  Proposition 1.4.

Let $X$ be a flat manifold, $Q\subset X$ a closed totally-geodesic submanifold
of dimension $n,$ and $U\subseteq X$ a tubular neighbourhood of $Q.$ Then
every compactly-supported stationary integral $n$-varifold in $U$ is parallel
to $Q,$ that is, a locally-constant multivalued graph over $Q.$ Such special
Lagrangians are always single-valued graphs if and only if $\pi_{1}Q$ is
abelian.

######  Example 1.5.

For instance, let $X=U=T^{*}Q$ with $Q$ a Klein bottle. Write $Q$ as the
mapping torus of the automorphism $S^{1}\to S^{1}$ given by $x\mapsto-x$ where
$x\in S^{1}={\mathbin{\mathbb{R}}}/{\mathbin{\mathbb{Z}}}.$ Then there are
natural double covers $T^{2}\to Q$ and $T^{*}T^{2}\to T^{*}Q,$ under which the
graphs of $dx$ and $-dx$ are pushed forward to a multivalued graph.

To make $Q$ a special Lagrangian we need that $Q$ be orientable. For instance,
let $Q$ be the mapping torus of the automorphism $T^{2}\to T^{2}$ given by
$(x,y)\mapsto(-x,-y)$ where $(x,y)\in
T^{2}={\mathbin{\mathbb{R}}}^{2}/{\mathbin{\mathbb{Z}}}^{2}.$ Then we get a
special Lagrangian multivalued graph in the same way.

######  Remark 1.6.

Although the hypothesis of Proposition 1.1 is stable under small perturbations
of the Ricci-flat Kähler metric (just because positivity is an open condition)
yet Proposition 1.4 will fail under perturbations of the ambient metric. For
instance, Kapouleas and Yang [37] construct a sequence of minimal surfaces
converging to the double of the Clifford flat torus $T^{2}\subset S^{3},$ and
a suitable rescaling of the ambient metric converges to a flat metric on the
tubular neighbourhood of $T^{2}\subset S^{3}.$

###  1.2 Unobstructed Lagrangians

The key notion we shall use is the Fukaya category ${\mathbin{\cal F}}(X)$ of
a Calabi–Yau manifold $X$ and its cohomology category $H{\mathbin{\cal F}}(X)$
with the same object set (in our notation ${\mathbin{\cal F}}(X)$ includes the
data of bounding cochains, which is an $A_{\infty}$ category without
curvature, and $H{\mathbin{\cal F}}(X)$ is an ordinary category). In both
${\mathbin{\cal F}}(X)$ and $H{\mathbin{\cal F}}(X)$ the morphism spaces are
${\mathbin{\mathbb{Z}}}$-graded vector spaces over a Novikov field $\Lambda$
with ground ring ${\mathbf{k}}.$

In any version of Fukaya categories, every object ${\mathbf{b}}$ consists of a
Lagrangian $L\subset X$ with additional data, and we say that ${\mathbf{b}}$
is supported on $L$ or that $L$ underlies ${\mathbf{b}}.$ In our category
${\mathbin{\cal F}}(X)$ we shall include in the data ${\mathbf{b}}$ a choice
of local system $E$ on $L$ with coefficients in $\Lambda^{0},$ the ring of
integers of the Novikov ring (we call such a local system _unitary_). The key
remaining datum for an object of ${\mathbin{\cal F}}(X)$ is a bounding cochain
in the sense of Fukaya, Oh, Ohta and Ono [22]. Whenever $L$ cannot be equipped
with a local system which admits a bounding cochain, we say that it is
$HF^{*}$-obstructed. Conversely, if there is a local system $E$ on $L$, and a
choice of bounding cochain $b$ for this pair, we say that $L$ is
$HF^{*}$-unobstructed, and underlies the object $(L,E,b)\in H{\mathbin{\cal
F}}(X).$

We shall denote by $HF^{*}({\mathbf{b}},{\mathbf{b}}^{\prime})$ the morphism
spaces in the cohomological category $H{\mathbin{\cal F}}(X)$ between two
objects ${\mathbf{b}},{\mathbf{b}}^{\prime}$. We shall write
${\mathbf{b}}\cong{\mathbf{b}}^{\prime}$ in $H^{0}{\mathbin{\cal F}}(X)$ if
there exists an element of $HF^{0}({\mathbf{b}},{\mathbf{b}}^{\prime})$ which
induces under right multiplication a graded linear isomorphism $HF^{*}({\bf
a},{\mathbf{b}})\cong HF^{*}({\bf a},{\mathbf{b}}^{\prime})$ for every object
${\bf a}\in H{\mathbin{\cal F}}(X).$

In practice, we will work in a Weinstein neighbourhood $U$ of the given closed
embedded Lagrangian $Q\subset X$, i.e. a neighbourhood equipped with a
symplectomorphism to a neighbourhood of the zero section in $T^{*}Q$. The
advantage of considering this subset is that $Q$ (as well as any Lagrangian
which is $C^{1}$-close to it) is necessarily unobstructed when considered as a
Lagrangian submanifold of this neighbourhood because it does not bound any
holomorphic disc in $U$; this need not be the case if we consider it as a
Lagrangian submanifold of $X$.

As $U$ is spherically positive in the sense of Fukaya, Oh, Ohta and Ono [24]
we can take ${\mathbf{k}}={\mathbin{\mathbb{Z}}}$ and then pass to
${\mathbf{k}}={\mathbin{\mathbb{Z}}}/p{\mathbin{\mathbb{Z}}}$ with $p$ prime.
We can thus work over the ground ring ${\mathbf{k}}$ of arbitrary
characteristic (whereas in general, we must choose ${\mathbf{k}}$ of
characteristic $0$ because of the virtual counts of Kuranishi spaces). The
advantage of having arbitrary characteristic is that we can cancel out the
effect of pseudoholomorphic discs if their contribution is a multiple of $p;$
moreover, with $p=2$ we can do without relative spin structures on
Lagrangians.

But when the characteristic is positive, we shall be more careful because we
use Lie–Kolchin’s theorem, Theorem 4.12, for which we want $\Lambda$ to be
algebraically closed. The ordinary Novikov field in the positive-
characteristic case is not algebraically closed because of Artin–Schreier
extensions. We shall therefore extend the definition of Novikov fields so that
we can choose a complete algebraically-closed Novikov field $\Lambda$ in
positive characteristic, as we show in §A.

We explain now how we treat badly immersed Lagrangians by means of
approximations. In the known construction of Fukaya categories, the underlying
Lagrangians are required at least to be cleanly immersed. On the other hand,
in our nearby problem we want to include as many (special) Lagrangians as
possible, so we introduce the next notion:

######  Definition 1.7.

Let $L\subset X$ be a closed immersed Lagrangian. An isomorphism class
$[{\mathbf{b}}]$ in $H^{0}{\mathbin{\cal F}}(X)$ is supported near $L$ if
there exist arbitrarily-small Lagrangian perturbations of $L$ which underlie
some objects in $[{\mathbf{b}}].$

More concretely, this means that for every $\epsilon>0$ there exists a
perturbation of the immersion $\hat{L}\to X$ with $C^{k}$ norm bounded by
$\epsilon$ for all $k$, and which supports an object representing the
isomorphism class $[{\mathbf{b}}].$ We make two more remarks about Definition
1.7; it is a strong condition but we shall need it for our application:

######  Remark 1.8.

Consider an arbitrary closed immersed Lagrangian with immersion map
$\iota:\hat{L}\to L\subset X$. If it is cleanly immersed or if everything is
real analytic, we can approximate it by a smooth family $\iota^{t}:\hat{L}\to
X,$ $t\in[0,1],$ such that $\iota^{0}=\iota$ and the other $\iota^{t},$
$t\in(0,1],$ are all generic and mutually isotopic under domain Hamiltonians
(on $\hat{L}$). This is a family version of the classical transversality
theorem in differential topology.

If it were possible to lift such an isotopy to an ambient Hamiltonian isotopy,
then we would know that any isomorphism class in $H^{0}{\mathbin{\cal F}}(X)$
supported on the immersion $\iota^{1}$ is also supported on $\iota^{t}$ for
all $t,$ generalising the Hamiltonian invariance theoreom by Fukaya, Oh, Ohta
and Ono [22]. In general, such a lift does not exist because then those strips
which bound arcs between self-intersection points will have constant area
under the isotopy. We can perhaps use the methods of [47] to prove invariance
statements for this perturbation problem.

######  Remark 1.9.

The condition that the isomorphism classes in $H^{0}{\mathbin{\cal F}}(X)$
should be the same under the perturbations will be crucial to our application
of the generalised Thomas–Yau theorem, Theorem 1.27 (ii), proved by the second
author [33]. It is a uniqueness theorem for special Lagrangians whose
perturbations underlie objects of the Fukaya category, and their isomorphism
classes tell us how the perturbed Lagrangians shall be close to special, which
we want to estimate in advance. This is why we want to have the same
isomorphism classes in $H^{0}{\mathbin{\cal F}}(X)$ during the perturbations.

###  1.3 Main Result

Let $X$ be a Calabi–Yau manifold and $Q\subset X$ a closed connected embedded
special Lagrangian. We make two definitions and state our main result:

######  Definition 1.10.

Let $U\subseteq X$ be a Weinstein neighbourhood of $Q.$ We say that a
Lagrangian immersion $\hat{L}\to L\subset U$ is unbranched if the composition
with the projection map $U\subset T^{*}Q\to Q$ is a finite cover $\hat{L}\to
Q$. This is equivalent to saying that there exists a finite cover $P\to Q$
such that the pull-back of $L\subset U$ under the induced map $T^{*}P\to
T^{*}Q$ is a union of smooth perturbations of $P$ (if $L$ is embedded, this
union is disjoint).

######  Definition 1.11.

We say that an object ${\mathbf{b}}\in H^{0}{\mathbin{\cal F}}(X)$ is
irreducible if those objects ${\bf a}\in H^{0}{\mathbin{\cal F}}(X)$
satisfying $HF^{0}({\bf a},{\mathbf{b}})\neq 0$ are all isomorphic to
${\mathbf{b}}.$

######  Theorem 1.12.

  * (i)

If $\pi_{1}Q$ is finite then $Q$ is, in any Weinstein neighbourhood $U$ of
$Q\subset X,$ a unique closed $HF^{*}$-unobstructed special Lagrangian
including those with nonzero phase.

  * (ii)

If $\pi_{1}Q$ has no nonabelian free subgroup, there exists a Weinstein
neighbourhood $U$ of $Q\subset X$ such that every closed $HF^{*}$-unobstructed
irreducibly-immersed special Lagrangian $L\subset U$ is unbranched.
Moreover, if $\pi_{1}Q$ is abelian, $L$ is a McLean perturbation of $Q;$ and
every irreducible object of $H^{0}{\mathbin{\cal F}}(U)$ supported near $L$ is
isomorphic to some object supported exactly on $L$ with the trivial bounding
cochain.

######  Remark 1.13.

Apart from the $HF^{*}$-unobstructed condition, the statement (i) above is
stronger than Proposition 1.1; for by Myers’ theorem, $\pi_{1}Q$ is
necessarily finite under the hypothesis of Proposition 1.1.

In (ii) the weaker hypothesis on $\pi_{1}Q$ is essential; for instance, if $Q$
is a compact Riemann surface of genus $>1$ then $\pi_{1}Q$ has free nonablian
subgroups and the cotangent bundle of such a surface admits many branched
nearby special Lagrangians given by the hyperKähler rotations of Hitchin
spectral curves [31, §5.1] where the hyperKähler metric is given by Feix [18]
and Kaledin [35]. In higher dimensions He [29] constructs branched special
Lagrangians which are locally modelled upon the product of $S^{1}$ and a
$2$-fold spectral curve. His construction applies to many
${\mathbin{\mathbb{Q}}}$-holomology $3$-spheres $Q$ [30] but they do not
include finite quotients of $S^{3}$ or
${\mathbin{\mathbb{R}}}P^{3}\\#{\mathbin{\mathbb{R}}}P^{3}$ (c.f. Proposition
2.8) so $\pi_{1}Q$ will have nonabelian free subgroups and this is compatible
with our statement (ii) (we shall not discuss whether He’s examples have
$HF^{*}$ obstructed or not).

Also $\pi_{1}Q$ being abelian is essential to the latter part of (ii), because
as in Proposition 1.4 there may be multivalued graphs over $Q.$ Note that in
Proposition 1.4 the manifold $Q$ is flat, which has a finite cover by $T^{n};$
so $\pi_{1}Q$ has a finite-index abelian subgroup, which is amenable and
accordingly satisfies the hypothesis of Theorem 1.12 (ii).

######  Remark 1.14.

By Theorem 1.12 those nearby special Lagrangians with $HF^{*}$ unobstructed in
the sense of Definition 1.7 are not very singular; in particular, in (i) (with
$\pi_{1}Q$ finite) or in (ii) with $\pi_{1}Q$ abelian, they are not singular
at all.

But this does not mean that we can deal with arbitrary singularities as in
Propositions 1.1 and 1.4, because Definition 1.7 itself is a strong
restriction on the Lagrangians.

###  1.4 Branched Examples

Here are examples of branched $C^{0}$-nearby closed Lagrangians, which will
all be Lagrangian surgeries. Let $Q$ be a closed connected orientable manifold
of dimension $n.$ By a Morse $1$-form on $Q$ we shall mean a closed $1$-form
on $Q$ which is locally of the form $df$ with $f$ a Morse function. We shall
also use the following notion:

######  Definition 1.15.

A standard Calabi–Yau structure near the zero-section $Q\subset T^{*}Q,$ is a
pair consisting of a compatible (integrable) complex structure $J$ and a
$J$-holomorphic volume form relative to which $Q$ is a special Lagrangian.

We begin with an example of nonspecial Lagrangian surgeries. In general, for
every transverse intersection point of two Lagrangians there are two types of
Lagrangian surgeries which we have to choose. In the following theorem we take
the same type for all the intersection points, and the proof will essentially
be an application of the result by Fukaya, Oh, Ohta and Ono [22, Chapter 10]:

######  Theorem 1.16.

If $Q$ admits a Morse $1$-form all of whose critical points have the same
index, then there exist a closed embedded branched
${\mathbin{\mathbb{Z}}}$-graded Lagrangian $L$ and an object of
$H{\mathbin{\cal F}}(T^{*}Q)$ supported on $L$ and not isomorphic to any
object of $H{\mathbin{\cal F}}(T^{*}Q)$ supported near a closed special
Lagrangian $($of any phase relative to any standard Calabi–Yau structure$).$

######  Remark 1.17.

By Poincaré duality in Morse–Novikov homology, the hypothesis implies that $n$
is even and the critical points have index $n/2.$

######  Example 1.18.

Take $Q$ to be the selfconnected sum of $S^{n/2}\times S^{n/2}$ with
$n\geqslant 4$ even. Starting with a Morse function $S^{n/2}\times
S^{n/2}\to{\mathbin{\mathbb{R}}}$ with one critical point of index $0,$ two of
index $n/2$ and one of index $n$, and performing the connected sum
construction at the critical point of index $0$ and that of index $n,$ we
obtain a $1$-form $\alpha$ on $Q$ satisfying the hypothesis. Since
$\pi_{1}Q\cong{\mathbin{\mathbb{Z}}}$, this shows that we do need the nearby
Lagrangians to be special in part (ii) of Theorem 1.12.

We turn now to an example of special Lagrangian surgeries, for which we take
two different types of Lagrangian surgeries at every pair of intersection
points:

######  Theorem 1.19.

Let $n\geqslant 3$ and let $Q$ have a Morse $1$-form $\alpha$ with at least
one critical point of index $1,$ at least one of index $n-1$ and none of index
$0$ or $n.$ Then near $Q\subset T^{*}Q$ there exist a standard Calabi–Yau
structure and relative to it a closed branched special Lagrangian.

The branched special Lagrangian in the preceding theorem is possibly immersed;
any pairs of critical points of $\alpha$ of indices $1$ and $n-1$ may be
removed by surgery but the other critical points become transverse double
points.

######  Example 1.20.

Take $Q$ to be the selfconnected sum of $S^{1}\times S^{n-1};$ then $Q$ has a
Morse $1$-form with exactly two critical points, one of index $1$ and the
other of index $n-1.$ So with $n\geqslant 3$ we can apply Theorem 1.19 and if
$n\geqslant 4$ we have $b_{2}(L)=0$ so that $L$ has $HF^{*}$ unobstructed.

But there are also $HF^{*}$-obstructed special Lagrangian surgeries:

######  Corollary 1.21.

In addition to the hypotheses of Theorem 1.19, assume that $\pi_{1}Q$ has no
nonabelian free subgroup.

  * (i)

If $n=3$ then near $Q\subset T^{*}Q$ there exist a standard Calabi–Yau
structure and relative to it a closed embedded $HF^{*}$-obstructed special
Lagrangian.

  * (ii)

If $n\geqslant 4$ then near $Q\subset T^{*}Q$ there exist a standard
Calabi–Yau structure and relative to it a closed generically-immersed
$HF^{*}$-obstructed special Lagrangian $L.$

######  Example 1.22.

For (i) take $Q=S^{1}\times S^{2}$ and regard it as the selfconnected sum of
$S^{3}.$ According to Milnor [43, Lemma 8.2] for every Morse function we can
produce, by changing it locally, two additional critical points of any
adjacent indices. Applying this to the height function of $S^{3}$ we get a
Morse function with exactly one critical point of index $0,$ exactly one
critical point of index $3$ and arbitrarily many pairs of critical points of
indices $1$ and $2.$ So $Q$ has such a Morse $1$-form as in (i) above.

In the same way, on $Q=S^{1}\times S^{n-1}$ with $n\geqslant 4$ there is such
a Morse $1$-form as in (ii) above with arbitrarily many pairs of critical
points of adjacent indices $1,2$ or $n-2,n-1.$

###### Proof of Corollary 1.21.

For (i) recall from Poincaré–Hopf’s theorem that the numbers of critical
points of indices $1$ and $2$ are the same. So by Theorem 1.19 there is a
branched embedded special Lagrangian. By Theorem 1.12 this has $HF^{*}$
obstructed as asserted. Part (ii) may be proved in the same way. ∎

Corollary 1.21 (ii) implies a statement without Lagrangians in it and perhaps
of interest in itself:

######  Corollary 1.23.

Let $Q$ be a closed connected orientable manifold of dimension $n$ with
$\pi_{1}Q$ having no nonabelian free subgroup. Let $\alpha$ be a Morse
$1$-form on $Q$ with at least one critical point of index $1,$ at least one of
index $n-1$ and none of index $0$ or $n.$ Then:

  * (i)

if $n=2$ we have $Q\cong T^{2}$ and $\alpha$ has no critical point at all;

  * (ii)

if $n=3$ there exists a smooth function $f:Q\to{\mathbin{\mathbb{R}}}$ such
that $\alpha-df$ has no critical point at all; and

  * (iii)

if $n\geqslant 4$ then either $b_{2}(Q)>0$ or $\alpha$ has some critical point
of index $2$ or $n-2.$

###### Proof of (i) and (iii).

Since $\alpha$ has no critical points of index $0$ or $n$ it follows that
$\alpha$ is nonexact, so $b_{1}(Q)>0.$ If $n=2$ this condition and the
hypothesis on $\pi_{1}Q$ imply $Q\cong T^{2}$, but since the Euler
characteristic of $T^{2}$ vanishes, every Morse $1$-form on $T^{2}$ with no
critical point of index $0$ or $2$ has no critical point at all, as in (i).

We turn now to (iii). Since the immersed Lagrangian $L$ in Corollary 1.21 (ii)
has $HF^{*}$ obstructed, it follows that either $b_{2}(L)>0$ or $L$ has some
self-intersection points of index $2$ or $n-2.$ So $b_{2}(Q)>0$ (because
$n\geqslant 4$) or $\alpha$ has some critical points of index $2$ or $n-2.$ ∎

We shall prove (ii) in two different ways, one of which uses the
geometrisation theorem on $3$-manifolds, and the result is not really new; the
geometry of $Q$ will be $S^{2}\times{\mathbin{\mathbb{R}}},$ Euclidean, Nil or
Sol. But the other proof, using Corollary 1.21 (ii), may still be interesting.

###  1.5 Thomas–Yau’s Theorem

We recall first the easy part of Thomas–Yau’s uniqueness theorem [55, Theorem
4.3] which is about Maslov indices. Let $X$ be a Calabi–Yau manifold of
complex dimension $n;$ and $L,L^{\prime}\subseteq X$ two embedded
${\mathbin{\mathbb{Z}}}$-graded Lagrangians intersecting each other
transversely at a point $x,$ graded by $\phi:L\to{\mathbin{\mathbb{R}}}$ and
$\phi^{\prime}:L^{\prime}\to{\mathbin{\mathbb{R}}}$ respectively. The Maslov
index $\mu_{L,L^{\prime}}(x)\in{\mathbin{\mathbb{Z}}}$ then satisfies the
inequality

$\frac{\phi(x)-\phi^{\prime}(x)}{2\pi}<\mu_{L,L^{\prime}}(x)<\frac{\phi(x)-\phi^{\prime}(x)}{2\pi}+n.$
(1.1)

In particular, the index of an self-intersection point of an immersed
Lagrangian (still ${\mathbin{\mathbb{Z}}}$-graded) satisfies the same formula,
which implies the following fact, whose analogue for embedded
${\mathbin{\mathbb{Z}}}$-graded Lagrangians is well-known:

######  Proposition 1.24.

If ${\mathbf{b}}\in H{\mathbin{\cal F}}(X)$ is an object supported on a
immersed nearly-special Lagrangian (whose phase is $C^{0}$-close enough to a
constant), then we can make a Floer cochain group
$CF^{*}({\mathbf{b}},{\mathbf{b}})$ supported in degrees $0,\cdots,n;$ and
accordingly, so is the cohomology group $HF^{*}({\mathbf{b}},{\mathbf{b}}).$ ∎

We formulate in §3 a general nonvanishing result:

######  Proposition 1.25.

In the circumstances of Proposition 1.24 above, the object ${\mathbf{b}}\in
H{\mathbin{\cal F}}(X)$ is nonzero with $HF^{n}({\mathbf{b}},{\mathbf{b}})\neq
0.$

######  Remark 1.26.

The embedded case is proved already by Fukaya, Oh, Ohta and Ono [22, Theorem
E]; their statement is more complex because they include those Lagrangians
which need not be ${\mathbin{\mathbb{Z}}}$-graded.

For the proof that ${\mathbf{b}}$ is nonzero we shall use the homotopy unit of
$HF^{*}({\mathbf{b}},{\mathbf{b}})$ which we discuss in §3.4. The fact that
$HF^{n}({\mathbf{b}},{\mathbf{b}})\neq 0$ will follow from the duality result
we discuss in §3.5.

We have stated Proposition 1.25 because it is interesting in itself; although
we shall not really need it for our main result, for the following two
reasons:

  * •

The fact that ${\mathbf{b}}$ is nonzero will be used in the proof of Theorem
1.12 but in those circumstances we can also prove directly the same
nonvanishing result. Let $L\subset T^{*}Q$ be a $C^{0}$-nearby special
Lagrangian relative to a standard Calabi–Yau structure near the zero-section
$Q.$ By the constancy theorem for currents, the push-forward of $L$ under the
projection $T^{*}Q\to Q$ is a constant multiple of $Q$ and the constant is
positive because no cancellation happens to special Lagrangians. In
particular, the intersection number of $L$ with $T_{q}^{*}Q,$ $q\in Q$
generic, is nonzero. Suppose now that ${\mathbf{b}}\in H{\mathbin{\cal F}}(U)$
is an object supported near $L.$ Then the nonzero intersection number of $L$
with $T_{q}^{*}Q$ is equal up to sign to the alternating sum
$\sum_{i\in{\mathbin{\mathbb{Z}}}}(-1)^{i}\dim
HF^{i}(T_{q}^{*}Q,{\mathbf{b}}).$ So $HF^{*}(T_{q}^{*}Q,{\mathbf{b}})\neq 0.$

  * •

The fact that $HF^{n}({\mathbf{b}},{\mathbf{b}})\neq 0$ will be used only
implicitly in the proof of Theorem 1.27 below, and nowhere else in the present
paper. Theorem 1.27 is proved by the second author [33] who gives other proofs
that $HF^{n}({\mathbf{b}},{\mathbf{b}})\neq 0.$ But our treatment in §3.5 will
be better than those.

We turn now to the central part of Thomas–Yau’s theorem (which involves
analysis in the proof). There is a generalized Thomas–Yau theorem proved by
the second author [33, Corollary 6.3 (ii) and Theorem 6.2] and we shall need
only the following simpler version (in which either $L$ or $L^{\prime}$ is
assumed to be embedded):

######  Theorem 1.27.

Let $X$ be a Calabi–Yau manifold and
${\mathbf{b}},{\mathbf{b}}^{\prime}\in{\mathbin{\cal F}}(X)$ be two objects
supported near two closed irreducibly-immersed special Lagrangians with either
$L$ or $L^{\prime}$ embedded.

  * (i)

If ${\mathbf{b}}$ and ${\mathbf{b}}^{\prime}$ are isomorphic, then
$L=L^{\prime}\subset X$.

  * (ii)

If $L$ and $L^{\prime}$ both have phase $0$ then either $L=L^{\prime}\subset
X$ or $HF^{*}({\mathbf{b}},{\mathbf{b}}^{\prime})$ is supported in degrees
$1,\cdots,n-1.$

######  Remark 1.28.

The point of (ii) is that the degrees are strictly $>0$ and $<n,$ because we
can always make the cochains supported in $[0,n]$ (in the same way as in
Proposition 1.24).

We shall also use the following version [33, Remark 5.2 (i)]:

######  Theorem 1.29.

Let $X$ be a real analytic Calabi–Yau manifold and $L,L^{\prime}\subset X$ two
closed irreducibly-immersed Lagrangians which are, near $L\cap L^{\prime},$
both special and graded by phase $0.$ Then either $L,L^{\prime}$ agree with
each other near $L\cap L^{\prime}$ or any pair of objects supported on them
have Floer cohomology supported in degrees $1,\cdots,n-1.$

######  Remark 1.30.

Here real analytic means that the Kähler form of $X$ is real analytic with
respect to the underlying real analytic structure of the complex manifold $X.$

###  1.6 Floer Theory in $T^{*}Q$

For the proof of Theorem 1.12 we shall use some more general facts about the
Fukaya category ${\mathbin{\cal F}}(T^{*}Q),$ where $Q$ is a closed connected
manifold. Apart from Lemma 1.36 the material in what follows will be entirely
topological, in which therefore we can leave out the hypothesis that $Q$ be
orientable when the ground ring ${\mathbf{k}}$ has characteristic $2.$
Otherwise we shall suppose $Q$ orientable.

The key fact we shall use is a version of the generation theorem [3] on the
Fukaya category ${\mathbin{\cal F}}(T^{*}Q).$ This is well known for exact
Lagrangians and it extends to nonexact Lagrangians as we state now. Consider
the wrapped category ${\mathbin{\cal W}}(T^{*}Q)$ which includes objects
supported on fibres in $T^{*}Q,$ and also contains ${\mathbin{\cal
F}}(T^{*}Q)$ as a full subcategory (because the objects of ${\mathbin{\cal
F}}(T^{*}Q)$ are by definition supported on closed Lagrangians). We work in
the derived category $D{\mathbin{\cal
W}}(T^{*}Q):=H^{0}\operatorname{Tw}{\mathbin{\cal W}}(T^{*}Q)$ (where
$\operatorname{Tw}$ means taking twisted complexes) so that we can speak of
generators and split generators in it. In §B.2 we show how the methods to
study exact Fukaya categories can be adapted to prove the following theorem,
in which $T_{q}^{*}Q$ refers to the object of the wrapped Fukaya category
associated to the cotangent fibre a point $q\in Q$, equipped with the trivial
rank-one local system and with the trivial bounding cochain:

######  Theorem 1.31.

The Fukaya category $H^{0}{\mathbin{\cal F}}(T^{*}Q)\subset D{\mathbin{\cal
W}}(T^{*}Q)$ lies in the category split-generated by $T^{*}_{q}Q$.

Recall also that the endomorphism group $HW^{*}(T_{q}^{*}Q,T_{q}^{*}Q)$ in
${\mathbin{\cal W}}(T^{*}Q)$ is computed by Abbondandolo, Portaluri and
Schwarz [1]; they construct a ${\mathbin{\mathbb{Z}}}$-graded ring isomorphism

$HW^{*}(T_{q}^{*}Q,T_{q}^{*}Q)\cong H_{-*}(\Omega_{q}Q)$ (1.2)

where $\Omega_{q}Q$ denotes the based-loop space and $H_{-*}(\Omega_{q}Q)$ is
negatively graded; experts should note that our Fukaya category is twisted by
the second Stiefel–Whitney class of $Q.$ Although (1.2) lifts to the
$A_{\infty}$ level [4] we shall not need that.

In practice, we shall define our Floer groups by virtual counts and shall
therefore check (in Proposition 3.42) that they are isomorphic to those used
for exact Lagrangians in the literature (without virtual counts) including the
left-hand side on (1.2).

Applying to Theorem 1.31 the $A_{\infty}$ version of Yoneda’s lemma, and using
(1.2), we get a fully faithful functor

$H{\mathbin{\cal F}}(T^{*}Q)\to\operatorname{mod}H_{-*}(\Omega_{q}Q)$ (1.3)

which on objects maps ${\mathbf{b}}$ to $HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$.
Here $\operatorname{mod}H_{-*}(\Omega_{q}Q)$ denotes the category of
${\mathbin{\mathbb{Z}}}$-graded left $H_{-*}(\Omega_{q}Q)$ modules (of finite
dimension over $\Lambda,$ because $H{\mathbin{\cal F}}(T^{*}Q)$ is made from
compact Lagrangians). We make now a definition which we shall often use in
this paper:

######  Definition 1.32.

By a single-degree object of $H{\mathbin{\cal F}}(T^{*}Q)$ we mean an object
${\mathbf{b}}$ of $H{\mathbin{\cal F}}(T^{*}Q)$ such that the Floer cohomology
group $HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$ is supported in a single degree.

With this definition we can state an important property of the Yoneda functor
(1.3):

######  Proposition 1.33.

Those single-degree objects ${\mathbf{b}}\in H^{0}{\mathbin{\cal F}}(T^{*}Q)$
with $HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$ supported in some fixed degree are
mapped under (1.3) to the category of finite-dimensional $\Lambda$-linear
representations of $\pi_{1}Q$; this functor induces an isomorphism on
morphisms of degree $0$.

###### Proof.

Since the self-Floer cohomology of a cotangent fibre is nonpositively graded
it follows that every single-degree $A_{\infty}$ module over it has trivial
higher actions. The only one nontrivial action is by degree $0$ morphisms,
which is isomorphic to $\Lambda[\pi_{1}Q],$ which gives a representation of
$\pi_{1}Q.$ This defines the desired functor for those single-degree objects
in the given degree. ∎

Since $HW^{*}(T_{q}^{*}Q,T_{q}^{*}Q)$ is supported in nonpositive degrees it
follows (by the degree-filtration argument [5, Appendix A]) that every object
of it category of finite rank modules is obtained as a repeated extension of
ungraded modules over $\Lambda[\pi_{1}Q],$ that is, representations of
$\pi_{1}Q.$ This extension problem is nontrivial; in Example 1.18, for
instance, the direct sum of $Q$ and $\operatorname{graph}\alpha$ are different
from their surgery, so there exist indeed two distinct objects made up from
the same representations but by different extensions.

But this will not matter to us in the proof of Theorem 1.12. For in (i) of
Theorem 1.12, after we pass to the universal cover of $Q$ the extension
problem will be resolved as follows:

######  Proposition 1.34.

If $\pi_{1}Q$ is trivial, the derived category $D{\mathbin{\cal F}}(T^{*}Q)$
is generated by the object supported on $Q$ $($with any grading and with the
trivial bounding cochain$);$ and every nonzero object ${\mathbf{b}}\in
D{\mathbin{\cal F}}(T^{*}Q)$ with $HF^{*}({\mathbf{b}},{\mathbf{b}})$
supported in nonnegative degrees is isomorphic to the direct sum of copies of
this generator.

######  Remark 1.35.

By Proposition 1.24 the condition on $HF^{*}({\mathbf{b}},{\mathbf{b}})$ will
be satisfied for those ${\mathbf{b}}$ supported on special Lagrangians.

On the other hand, in (ii) of Theorem 1.12 we can prove that the extension
problem becomes trivial for nearby special Lagrangians, as we state now. Fix
any point $q\in Q,$ grade $T_{q}^{*}Q$ by phase $n\pi/2$ at $q$ and make it
into an object the wrapped Fukaya category by giving the trivial rank-one
local system and the trivial bounding cochain. Then the following holds, for
which we do not need any hypothesis on $\pi_{1}Q$:

######  Lemma 1.36.

For every standard Calabi–Yau structure near $Q\subset T^{*}Q$ there exists a
neighbourhood $U$ of $Q\subset T^{*}Q$ such that every object ${\mathbf{b}}\in
H{\mathbin{\cal F}}(U)$ supported near a closed special Lagrangian $($graded
by phase $0)$ is of single degree, supported always in the degree $n=\dim Q.$

######  Remark 1.37.

Here the speciality is essential because otherwise
$HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$ may be supported in two or more degrees (as
in the example of Theorem 1.16 with $n\geqslant 4$). There is in fact an
analytical theorem by Almgren [11], explained also by De Lellis and Spadaro
[15, Theorem 2.4], which applies to general area-minimising currents and has
moreover the same conclusion at the chain level. But this applies only to
those nearby currents of a given homological degree and also the cohomology-
level result will do for us.

The heart of the proof of Lemma 1.36 is to make some compactly-supported
Hamiltonian perturbations of $T_{q}^{*}Q,$ indexed by
$k\in\\{1,\cdots,n-1\\},$ all special Lagrangians near $q$. To these we can
apply Theorem 1.29 (after reduction to the real analytic case) which will
imply that $HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$ is supported in the degrees
$1+k,\cdots,n-1+k.$ But $\bigcap_{k=1}^{n-1}[1+k,n-1+k]=\\{n\\}$ as we want.

By Lemma 1.36 we can use Proposition 1.33 so the problem becomes that of
representation theory, for which we shall use the hypothesis on $\pi_{1}Q.$
The key result is the following, which is again entirely topological:

######  Theorem 1.38.

  * (i)

If $\pi_{1}Q$ has no nonabelian free subgroup, then for every nonzero single-
degree object ${\mathbf{b}}\in H{\mathbin{\cal F}}(T^{*}Q)$ there exist a
finite cover $f:P\to Q,$ a closed $1$-form $\alpha$ on $P$ and a unitary local
system $E$ on $P^{\alpha}$ such that $HF^{0}(E,f^{*}{\mathbf{b}})\neq 0.$
Moreover, if $\pi_{1}Q$ is abelian, we can take $P=Q.$

  * (ii)

In these circumstances, if $Q$ is given a Riemannian metric, we can choose
$\alpha$ to be harmonic, with its norm bounded above by $c\delta;$ where $c>0$
is independent of ${\mathbf{b}},$ and $\delta>0$ is such that the underlying
Lagrangian of ${\mathbf{b}}$ is within distance $\delta$ from $Q\subset
T^{*}Q.$

######  Remark 1.39.

For (i) we shall use some standard facts from representation theory. For (ii)
we shall use action completed Floer groups as Seidel and others do [52, 59,
60].

In passing to finite covers (as in Theorem 1.38 (i) above) we shall use the
fact that they are compatible with Yoneda functors:

######  Proposition 1.40.

For every finite cover $(P,p)\to(Q,q)$ there is a commutative diagram

${H{\mathbin{\cal F}}(T^{*}Q)}$${H{\mathbin{\cal
F}}(T^{*}P)}$${\operatorname{mod}H_{-*}(\Omega_{q}Q)}$${\operatorname{mod}H_{-*}(\Omega_{p}P)}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$
(1.4)

where the two vertical arrows are the Yoneda functors, the top horizontal
$f^{*}$ corresponds at the object level to pulling back Lagrangians under the
finite cover $T^{*}P\to T^{*}Q,$ and the bottom horizontal $f^{*}$ is induced
by the push-forward map $\Omega_{p}P\to\Omega_{q}Q.$

Theorem 1.38 and Proposition 1.40 will be proved after the preparation in §3.

###  1.7 Plan of the Paper

In §2 we prove all the results but Theorem 1.38 and Proposition 1.40; the
proofs in §2 are relatively short, for which we shall not need so much from
general Floer theory as we do in §3. In §4 we prove Theorem 1.38 and
Proposition 1.40 using the results of §3. In §A we recall the relevant facts
about Novikov fields because in Floer theory the treatment of the positive
characteristic case is less standard. In §B we sketch the proof of Theorem
1.31.

###  1.8 Acknowledgements

This paper is based upon the discussions we had in Fall 2016 at the Institute
for Advanced Study in Princeton. We are both grateful to David Treumann for
pointing out that spectral curves show the necessity of the hypothesis on the
fundamental group in Theorem 1.12 (ii), and to Dominic Joyce for his help with
Example 1.22. The second author would also like to thank Mark Haskins and
Martin Li for interesting conversations which inspired him to begin this work.

The first author was supported by the Erik Ellentuck Fellowship, the IAS Fund
of Math, NSF grants DMS-1308179, DMS-1609148 and DMS-1564172, DMS-2103805 and
by the Simons Foundation through its “Homological Mirror Symmetry”
Collaboration grant. The second author was supported by the WPI project at the
Kavli Institute for Physics and Mathematics of Universe, by the JSPS grants
16K17587, 18J00075 and 21K13788, and by the NSFC grant 11950410501.

##  2 Short Proofs

In §2.1 we prove the two analytical facts, Propositions 1.1 and 1.4. In §2.2
we recall the relevant facts about Lagrangian surgeries, which we use in §2.3
to prove Theorems 1.16 and 1.19. In §2.4 we prove Corollary 1.23; in §2.5 we
prove Proposition 1.34; in §2.6 we prove Theorem 1.12 (i); in §2.7 we prove
Lemma 1.36; and in §2.8 we prove Theorem 1.12 (ii).

###  2.1 Proofs of Propositions 1.1 and 1.4

Let $X$ be a Riemannian manifold and $Q\subset X$ a closed area-stationary
submanifold of dimension $n.$ Given a vector field $v$ on $X$, a point $x\in
X$, and an $n$-dimensional plane $S\subseteq T_{x}X$ we denote by
$\operatorname{div}_{S}v$ the trace of the map $S\subseteq T_{x}X\to T_{x}X\to
S$ where the two arrows are the covariant derivative $\nabla v:T_{x}X\to
T_{x}X$ and the orthogonal projection $T_{x}X\to S.$ In particular, if $v$ is
the gradient vector field of a smooth function $f:X\to{\mathbin{\mathbb{R}}}$
then
$\operatorname{div}_{S}\operatorname{grad}f=\operatorname{tr}_{S}\nabla^{2}f.$
We use also the notion of rectifiable integral varifolds in $X$ which we
treat, following Simon [53], as Radon measures on $X.$ Recall that if $V$ is a
compactly-supported area-stationary integral $n$-varifold in $X$ then for
every vector field $v$ on $X$ we have
$\int_{X}\operatorname{div}_{TV}v\,dV=0.$ In particular,

$\int_{X}\operatorname{tr}_{TV}\nabla^{2}f\,dV=0$ (2.1)

for every smooth function $f:X\to{\mathbin{\mathbb{R}}}.$

We come now to the proof of Proposition 1.1. Tsai and Wang [58, Proposition
A(iii)] prove that in the circumstances of Proposition 1.1 the special
Lagrangian $Q\subset X$ is strongly stable in their sense. We prove that there
exists about $Q\subset X$ a neighbourhood $U$ in which every compactly-
supported stationary integral $n$-varifold is supported in $Q.$ According to
Tsai and Wang [58, Proposition 4.1] we can take $U$ such that for every point
$x\in U$ and for every $n$-dimensional plane $S\subseteq T_{x}U$ we have at
$x$

$\operatorname{tr}_{S}\nabla^{2}d_{Q}^{2}\geqslant c\,d_{Q}^{2}$ (2.2)

where $d_{Q}:U\to[0,\infty)$ is the distance from $Q$ and $c>0$ depends only
upon $Q.$ Applying (2.1) with $f=\operatorname{grad}d_{Q}^{2}$ and using (2.2)
we find $d_{Q}^{2}=0$ on the support of every nearby varifold about $Q\subset
X,$ as we have claimed.

We go next to the proof of Proposition 1.4. We use the following lemma from
geometric measure theory:

######  Lemma 2.1.

For any two nonnegative integers $n,k,$ in the flat manifold
$X:=T^{n}\times{\mathbin{\mathbb{R}}}^{k}$ every compactly-supported
stationary integral $n$-varifold has support parallel to
$Q:=T^{n}\times\\{0\\}.$

###### Proof.

Letting $V$ be a compactly-supported stationary integral $n$-varifold in $X$
and applying (2.1) again with $v=\operatorname{grad}d_{Q}^{2}$ we get
$\int_{X}\operatorname{tr}_{TV}\nabla^{2}d_{Q}^{2}\,dV=0.$ On the other hand
$\nabla^{2}d_{Q}$ is nonnegative and degenerates only in the direction of
$T^{n}\times\\{0\\}$ so $V$-almost everywhere in $X$ we have
$\operatorname{tr}_{TV}\nabla^{2}d_{Q}^{2}=0$ and $TV$ is parallel to
$T^{n}\times\\{0\\}.$

We use now Simon’s lemma [53, Lemma 19.5]: let $x$ be a point of the support
of $V;$ let $\epsilon>0$ be so small that for every point $y$ with distance
$<\epsilon$ from $x$ we can define
$y-x\in{\mathbin{\mathbb{R}}}^{n}\times{\mathbin{\mathbb{R}}}^{k};$ let
$\delta\in(0,1);$ and let $y$ be a point of the support of $V$ with distance
$\epsilon\delta/4$ from $x$ and such that $y-x$ is not parallel to
$T^{n}\times\\{0\\};$ then,

$\Theta_{V}(x)+\Theta_{V}(y)\leqslant\nobreak(1-\delta)^{-n}\epsilon^{-n}\omega_{n}^{-1}V(B_{\epsilon})$
(2.3)

where $\Theta_{V}$ denote the density of $V$ and $B_{\epsilon}$ denotes the
ball of radius $\epsilon$ about $x.$ Simon’s formula has in fact another term
but in our circumstances it vanishes because $TV$ is parallel to
$T^{n}\times\\{0\\}.$

As $\epsilon,\delta$ tend to $0$ the right-hand side of (2.3) tends to
$\Theta_{V}(x)$ and in particular becomes less than $\Theta_{V}(x)+1$ but then
by (2.3) we have $\Theta_{V}(y)<1.$ This is impossible, so Simon’s hypothesis
fails; and accordingly, if $y$ is a point of the support of $V$ close enough
to $x$ then $y-x$ is parallel to $T^{n}\times\\{0\\}$ as we want. ∎

Since $Q$ is a closed flat manifold we get, according to Bieberbach, a finite
Galois cover $T^{n}\to Q$ such that $\pi_{1}T^{n}$ is a maximal abelian
subgroup of $\pi_{1}Q.$ If $\pi_{1}Q$ is abelian, we have $Q=T^{n}$ so no
constant graph over $Q$ is truly multivalued. If $\pi_{1}Q$ is nonabelian,
there is an element $\gamma\in\pi_{1}Q\setminus\pi_{1}T^{n}$ which acts
nontrivially upon $H^{1}(T^{n},{\mathbin{\mathbb{R}}})$ so there is a constant
$1$-form that varies under $\gamma.$ The graph of this $1$-form then maps,
under the covering map $T^{*}T^{n}\to T^{*}Q,$ onto a truly-multivalued graph
over $Q.$ ∎

###  2.2 Lagrangian Surgeries

We state a generalization of a result by Fukaya, Oh, Ohta and Ono [22, Chapter
10] which describes the effect of applying Lagrangian surgeries to a pair of
objects of the Fukaya category. For the underlying
${\mathbin{\mathbb{Z}}}$-graded Lagrangians we use the same symbol
$L^{\prime}\\#L$ as Seidel does [49, §2e], fixing the type of surgery
performed at each intersection point to be the one described in [22, Chapter
10, §54.1].

######  Theorem 2.2 (cf. [22, Chapter 10]).

Let $L,L^{\prime}\subseteq X$ be two closed embedded
${\mathbin{\mathbb{Z}}}$-graded Lagrangians which intersect each other
transversely at least at one point, with the Maslov index function
$\mu_{L,L^{\prime}}:L\times_{X}L^{\prime}\to{\mathbin{\mathbb{Z}}}$
identically equal to $1.$ Let $L^{\prime}\\#L$ be a
${\mathbin{\mathbb{Z}}}$-graded Lagrangian surgery which is, outside $L\cap
L^{\prime},$ $C^{\infty}$-close enough to $L\cup L^{\prime}.$ Then for any two
objects ${\mathbf{b}},{\mathbf{b}}^{\prime}\in H{\mathbin{\cal F}}(X)$
supported on $L,L^{\prime}$ respectively, there exist an object
${\mathbf{b}}^{\prime\prime}\in{\mathbin{\cal F}}(X)$ supported on
$L^{\prime}\\#L$ and for every other object ${\bf a}\in H{\mathbin{\cal
F}}(X)$ a long exact sequence

$\cdots\to HF^{*}({\bf a},{\mathbf{b}}^{\prime})\to HF^{*}({\bf
a},{\mathbf{b}}^{\prime\prime})\to HF^{*}({\bf a},{\mathbf{b}})\to\cdots$
(2.4)

where the connecting homomorphism $HF^{*}({\bf a},{\mathbf{b}})\to
HF^{*+1}({\bf a},{\mathbf{b}}^{\prime})$ is the right multiplication by some
element of $HF^{1}({\mathbf{b}},{\mathbf{b}}^{\prime}).$

######  Remark 2.3.

There are many objects ${\mathbf{b}}^{\prime\prime}$ of this form according to
different scales of surgeries at different intersection points of
$L,L^{\prime}.$ In fact, in [22, Chapter 10] the result is stated only for
Lagrangians meeting at a single point, and their proof contains a cobordism
argument for the moduli spaces of holomorphic discs with boundary on the
surgery; the cobordism will be associated to stretching the neck (degenerating
the almost complex structure) near the intersection points (which may be more
than one in our case).

The method of [19] provide an easier proof which does not rely on a
degeneration argument: one can associate to the surgery a Lagrangian cobordism
in $X\times{\mathbin{\mathbb{C}}}$, and the count of holomorphic triangles
with boundary on this cobordism yields the desired long exact sequence. At the
technical level, the results of [19] are stated for monotone or exact
Lagrangians, but these assumptions are only imposed to avoid using the theory
of virtual chains in case the Lagrangians bound holomorphic discs. In practice
we shall take $L,L^{\prime}$ to be graphical Lagrangians in a cotangent
bundle, which do not bound any holomorphic discs; so the proof of [19] can be
implemented with no difficulty.

For special Lagrangians we have:

######  Corollary 2.4.

In the circumstances of Theorem 2.2 let $L,L^{\prime}$ be both special $($of
phase $0$ modulo $2\pi{\mathbin{\mathbb{Z}}}).$ Then the object
${\mathbf{b}}^{\prime\prime}\in H{\mathbin{\cal F}}(X)$ is not isomorphic to
any other object supported near a closed special Lagrangian of any phase in
${\mathbin{\mathbb{R}}}.$

###### Proof.

By shifting $L,L^{\prime}$ we may suppose that $L$ has phase $0$ and
$L^{\prime}$ has phase $2\pi i\in 2\pi{\mathbin{\mathbb{Z}}}.$ Then by (1.1)
we have $-i\in(1-n,1)$ so $i\geqslant 0.$ Applying (2.4) to
${\mathbf{b}},{\mathbf{b}}^{\prime}$ in place of ${\bf a}$ we get two exact
sequences

$\displaystyle HF^{n}({\mathbf{b}},{\mathbf{b}}^{\prime\prime})\to
HF^{n}({\mathbf{b}},{\mathbf{b}})\to
HF^{n+1}({\mathbf{b}},{\mathbf{b}}^{\prime})=0,$ (2.5) $\displaystyle
0=HF^{-1}({\mathbf{b}}^{\prime},{\mathbf{b}})\to
HF^{0}({\mathbf{b}}^{\prime},{\mathbf{b}}^{\prime})\to
HF^{0}({\mathbf{b}}^{\prime},{\mathbf{b}}^{\prime\prime}).$ (2.6)

On the other hand $HF^{n}({\mathbf{b}},{\mathbf{b}})$ and
$HF^{0}({\mathbf{b}}^{\prime},{\mathbf{b}}^{\prime})$ are both nonzero, so
$HF^{n}({\mathbf{b}},{\mathbf{b}}^{\prime\prime})$ and
$HF^{0}({\mathbf{b}}^{\prime},{\mathbf{b}}^{\prime\prime})$ are both nonzero.
Suppose contrary to our conclusion that ${\mathbf{b}}^{\prime\prime}$ is
isomorphic to an object ${\bf a}\in{\mathbin{\cal F}}(X)$ supported near a
closed special Lagrangian $L^{\prime\prime}$ of phase
$\theta\in{\mathbin{\mathbb{R}}}.$ By perturbing $L^{\prime\prime}$ we may
suppose that ${\bf a}$ is supported on a nearly-special Lagrangian $L^{*}$
which intersect $L$ and $L^{\prime}$ generically. Since
$HF^{n}({\mathbf{b}},{\bf a})\cong
HF^{n}({\mathbf{b}},{\mathbf{b}}^{\prime\prime})\neq 0$ we get an intersection
point $x\in L\cap L^{*}$ with Maslov index $n.$ Hence denoting by
$\phi:L^{*}\to{\mathbin{\mathbb{R}}}$ the phase function and using (1.1) we
find $-\phi(x)/2\pi\in(0,n)$ and in particular $\phi(x)<0.$ Also since
$HF^{0}({\mathbf{b}}^{\prime},{\bf a})\neq 0$ we get likewise an intersection
point $x^{\prime}\in L^{\prime}\cap L^{*}$ with
$i-\phi(x^{\prime})/2\pi\in(-n,0)$ and in particular
$\phi(x^{\prime})>i\geqslant 0.$ But both $\phi(x)$ and $\phi(x^{\prime})$ are
near $\theta$ so $i=\theta=0;$ that is, $L,$ $L^{\prime}$ and
$L^{\prime\prime}$ have all phase $0.$ Since $HF^{n}({\mathbf{b}},{\bf a})$
and $HF^{0}({\mathbf{b}}^{\prime},{\bf a})$ are both nonzero it follows by
Theorem 1.27 (ii) that these three Lagrangians have a common irreducible
component. But this is impossible because of our initial hypothesis that $L$
and $L^{\prime}$ are mutually transverse. ∎

###  2.3 Proof of Theorems 1.16 and 1.19

The key theorem we use is:

######  Theorem 2.5 (Honda [32]).

Let $Q$ be a closed manifold of dimension $n,$ and $\alpha$ a Morse $1$-form
on $Q$ with no critical point of index $0$ or $n.$ Then there exists on $Q$ a
Riemannian metric $g,$ and in the de-Rham class $[\alpha]\in
H^{1}(Q,{\mathbin{\mathbb{R}}}),$ a $g$-harmonic Morse $1$-form $\beta$ which
has the same distribution of indices as $\alpha$ has; that is, for each
$i\in\\{0,\cdots,n\\}$ the number of index-$i$ critical points of $\beta$ is
equal to that of $\alpha.$ ∎

If $Q$ is equipped with any real analytic structure, we may assume, in the
above result, that $g$ and $\beta$ both real analytic. To see this, perturb
the metric from Honda’s theorem to a real analytic Riemannian metric, which we
denote still by the same $g$. Choosing the perturbation to be sufficiently
small, there exists a unique $g$-harmonic $1$-form near $\beta$ in the same
de-Rham class, which we denote again by the same $\beta.$ This $\beta$ is
analytic by elliptic regularity.

We apply to $(Q,g)$ the following well-known fact:

######  Lemma 2.6.

Let $(Q,g)$ be a closed oriented real analytic Riemannian manifold. Then there
exist a disc subbundle $X\subseteq T^{*}Q$ and a standard Calabi–Yau structure
$(J,\Omega)$ such that the restriction of the Kähler metric to $Q$ is equal to
$g,$ and the complex structure is compatible with the Liouville structure of
$X$ (in the sense of Definition 3.1). ∎

######  Remark 2.7.

The real analytic structure induced from this Calabi–Yau complex structure is
possibly different from that induced from the real analytic structures of $Q$
and $T^{*}Q.$

Applying McLean’s theorem to the $\beta$ above, we get a $C^{1}$-nearby
special Lagrangian $Q^{\prime}$ such that the pair $(Q,Q^{\prime})$ has the
same indices as $\alpha.$

Finally we have:

###### Proof of Theorems 1.16 and 1.19.

For Theorem 1.16 consider $Q^{\prime}[n/2-1]\\#Q$ as in Lemma 2.2. Then by
Corollary 2.4 this cannot be represented by a special Lagrangian, as claimed.
For Theorem 1.19 we have only to apply to $Q,Q^{\prime}$ the construction by
Joyce [34, Theorem 9.7]. ∎

###  2.4 Proof of Corollary 1.23 (ii)

We give two proofs of Corollary 1.23 (ii). The first proof goes as follows.
Let $x,y$ be two critical points of $\alpha,$ one of index $1$ and the other
of index $2;$ such a pair exists by Poincaré duality of Novikov homology. As
in Corollary 1.21 (ii) there is a double point surgery $L$ with $HF^{*}$
obstructed. In dimension $n=3$ the bounding cochains on $L$ are critical
points of the disc potential function. The relevant homology classes come from
those of strips between $x,y$ with the right orientation. For this we use
again Fukaya, Oh, Ohta Ono’s result; the number of holomorphic discs
homologous to this strip with multiplicity one is conserved under the surgery.
We prove that the number of holomorphic discs homologous to the multiplicity-
one strip is exactly one. It is certainly an integer because $T^{*}Q$ is
spherically positive. Passing to
${\mathbin{\mathbb{Z}}}/p{\mathbin{\mathbb{Z}}}$ we see that it is nonzero
modulo $p,$ so it is really one. As the Floer complex of $(Q,Q^{\alpha})$ and
the Novikov complex of $\alpha$ are quasi-isomorphic, the number of gradient
trajectories from $y$ to $x$ is also one. We can then use the cancellation
theorem in Morse theory [43, Theorem 5.4]; that is, we can cancel out $x,y$
with each other by changing $\alpha$ in its de-Rham cohomology class.
Repeating this process we get a Morse $1$-form with no critical point at all,
as claimed.

For the other proof we use the geometrisation theorem on $3$-manifolds, from
which we deduce:

######  Proposition 2.8.

Every closed orientable $3$-manifold $Q$ satisfies one of the following five
statements: (i) $\pi_{1}Q$ is finite; (ii) $Q\cong S^{1}\times S^{2};$ (iii)
$Q\cong{\mathbin{\mathbb{R}}}P^{3}\\#{\mathbin{\mathbb{R}}}P^{3};$ (iv) there
exists a finite cover $P\to Q$ where $P$ is the mapping torus of an
orientation-preserving diffeomorphism $T^{2}\to T^{2};$ or (v) $\pi_{1}Q$ has
a nonabelian free subgroup.

###### Proof.

We suppose that $\pi_{1}Q$ is infinite and prove that one of (ii)–(v) holds;
we shall refer to the book [12] for those results about $3$-manifolds which we
use, without tracing back their origins in the literature. As $\pi_{1}Q$ is
infinite, $Q$ has no Riemannian metric of positive constant sectional
curvature, so $Q$ is not spherical [12, §1.7]. According to Theorem 1.11.1 of
[12], either one of (ii)–(iv) above holds; or $\pi_{1}Q$ has no finite-index
solvable subgroup. It thus remains to show that, if $Q$ is an infinite
$3$-manifold group with no finite-index solvable subgroup, then it must have a
nonabelian free subgroup.

The first step in this direction is to note that [12, p50, (C.6)] asserts that
our assumption on the fundamental group implies that one of the following
(a)–(d) holds: (a) $Q$ is reducible [12, p3]; (b) $Q$ is a hyperbolic manifold
and $\pi_{1}Q$ is a linear group over $\overline{\mathbin{\mathbb{Q}}}$ [12,
p51, (C.7)]; (c) $Q$ is a Seifert manifold and $\pi_{1}Q$ is a linear group
over ${\mathbin{\mathbb{Z}}}$ [12, p52, (C.10) and (C.11)]; or (d) $Q$ is a
Haken manifold with an incompressible torus and (v) holds [12, p58, (C.24)].
Moreover:

  * •

In Case (a) it is well known that either (ii) holds or $Q$ is a connected sum
of two closed $3$-manifolds $P_{1},P_{2}$ neither of which is a sphere. In the
latter case $\pi_{1}P_{1},\pi_{1}P_{2}$ are both nontrivial (because
Poincaré’s conjecture is true). If $\\#\pi_{1}P_{1}=\\#\pi_{1}P_{2}=2$ then
$P_{1}\cong P_{2}\cong{\mathbin{\mathbb{R}}}P^{3}$ and (iii) holds. Otherwise,
$\pi_{1}Q\cong\pi_{1}P_{1}*\pi_{1}P_{2}$ has so many elements that (v) holds
as follows: one of $\\{\pi_{1}P_{1},\pi_{1}P_{2}\\}$ has at least two
nontrivial elements $x,y$ and the other of $\\{\pi_{1}P_{1},\pi_{1}P_{2}\\}$
has at least one nontrivial element $z$ so there is a nonabelian free group
generated by $xz,yz.$

  * •

In Cases (b) and (c) we use Tits’ theorem [56, Theorem 1]: in characteristic
$0$ every linear group has either a finite-index subgroup or a nonabelian free
subgroup (there is also a positive-characteristic version, Theorem 4.11, which
we shall use later). Since we have supposed that $\pi_{1}Q$ has no finite-
index solvable subgroup, it follows that (v) holds in Cases (b) and (c).

This completes the proof. ∎

Proposition 2.8 implies Corollary 1.23 (ii) as follows:

######  Corollary 2.9.

Let $Q$ be a closed orientable $3$-manifold with $\pi_{1}Q$ having no
nonabelian free subgroup. Then every nonzero element of
$H^{1}(Q,{\mathbin{\mathbb{R}}})$ may be represented by a nowhere-vanishing
closed $1$-form on $Q.$

###### Proof.

By Proposition 2.8 one of (i)–(iv) holds. In Cases (i) and (iii) we have
$H^{1}(Q,{\mathbin{\mathbb{R}}})=0$ and there is nothing to prove. In Case
(ii) we have $Q\cong S^{1}\times S^{2}$ and letting $x$ be the coordinate of
$S^{1}$ we have
$H^{1}(Q,{\mathbin{\mathbb{R}}})\cong{\mathbin{\mathbb{R}}}[dx],$ which has
certainly the desired property. In Case (iv) the mapping torus $P$ is
determined by the monodromy matrix $M\in SL_{2}({\mathbin{\mathbb{Z}}})$
acting upon
$H^{1}(T^{2},{\mathbin{\mathbb{Z}}})\cong{\mathbin{\mathbb{Z}}}^{2}$ which we
can classify by direct computation into the following three cases; (a) there
is an integer $k>0$ such that $M^{k}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix};$
(b) $M^{2}$ is conjugate in $GL_{2}({\mathbin{\mathbb{Z}}})$ to
$\begin{pmatrix}1&l\\\ 0&1\end{pmatrix}$ for some
$l\in{\mathbin{\mathbb{Z}}}\setminus\\{0\\};$ or (c) $M$ has two distinct real
eigenvalues, neither equal to $1.$ We study these respectively as follows:

  * •

In Case (a) we can rechoose the finite cover $P\to Q$ so that $P=T^{3},$ from
which we get a natural embedding $H^{1}(P,{\mathbin{\mathbb{R}}})\subseteq
H^{1}(T^{3},{\mathbin{\mathbb{R}}})={\mathbin{\mathbb{R}}}[dx]\oplus{\mathbin{\mathbb{R}}}[dy]\oplus{\mathbin{\mathbb{R}}}[dz]$
where $x,y,z$ are the three coordinates of $T^{3}=S^{1}\times S^{1}\times
S^{1}.$ This expression implies the desired property of
$H^{1}(P,{\mathbin{\mathbb{R}}}).$

  * •

In Case (b) we can rechoose the finite cover $P\to Q$ so that $M$ will have
two eigenvalues both equal to one. Trivialising $Q$ over two intervals in
$S^{1}$ and applying to them the Mayer–Vietoris sequence, we get an
isomorphism

$H^{1}(P,{\mathbin{\mathbb{R}}})/{\mathbin{\mathbb{R}}}[dx]\to
H^{1}(T^{2},{\mathbin{\mathbb{R}}})^{M}$ (2.7)

where $[dx]$ is obtained from the projection $x:P\to S^{1}$ and the right-hand
side is the $M$-invariant subspace of $H^{1}(T^{2},{\mathbin{\mathbb{R}}}).$
As $M$ has two eigenvalues both equal to one, there is a coordinate
$y:T^{2}\to S^{1}$ such that $dy$ is invariant under $M$ and lifts to $P.$
Then by (2.7) we have
$H^{1}(P,{\mathbin{\mathbb{R}}})={\mathbin{\mathbb{R}}}[dx]\oplus{\mathbin{\mathbb{R}}}[dy],$
in which $H^{1}(Q,{\mathbin{\mathbb{R}}})$ has certainly the desired property.

  * •

In Case (c) there is again an isomorphism of the form (2.7) but the right-hand
side vanishes because $M$ has no eigenvalue equal to one. So
$H^{1}(P,{\mathbin{\mathbb{R}}})={\mathbin{\mathbb{R}}}[dx],$ in which
$H^{1}(Q,{\mathbin{\mathbb{R}}})$ has certainly the desired property.

This completes the proof. ∎

###  2.5 Proof of Proposition 1.34

We begin by recalling the first part of Proposition 1.34 and proving it:

######  Proposition 2.10.

If $\pi_{1}Q$ is trivial, the derived category $D{\mathbin{\cal F}}(T^{*}Q)$
is generated by the single object supported on $Q.$

###### Proof.

Since $Q$ intersects $T_{q}^{*}Q$ transversely at a single point, it follows
that $HF^{*}(Q,T_{q}^{*}Q)$ is one-dimensional over $\Lambda.$ Since
$\pi_{1}Q$ is trivial we see, using the Yoneda functor (1.3), that every
single-degree object of $H{\mathbin{\cal F}}(T^{*}Q)$ is isomorphic up to
shift to $Q\otimes V$ for some finite-dimensional $\Lambda$-vector space,
which completes the proof. ∎

We use an algebraic lemma by the first author [5, Lemma A.4], which goes back
to the first general results on Arnold’s nearby Lagrangian problem [26]; the
original statement is over ${\mathbin{\mathbb{Z}}}_{2}$ but it applies also to
$\Lambda$:

######  Lemma 2.11.

Let ${\mathbin{\cal A}}$ be a ${\mathbin{\mathbb{Z}}}$-graded $\Lambda$-linear
triangulated $A_{\infty}$ category generated by one object
$\mathbf{a}\in{\mathbin{\cal A}}$ whose endomorphisms are supported in
nonnegative degrees. Then every object ${\mathbf{b}}\in{\mathbin{\cal A}}$
with $\hom^{*}({\mathbf{b}},{\mathbf{b}})$ supported in nonnegative degrees is
isomorphic up to shift to a direct sum of copies of $\mathbf{a}$. ∎

By Proposition 2.10 we can take ${\mathbin{\cal A}}=D{\mathbin{\cal
F}}(T^{*}Q)$ and $\mathbf{a}=Q$ in this lemma, which implies the second part
of Proposition 1.34. ∎

###  2.6 Proof of Theorem 1.12 (i)

Let $U\subseteq X$ be a Weinstein neighbourhood of $Q$ and regard it as
embedded in $T^{*}Q$ with a standard Calabi–Yau structure. Suppose first that
$\pi_{1}Q$ is trivial. Let ${\mathbf{b}}\in H{\mathbin{\cal F}}(U)$ be an
object supported near a closed special Lagrangian $L\subset T^{*}Q$ relative
to a standard Calabi–Yau structure. By Proposition 1.25 (or as mentioned in
Remark 1.26) the object ${\mathbf{b}}$ is nonzero and by Proposition 1.24 the
graded group $HF^{*}({\mathbf{b}},{\mathbf{b}})$ is supported in nonnegative
degrees. So by Proposition 1.34 this object is isomorphic to some object
supported on $Q$ with the trivial bounding cochain. Hence by Theorem 1.27 (i)
we find $L=Q$ as we want.

If $\pi_{1}Q$ is not trivial but finite, taking the universal cover
$f:T^{*}P\to T^{*}Q$ we find $f^{*}L=P$ so $L=Q.$ ∎

###  2.7 Proof of Lemma 1.36

Suppose first that the Calabi–Yau structure near $Q\subset T^{*}Q$ is real
analytic. For each $k=1,\cdots,n-1$ we shall make a compactly-supported
Hamiltonian perturbation $L$ of $T_{q}^{*}Q$, which is special Lagrangian of
phase $-2\pi k$ near $q\in T^{*}Q$. To start with, take a
${\mathbin{\mathbb{C}}}$-linear isomorphism
$T_{q}(T^{*}Q)\cong{\mathbin{\mathbb{C}}}^{n}$ which maps $T_{q}Q$ to
${\mathbin{\mathbb{R}}}^{n}\subseteq{\mathbin{\mathbb{C}}}^{n},$ $T_{q}^{*}Q$
to $(i{\mathbin{\mathbb{R}}})^{n},$ the symplectic form to
$(i/2)\sum_{j=1}^{n}dz_{j}\wedge d\bar{z}_{j}$ and the holomorphic volume form
to $\bigwedge_{j=1}^{n}dz_{j}$ where $z_{1},\cdots,z_{n}$ are the coordinates
of ${\mathbin{\mathbb{C}}}^{n}.$ Take
$\theta_{1},\cdots,\theta_{k}\in(-\pi/2,0)$ and
$\theta_{k+1},\cdots,\theta_{n}\in(0,\pi/2)$ with
$\theta_{1}+\cdots+\theta_{n}=0$ so that there is a linear special Lagrangian
$\Pi:=\\{(e^{i\theta_{1}}x_{1},\cdots,e^{i\theta_{n}}x_{n})\in{\mathbin{\mathbb{C}}}^{n}:x_{1},\cdots,x_{n}\in{\mathbin{\mathbb{R}}}\\}.$

There are two ways of going from this special Lagrangian plane in
$T_{q}(T^{*}Q)$ to special Lagrangian near $q\in T^{*}Q.$ The first is to use
Cartan–Kähler’s theorem as Harvey and Lawson do [28, Chapter III, the proof of
Theorem 5.5], which we can do because we have supposed that our Calabi–Yau
structure is real analytic. The second is to use the implicit function theorem
again as Harvey and Lawson do [28, Chapter III, the proof of Corollary 2.14].

In either case we get a special Lagrangian near $q\in T^{*}Q$, which we cut
off outside $Q\subset T^{*}Q$ to obtain a compactly-supported Hamiltonian
perturbation $L$ of $T_{q}^{*}Q.$ So there is a neighbourhood $U$ of $Q\subset
T^{*}Q$ in which $L$ is special. Grade the Lagrangians $L$ and $\Pi$ by phase
$0$ and the Lagrangians $T_{q}^{*}Q$ and $(i{\mathbin{\mathbb{R}}})^{n}$ by
phase $n\pi/2$ at $q.$ Then by the definition of $\Pi$ there is a
${\mathbin{\mathbb{Z}}}$-graded Hamiltonian isotopy between $L[k]$ and
$T_{q}^{*}Q.$ Suppose now that ${\mathbf{b}}\in H{\mathbin{\cal F}}(U)$ is
supported near a closed special Lagrangian $N.$ By varying
$\theta_{1},\cdots,\theta_{k}\in(-\pi/2,0)$ and
$\theta_{k+1},\cdots,\theta_{n}\in(0,\pi/2)$ if need be, we may suppose $N\neq
L[k]$ near $N\cap L[k].$ Applying Theorem 1.29 to $N,L[k]$ we see that
$HF^{*}(L[k],{\mathbf{b}})$ is supported in degrees $1,\cdots,n-1.$ So
$HF^{*}(L,{\mathbf{b}})$ is supported in degrees $1+k,\cdots,n-1+k.$ This
holds for every $k\in\\{1,\cdots,n-1\\},$ and
$\bigcap_{k=1}^{n-1}[1+k,n-1+k]=\\{n\\}$ as we want.

Finally, if the Kähler form $\omega$ near $Q\subset T^{*}Q$ is not real
analytic we can approximate it by a real analytic one (after taking a Kähler
potential) and find a compactly-supported Hamiltonian diffeomorphism
$\phi:T^{*}Q\to T^{*}Q$ such that $\phi_{*}\omega$ is real analytic. Applying
to this the result above, we get a corresponding neighbourhood $V$ of
$Q\subset T^{*}Q.$ Let $U:=\phi^{*}V,$ ${\mathbf{b}}\in H{\mathbin{\cal
F}}(U)$ an object supported near a closed special Lagrangian (relative to
$\omega$) and $\phi_{*}{\mathbf{b}}\in H{\mathbin{\cal F}}(V)$ the push-
forwarded object. Then $HF^{*}(T_{q}^{*}Q,\phi_{*}{\mathbf{b}})$ is supported
in the single degree $n.$ But as $\phi$ is compactly supported, we have
$HF^{*}(T_{q}^{*}Q,\phi_{*}{\mathbf{b}})\cong
HF^{*}(T_{q}^{*}Q,{\mathbf{b}}),$ which completes the proof. ∎

###  2.8 Proof of Theorem 1.12 (ii)

Let $U$ be a Weinstein neigbourhood of $Q\subset X,$ embedded in $T^{*}Q$ with
a standard Calabi–Yau structure near $Q\subset T^{*}Q.$ We shall make $U$ so
small that we can use McLean’s theorem below. Let $L\subset U$ be a closed
$HF^{*}$-unobstructed special Lagrangian, and ${\mathbf{b}}\in{\mathbin{\cal
F}}(U)$ an object supported near $L.$ Then ${\mathbf{b}}$ is again nonzero,
and by Lemma 1.36, of single degree. Applying Theorem 1.38 to ${\mathbf{b}}$
we get a finite cover $f:P\to Q,$ a harmonic $1$-form $\alpha$ on $P$ and a
unitary local system $E$ on $P^{\alpha}$ such that
$HF^{0}(E,f^{*}{\mathbf{b}})\neq 0.$ Also as in Theorem 1.38, with $U$ small
enough, we can make $\alpha$ as small as we like. So we can apply to $\alpha$
the McLean theorem, to make a Hamiltonian perturbation of $P^{\alpha}$ which
is special Lagrangian; and by an abuse of notation we may suppose that
$P^{\alpha}$ itself is a special Lagrangian. Then by Theorem 1.27 (ii) this
$P^{\alpha}$ is an irreducible component of $f^{*}L$ and $L$ is unbranched.

If $\pi_{1}Q$ is abelian then by Theorem 1.38 we can do without the finite
cover of $Q.$ In particular, we have $HF^{0}(E,{\mathbf{b}})\neq 0,$ which
implies $E\cong{\mathbf{b}}$ in $H^{0}{\mathbin{\cal F}}(U)$ if ${\mathbf{b}}$
is irreducible. ∎

##  3 General Floer Theory

In this section we recall and prove the relevant facts in general Floer
theory. We begin in §§3.1 and 3.2 by constructing the curved Fukaya category
${\mathbin{\cal C}}(X).$ In §§3.3–3.5 we discuss bounding cochains, units and
duality of Floer cohomology groups. In §§3.6–3.8 we introduce the wrapped
Fukaya category ${\mathbin{\cal W}}(X)$ by a certain localisation.

###  3.1 Fukaya Categories

We begin with the definition of ambient spaces in which we do Floer theory.
The example we have in mind is cotangent bundles, but we make a more general
definition:

######  Definition 3.1.

A compact symplectic manifold $(X,\omega)$ with contact boundary is a
symplectic manifold which is given a $1$-form $\lambda$ near
${\partial}X\subseteq X$ such that $d\lambda=\omega$ and the vector field
$v_{\lambda}$ given by $v_{\lambda}\lrcorner\,\omega=\lambda$ is outwards
pointing on ${\partial}X$.

In these circumstances there exists near ${\partial}X\subseteq X$ a unique
smooth function $r_{\lambda}$ with values in $(0,1]$ such that

$r_{\lambda}|_{{\partial}X}=1\text{ and
}dr_{\lambda}(v_{\lambda})=r_{\lambda}.$ (3.1)

We say that an almost complex structures $J$ is compatible with the contact
boundary if it is compatible with $\omega$ in the usual sense and if
$\lambda=Jdr_{\lambda}$ near ${\partial}X.$

A choice of global $1$-form $\lambda$ with $d\lambda=\omega$ satisfying the
conditions in Definition 3.1 is called a Liouville structure on $X$, and we
say in this case that $X$ is a Liouville domain.

######  Remark 3.2.

At the moment we can take $X$ to be a compact symplectic manifold without
boundary. But in §3.7 we shall take $X$ to be a Liouville domain, and in §4 we
shall take $X$ to be a constant-radius disc subbundle of $T^{*}Q,$ $Q$ a
closed Riemannian manifold, and $\lambda$ to be the canonical $1$-form on
$T^{*}Q.$

We consider the following class of Lagrangians:

######  Definition 3.3.

Let $(X,\omega)$ and $\lambda$ be as in Definition 3.1, and $\iota:\hat{L}\to
X$ a Lagrangian immersion. By an abuse of notation we shall usually mention
only the image $L:=\iota(\hat{L})\subseteq X$ in place of the immersion map,
and say that $L$ is compact if so is $\hat{L}.$

We say that $L$ is generic if its every self-intersection point is a
transverse double point outside ${\partial}X.$ We say that $L$ has Legendrian
collar if $\iota({\partial}\hat{L})\subseteq{\partial}X,$ if
$\iota^{*}\lambda=0$ near ${\partial}\hat{L}\subseteq\hat{L}$ and if
$\iota^{*}r_{\lambda}$ has regular value $1$ (this is automatically true if
$\hat{L}$ is closed).

Suppose now that $L_{0}$ and $L_{1}$ are two Lagrangians in $X$. We say that
they are generic if each of them is generic and they intersect each other
transversely, outside their self-intersection points and their boundaries.

We recall the notion of ${\mathbin{\mathbb{Z}}}$-graded Lagrangians in
${\mathbin{\mathbb{Z}}}$-graded symplectic manifolds. There are several ways
of doing this, as discussed in [49]. We work with the following definition in
which we use almost complex structures on the ambient symplectic manifold,
because this leads directly to the definition of special Lagrangians (but this
section is topological so we do not have to take integrable complex
structures):

######  Definition 3.4.

A ${\mathbin{\mathbb{Z}}}$-grading on $(X,\omega)$ is choice up to homotopy of
a nowhere vanishing section $\Omega^{2}$ of the squared canonical bundle of
$X$ relative to some $\omega$-compatible almost complex structure $J.$

We note that a choice of ${\mathbin{\mathbb{Z}}}$-grading on $(X,\omega)$
exists if and only if $2c_{1}(X)=0\in H^{2}(X,{\mathbin{\mathbb{Z}}})$ where
$c_{1}(X)$ denotes the first Chern class of $X.$

Given a quadratic volume form $\Omega^{2}$ representing a
${\mathbin{\mathbb{Z}}}$-grading on $X$, we associate to each Lagrangian
immersion $\iota:L\to X$ a unique smooth function
$\phi:L\to{\mathbin{\mathbb{R}}}/2\pi{\mathbin{\mathbb{Z}}}$ with $\mathop{\rm
Im}e^{-i\phi}\iota^{*}\Omega^{\otimes 2}=0$.

######  Definition 3.5.

We say that a Lagrangian immersion $\iota:L\to X$ is
${\mathbin{\mathbb{Z}}}$-graded if it is equipped with a lift of $\phi$ to the
universal cover
${\mathbin{\mathbb{R}}}\to{\mathbin{\mathbb{R}}}/2\pi{\mathbin{\mathbb{Z}}}.$
We say that $\iota$ is special of phase $\phi$ with respect to
$\Omega^{\otimes 2}$ if $\phi$ is locally constant. The shift operator $[1]$
maps the ${\mathbin{\mathbb{Z}}}$-graded Lagrangian $(\iota,\phi)$ to
$(\iota,\phi+2\pi).$

A Lagrangian $L$ admits a ${\mathbin{\mathbb{Z}}}$-grading if and only if the
Maslov class in $H^{1}(L,{\mathbin{\mathbb{Z}}})$, which is represented by
$\phi,$ vanishes.

We now consider the case in which we have a ${\mathbin{\mathbb{Z}}}$-graded
Lagrangian immersion $\iota:\hat{L}\to L\subset X$, which is generic. We
consider the fibre product

$L\times_{X}L:=\\{(x,y)\in\hat{L}\times\hat{L}:\iota(x)=\iota(y)\\}$ (3.2)

which consists of the diagonal $\\{(x,x)\in\hat{L}\times\hat{L}\\},$
identified with $\hat{L},$ and the self-intersection pairs in $\hat{L}.$

Associated to each self intersection $(x_{1},x_{2})\in L\times_{X}L$ is an
integer (the Maslov index), defined as follows: denote by $x\in X$ the image
of $x_{i}$, and take a ${\mathbin{\mathbb{C}}}$-linear isomorphism
$T_{x}X\cong{\mathbin{\mathbb{C}}}^{n}$ which maps $\omega$ to
$\frac{i}{2}\sum_{j=1}^{n}dz_{j}\wedge d\bar{z}_{j},$ $T_{x_{1}}L$ to
${\mathbin{\mathbb{R}}}^{n}\subseteq{\mathbin{\mathbb{C}}}^{n}$ and
$T_{x_{2}}L$ to
$\\{(e^{i\theta_{1}}x_{1},\cdots,e^{i\theta_{n}}x_{n})\in{\mathbin{\mathbb{C}}}^{n}:x_{1},\cdots,x_{n}\in{\mathbin{\mathbb{R}}}\\}$
for some $\theta_{1},\cdots,\theta_{n}\in(0,\pi).$ These
$\theta_{1},\cdots,\theta_{n}$ are unique up to order, so we can define

$\mu_{L}(x_{1},x_{2}):=\frac{1}{2\pi}(2\theta_{1}+\cdots+2\theta_{n}+\phi_{1}-\phi_{2})\in{\mathbin{\mathbb{Z}}}$
(3.3)

where $\phi_{i}$ is the phase of $L$ at $x_{i}$ for $i=1,2.$

######  Definition 3.6.

The Maslov index function

$\mu_{L}:L\times_{X}L\to{\mathbin{\mathbb{Z}}}$ (3.4)

identically vanishes on the diagonal and is given by $\mu_{L}(x,y)$ for a
self-intersection pair $(x,y)\in L\times_{X}L$.

Suppose now that $L_{0},L_{1}\subseteq X$ are two generic Lagrangians. The
fibre product $L_{0}\times_{X}L_{1}$ is then a finite set, and since
$L_{0},L_{1}$ are ${\mathbin{\mathbb{Z}}}$-graded we get again a Maslov index
function $\mu_{L_{0},L_{1}}:L_{0}\times_{X}L_{1}\to{\mathbin{\mathbb{Z}}}$, by
the same formula as above.

######  Remark 3.7.

Since $\theta_{1},\cdots,\theta_{n}\in(0,\pi)$ it follows by (3.3) that

$\mu_{L}(x_{1},x_{2})-n<\frac{\phi_{1}-\phi_{2}}{2\pi}<\mu_{L}(x_{1},x_{2}).$
(3.5)

In particular, if $\phi_{1}-\phi_{2}$ is nearly zero then
$\mu_{L}(x_{1},x_{2})\in[0,n]$ and if $\phi_{1}=\phi_{2}$ then
$\mu_{L}(x_{1},x_{2})\in[1,n-1].$

We consider then the following class of local systems:

######  Definition 3.8.

We choose a complete algebraically-closed Novikov field $\Lambda$ in the sense
of Definition A.5 (depending on a choice of the ground ring ${\mathbf{k}}$).
By a unitary local system on a manifold $\hat{L}$ we mean a local system on
$\hat{L}$ of nonzero finite-rank $\Lambda$ modules, equipped with valuations
that are preserved by parallel transports.

We recall that a valuation on a module $E$ over a valued field $K$ is a
function

${\mathbf{v}}:E\to{\mathbin{\mathbb{R}}}\cup\\{\infty\\},$ (3.6)

so that

  * •

for $a\in E$, ${\mathbf{v}}a=+\infty$ if and only if $a=0$,

  * •

for $a$ and $b$ in $E$, we have
${\mathbf{v}}(a+b)\geqslant\min\\{{\mathbf{v}}a,{\mathbf{v}}b\\}$.

  * •

for $c\in K$ and $a\in E$, we have ${\mathbf{v}}(c\cdot
a)={\mathbf{v}}c+{\mathbf{v}}a$ (${\mathbf{v}}$ denotes the valuations on both
$K$ and $E$).

######  Remark 3.9.

We denote by $\Lambda^{0}\subseteq\Lambda$ the Novikov ring, the ring of
integers in $\Lambda.$ An equivalent way to describe a unitary local system is
to consider the (free) $\Lambda^{0}$ local system $E^{0}$ obtained by
considering those elements of strictly positive valuation. The local system
$E$ can be recovered as $E\cong E^{0}\otimes_{\Lambda^{0}}\Lambda$.

We introduce now the notion of branes:

######  Definition 3.10.

Let $(X,\omega)$ and $\lambda$ be as in Definition 3.6 and ${\mathbf{k}}$ and
$\Lambda$ be as in Definition 3.8. If ${\mathbf{k}}$ has characteristic $\neq
2,$ suppose we are given on $X$ a background class for relative spin
structures on Lagrangians (c.f. [22]). By a brane on $X$ we mean a pair
$(L,E)$ where:

  * •

$L\subset X$ is a compact ${\mathbin{\mathbb{Z}}}$-graded generically-immersed
Lagrangian with Legendrian collar, given also a relative spin structure if
${\mathbf{k}}$ has characteristic $\neq 2;$ and

  * •

$E$ is a unitary local system on $\hat{L}$ in the notation of Definition 3.3.

We shall often write just $E$ in place of $(L,E).$

In our Fukaya category construction we have to choose first a countable set of
branes:

######  Hypothesis 3.11.

Let $((X,\omega),\lambda,{\mathbf{k}},\Lambda)$ be as in Definition 3.10, and
consider a countable set of branes ${\mathbin{\cal C}}(X)$ with the property
that any two distinct underlying Lagrangians of elements of ${\mathbin{\cal
C}}(X)$ are generic in the sense of Definition 3.3.

We introduce also the homologically perturbed Floer complexes:

######  Definition 3.12.

Let ${\mathbin{\cal C}}(X)$ be as in Hypothesis 3.11. The homologically
perturbed Floer complex associated to a pair of elements $(L_{0},E_{0})$ and
$(L_{1},E_{1})$ of ${\mathbin{\cal C}}(X)$, is the
${\mathbin{\mathbb{Z}}}$-graded finite-dimensional $\Lambda$-vector space

$CF^{*}(E_{0},E_{1}):=H^{*}(L_{0}\times_{X}L_{1},\hom(E_{0},E_{1})[-\mu_{L_{0},L_{1}}])$
(3.7)

where $E_{0},E_{1}$ are pulled back by the projections from
$L_{0}\times_{X}L_{1}$ to $L_{0},L_{1}$ respectively; and
$[-\mu_{L_{0},L_{1}}]$ denotes the degree shift by (3.4).

######  Remark 3.13.

The definition (3.7) is actually the result of applying the filtered
homological perturbation Lemma [22, Theorem A], to the singular or de Rham
model in the literature. More precisely, in any construction of Lagrangian
Floer cohomology groups we have to choose a chain model for Floer complexes
$CF^{*}_{model}(E_{0},E_{1})$ with $L_{0}=L_{1}.$ Fukaya, Oh, Ohta and Ono
[22] and Akaho and Joyce [10] both take the singular chain model. More
precisely, they take for every $(E_{0},E_{1})$ a subcomplex of
$L_{0}\times_{X}L_{1}$ relative to the boundary, which depends upon many other
data in their construction. Fukaya [20] takes the de-Rham model with
${\mathbf{k}}={\mathbin{\mathbb{R}}}.$

The homologically perturbed Floer complexes above have natural valuations
which are inherited from those of $E_{0}$ and $E_{1}$ and the infimum
valuation

${\mathbf{v}}\phi=\inf_{a\neq 0}{\mathbf{v}}\phi(a)-{\mathbf{v}}a.$ (3.8)

on the space of $\Lambda$-linear maps between valued vector spaces.

######  Theorem 3.14.

Let ${\mathbin{\cal C}}(X)$ be as in Hypothesis 3.11 and let
$n:={\frac{1}{2}}\dim X.$ Then ${\mathbin{\cal C}}(X)$ has the structure of a
${\mathbin{\mathbb{Z}}}$-graded $\Lambda^{0}$-linear $A_{\infty}$ category
whose morphism spaces are given by (3.7) and which satisfies the following
properties:

  * •

the $A_{\infty}$ structure is curved in such a way that if
$(L,E)\in{\mathbin{\cal C}}(X),$ if $L$ has no self-intersection point of
index $2$, and if $L$ does not bound any nonconstant pseudoholomorphic disc,
the curvature ${\mathfrak{m}}^{0}$ of $C^{*}(L,E)$ is zero;

  * •

the $A_{\infty}$ structure is filtered in such a way that for
$E_{0},\cdots,E_{k}\in{\mathbin{\cal C}}(X)$ with $E_{k+1}=E_{0}$ the
structure map

${\mathfrak{m}}^{k}:\prod_{i=0}^{k-1}CF^{*}(E_{i},E_{i+1})\to
CF^{*}(E_{0},E_{k})$ (3.9)

is of the form $\sum{\mathfrak{m}}^{k}_{\gamma}T^{\gamma},$ with
$\\{\gamma\\}$ a sequence in $[0,\infty)$ tending to $\infty;$

  * •

for a pair $E_{0},E_{1}\in{\mathbin{\cal C}}(X)$, the differential

${\mathfrak{m}}^{1}:CF^{*}(E_{0},E_{1})\to CF^{*}(E_{0},E_{1})$ (3.10)

has leading term ${\mathfrak{m}}^{1}_{0}=0;$ and

  * •

for a triple $E_{0},E_{1},E_{2}\in{\mathbin{\cal C}}(X)$ which are supported
on the same Lagrangian, the product

${\mathfrak{m}}^{2}:CF^{*}(E_{0},E_{1})\times CF^{*}(E_{1},E_{2})\to
CF^{*}(E_{0},E_{2})$ (3.11)

has leading term ${\mathfrak{m}}^{2}_{0}$ agreeing with the usual cup product

$H^{*}(L\times_{X}L,\hom(E_{0},E_{1}))\otimes
H^{*}(L\times_{X}L,\hom(E_{1},E_{2}))\to
H^{*}(L\times_{X}L,\hom(E_{0},E_{2}))$ (3.12)

on those components with $\mu_{L,L}=0$, and vanishes otherwise.

###  3.2 Proof of Theorem 3.14

We begin by taking the singular chain model of (3.7) as in Remark 3.13:

######  Definition 3.15.

Let $L$ be a manifold, and $E$ a unitary local system. The smooth singular
chain complex $C_{*}(L,E)$ is the $\Lambda^{0}$ module generated by pairs
$(\sigma,\epsilon)$ where $\sigma$ is a smooth singular $i$-simplex in $L,$
and $\epsilon$ a (flat) section of $\sigma^{*}E$ of nonnegative valuation,
modulo the relation

$a(\sigma,\epsilon)=(\sigma,a\epsilon)\text{ for }a\in\Lambda^{0}.$ (3.13)

The differential $d$ on $C_{*}(L,E)$ is defined by

$d(\sigma,\epsilon)=\sum_{j=0}^{i}(-1)^{j}(\sigma\circ\delta^{i},\epsilon\circ\delta^{i})$
(3.14)

where $\delta^{j}$ is the $j^{\rm th}$ face inclusion map.

If $L$ has boundary, we define
$C_{*}(L,{\partial}L;E):=C_{*}(L,E)/C_{*}({\partial}L,E).$ We regard this, by
Poincaré duality, as a cochain complex.

In our circumstances, to $(L_{i},E_{i})\in{\mathbin{\cal C}}(X),$ $i=0,1,$ we
assign

$C_{*}(L_{0}\times_{X}L_{1},{\partial}(L_{0}\times_{X}L_{1});\hom(E_{0},E_{1})).$
(3.15)

We introduce now the moduli spaces of pseudoholomorphic discs. Let
$L_{0},\cdots,L_{k}\subseteq X$ be $k+1$ underlying Lagrangians in
${\mathbin{\cal C}}(X).$

######  Definition 3.16.

We denote by ${\mathbin{\cal M}}(L_{1},\cdots,L_{k};L_{0})$ the compactified
moduli space of finite-area stable pseudoholomorphic discs in $X$ with $k+1$
boundary points, whose corresponding segments are ordered counterclockwise and
lifted to $L_{0},\cdots,L_{k}.$

For $N>0$ we denote by ${\mathbin{\cal M}}_{\leqslant\nobreak
N}(L_{1},\cdots,L_{k};L_{0})$ the subset of this of those elements of area
$\leqslant\nobreak N,$ and by ${\mathbin{\cal
M}}_{N}(L_{1},\cdots,L_{k};L_{0})$ that of area $N.$

We recall the maximum principle on Riemann surfaces with Neumann boundary
conditions:

######  Lemma 3.17 (Abouzaid–Seidel [8, Lemma 7.2]).

There exists in $X\setminus{\partial}X$ a compact set which contains the image
of every nonconstant element of ${\mathbin{\cal
M}}(L_{1},\cdots,L_{k};L_{0}).$ ∎

So by Gromov’s compactness theorem we have:

######  Corollary 3.18.

For every $N>0$ the space ${\mathbin{\cal M}}_{\leqslant\nobreak
N}(L_{1},\cdots,L_{k};L_{0})$ is compact. ∎

Suppose now that $L_{0},\cdots,L_{k}$ are given unitary local systems
$E_{0},\cdots,E_{k}$ respectively. Note that for every $i=1,\cdots,k$ there is
an evaluation map

${\rm ev}_{i}:{\mathbin{\cal M}}(L_{1},\cdots,L_{k};L_{0})\to
L_{i-1}\times_{X}L_{i}$ (3.16)

where $L_{-1}:=L_{k}.$ So there is on ${\mathbin{\cal
M}}(L_{1},\cdots,L_{k};L_{0})$ a sequence of local systems

$({\rm ev}_{0}^{*}E_{k},{\rm ev}_{0}^{*}E_{0};{\rm ev}_{1}^{*}E_{0},{\rm
ev}_{1}^{*}E_{1};\cdots;{\rm ev}_{k}^{*}E_{k-1},{\rm ev}_{k}^{*}E_{k}).$
(3.17)

There are also parallel transport isomorphisms ${\rm ev}_{i}^{*}E_{i}\cong{\rm
ev}_{i+1}^{*}E_{i}$ for $i=0,\cdots,k$ where ${\rm ev}_{k+1}:={\rm ev}_{0};$
and there is a natural map of local systems

$\bigotimes_{i=1}^{k}\hom({\rm ev}_{i}^{*}E_{i-1},{\rm
ev}_{i}^{*}E_{i})\to\hom({\rm ev}_{0}^{*}E_{0},{\rm ev}_{0}^{*}E_{k}).$ (3.18)

We sketch now the $A_{\infty}$ structure construction for Theorem 3.14. We
parametrise by positive integers the underlying Lagrangians in ${\mathbin{\cal
C}}(X)$ and denote by ${\mathbin{\cal L}}_{N}$ the set of the first $N$
Lagrangians. Let $\Gamma_{N}\subseteq[0,\infty)$ be the monoid of areas of
pseudoholomorphic discs with boundary on ${\mathbin{\cal L}}_{N}.$ Following
Akaho and Joyce [10, Definition 3.21] we assign to $\gamma\in\Gamma_{N}$ the
integer

$\|\gamma\|:=\max\\{i:\gamma=\gamma_{1}+\cdots+\gamma_{l}\text{ for some
}\gamma_{1},\cdots,\gamma_{l}\in\Gamma\setminus\\{0\\}\\}+[\gamma]$ (3.19)

where $[\gamma]$ denotes the greatest integer $\leqslant\nobreak\gamma;$ if
$\gamma=0$ this means $\|0\|:=0.$ We write $A_{K}:=A_{K,0}$ in Akaho–Joyce’s
notation for $A_{K,0}$ categories.

Here are the next five steps:

  * (I)

For each positive integer $K$ construct a curved $A_{K}$ category
${\mathbin{\cal C}}_{K}^{\rm sing}({\mathbin{\cal L}}_{N}).$ This consists of
some operations ${\mathfrak{m}}^{k}_{\gamma}$ for
$k-1+\|\gamma\|\leqslant\nobreak K,$ $\gamma\in\Gamma_{N},$ which satisfy the
$A_{\infty}$ relations in this range. The morphisms between
$E_{0},E_{1}\in{\mathbin{\cal C}}_{K}^{\rm sing}({\mathbin{\cal L}}_{N})$ is,
with respect to the leading-term differential, a subcomplex of (3.15).

  * (II)

Construct a homotopy equivalence from ${\mathbin{\cal C}}_{K}^{\rm
sing}({\mathbin{\cal L}}_{N})$ to the $A_{K}$ reduction of ${\mathbin{\cal
C}}_{K+1}^{\rm sing}({\mathbin{\cal L}}_{N})$ to those
${\mathfrak{m}}^{k}_{\gamma}$ with $k-1+\|\gamma\|\leqslant\nobreak K$.

  * (III)

Construct a curved filtered $A_{\infty}$ category ${\mathbin{\cal C}}^{\rm
sing}({\mathbin{\cal L}}_{N})$ on ${\mathbin{\cal C}}_{0}^{\rm
sing}({\mathbin{\cal L}}_{N})$ whose $A_{k}$ reduction is obtained from the
data of (i) the $A_{k}$ structure on ${\mathbin{\cal C}}_{K}^{\rm
sing}({\mathbin{\cal L}}_{N})$ and (ii) the homotopy equivalences between
${\mathbin{\cal C}}_{i}^{\rm sing}({\mathbin{\cal L}}_{N})$ and
${\mathbin{\cal C}}_{i+1}^{\rm sing}({\mathbin{\cal L}}_{N})$ for $0\leq i\leq
k$.

  * (IV)

Construct a homotopy equivalence from ${\mathbin{\cal C}}^{\rm
sing}({\mathbin{\cal L}}_{N})$ to the subcategory of ${\mathbin{\cal C}}^{\rm
sing}({\mathbin{\cal L}}_{N+1})$ of objects supported in ${\mathbin{\cal
L}}_{N}.$

  * (V)

Construct a curved filtered $A_{\infty}$ category ${\mathbin{\cal C}}^{\rm
sing}(X)$ as a colimit of ${\mathbin{\cal C}}^{\rm sing}({\mathbin{\cal
L}}_{N}).$

We do (I) as follows. By Corollary 3.18 every moduli space of the form
${\mathbin{\cal M}}_{\leqslant\nobreak K-k+1}(L_{1},\cdots,L_{k};L_{0})$ is a
finite union of Kuranishi spaces in the sense of Fukaya, Oh, Ohta and Ono
[22]. To any two Lagrangians $L_{0},L_{1}$ we assign a countable set
${\mathbin{\cal X}}_{K}(L_{0},L_{1})$ of smooth simplices to
$L_{0}\times_{X}L_{1},$ which is closed under the face maps, so that:

  * •

the subcomplex of (3.15) generated by those elements supported on
${\mathbin{\cal X}}_{K}(L_{0},L_{1})$ is quasi-isomorphic to the entire
complex under the inclusion map; and

  * •

given $(x_{1},\cdots,x_{k})\in\prod_{i=1}^{k}{\mathbin{\cal
X}}_{j_{i}}(L_{i-1},L_{i})$ there is a chosen multivalued perturbation of

${\mathbin{\cal M}}_{\gamma}(L_{1},\cdots,L_{k};L_{0})\times_{\rm
ev}(x_{1}\times\cdots\times x_{k}),$ (3.20)

where ${\rm ev}:=({\rm ev}_{1},\cdots,{\rm ev}_{k}),$ whenever

$\sum j_{i}+\|\gamma\|-k+1\leq K,$ (3.21)

and a triangulation of this perturbation whose image under ${\rm ev}_{0}$ lies
in ${\mathbin{\cal X}}_{K}(L_{0},L_{k}).$

Here some of $(x_{1},\cdots,x_{k})$ may be the same, so the perturbation of
(3.20) will depend upon $(x_{1},\cdots,x_{k}).$

Recall that an element of (3.15) supported on a singular chain $x$ is a
section $\epsilon$ of $x^{*}\hom(E_{0},E_{1}).$ Using also the map (3.18),
Akaho–Joyce’s theorem [10, Theorem 6.1], and the homological perturbation
argument from [10, Section 7], we get the structure maps

${\mathfrak{m}}^{k}_{\gamma}:\prod_{i=1}^{k}C_{K}^{\rm sing}(E_{i-1},E_{i})\to
C_{K}^{\rm sing}(E_{0},E_{k}).$ (3.22)

Here we can take the coefficients in ${\mathbf{k}}$ because we are in the
spherically positive case in the sense of Fukaya, Oh, Ohta and Ono [24].

We follow now the remaining key steps in the construction by Akaho and Joyce
[10]. All the statements referred in what follows are taken from this paper.
The proof that the structure maps given by (3.22) satisfy the $A_{K}$
relations is the same as that of Proposition 7.2. Step (II) is achieved by the
same arguments as in §§8–10. Step (III) is entirely algebraic as in Definition
11.1. Step (IV) is achieved by the same arguments as in Theorem 11.2 (a). We
can then construct for every $N$ a curved filtered $A_{\infty}$ category with
objects supported on ${\mathbin{\cal L}}_{N+1}$ that has ${\mathbin{\cal
C}}^{\rm sing}({\mathbin{\cal L}}_{N})$ as a subcategory; this is done in the
same way as in the construction in Definition 11.1, applied to $A_{\infty}$ in
place of $A_{K}.$ Step (V) is then achieved by taking the union of these.

Applying to ${\mathbin{\cal C}}^{\rm sing}(X)$ the homological perturbation
lemma [10, Corollary 11.4] we get on ${\mathbin{\cal C}}(X)$ an $A_{\infty}$
structure which satisfies the required properties. In particular, the fibre
products above have leading term equal to the intersection products of
singular chains, which are, by the usual Poincaré duality, cup products as we
want. ∎

###  3.3 Bounding Cochains

We recall the definition of bounding cochains in the $A_{\infty}$ category
${\mathbin{\cal C}}(X)$:

######  Definition 3.19.

For $E_{0},E_{1}\in{\mathbin{\cal C}}(X)$ we define

$CF^{*}(E_{0},E_{1})^{+}\subseteq CF^{*}(E_{0},E_{1})$ (3.23)

to be the subset of elements of valuation $>0.$ For
$E_{0},\cdots,E_{k}\in{\mathbin{\cal C}}(X)$ and for

$b:=(b_{0},\cdots,b_{k-1})\in\prod_{i=0}^{k-1}C^{1}(E_{i},E_{i+1})^{+}$ (3.24)

we denote by

${\mathfrak{m}}^{k}_{b}:\prod_{i=0}^{k-1}CF^{*}(E_{i},E_{i+1})\to
CF^{*}(E_{0},E_{k})$ (3.25)

the deformation of ${\mathfrak{m}}^{k}$ by $b$. For $E\in{\mathbin{\cal
C}}(X)$ we call $b\in CF^{1}(E,E)$ a bounding cochain if
${\mathfrak{m}}^{0}_{b}=0.$

We introduce now the uncurved Fukaya category ${\mathbin{\cal F}}_{\rm nc}(X)$
where we write ‘nc’ to mean noncompact, to distinguish it from ${\mathbin{\cal
F}}(X)$ in Definition 3.27 below:

######  Definition 3.20.

We define the ${\mathbin{\mathbb{Z}}}$-graded $\Lambda$-linear uncurved
$A_{\infty}$ category ${\mathbin{\cal F}}_{\rm nc}(X)$ as follows:

  * •

we define an object of ${\mathbin{\cal F}}_{\rm nc}(X)$ to be a triple
${\mathbf{b}}=(L,E,b)$ where $(L,E)$ is an object of ${\mathbin{\cal C}}(X)$
with $L$ a closed Lagrangian, and $b$ a bounding cochain in $CF^{*}(E,E);$

  * •

for ${\mathbf{b}}_{i}=(E_{i},b_{i})\in{\mathbin{\cal F}}(X),$ $i=0,1,$ we
define the morphism space between them to be $CF^{*}(E_{0},E_{1});$ and

  * •

for $k=1,2,3,\cdots,$ ${\mathbf{b}}_{i}=(E_{i},b_{i})\in{\mathbin{\cal
F}}_{\rm nc}(X),$ $i=0,\cdots,k,$ ${\mathbf{b}}_{k+1}={\mathbf{b}}_{0},$ we
take (3.25) to be the $A_{\infty}$ structure map.

We define the ${\mathbin{\mathbb{Z}}}$-graded $\Lambda$-linear associative
category $H{\mathbin{\cal F}}_{\rm nc}(X)$ as the associated cohomological
category, which has the same objects as ${\mathbin{\cal F}}(X),$ and in which
the morphism space between two objects ${\mathbf{b}}_{0},{\mathbf{b}}_{1}$ is
denoted by

$HF^{*}({\mathbf{b}}_{0},{\mathbf{b}}_{1}):=H^{*}(CF^{*}(E_{0},E_{1}),{\mathfrak{m}}^{1}_{b}).$
(3.26)

###  3.4 Units

We state a theorem on units in the Fukaya categories:

######  Theorem 3.21.

The $A_{\infty}$ categories ${\mathbin{\cal C}}(X),{\mathbin{\cal F}}_{\rm
nc}(X)$ are both homotopy unital. For $(L,E)\in{\mathbin{\cal C}}(X)$ the
homotopy unit in $CF^{*}(E,E)$ has leading term equal to the fundamental class
of $L.$

######  Remark 3.22.

This is proved for embedded Lagrangians by Fukaya, Oh, Ohta and Ono [22,
§7.3], which extends to immersed Lagrangians; we shall omit the details.

If ${\mathbf{k}}$ contains ${\mathbin{\mathbb{R}}}$ we can make the
$A_{\infty}$ structures strictly unital, proved also for embedded Lagrangians
by Fukaya [20], taking the de-Rham model for (3.7). The strict unitality fails
if we take the singular chain model as we do in the proof below of Theorem
3.14, because the perturbations of Kuranishi spaces of the form (3.20) need
not be compatible with forgetting marked points.

The homotopy unitality [22, §7.3] is proved by taking fundamental chains of
the underlying Lagrangians in ${\mathbin{\cal C}}(X)$ and constructing in a
consistent way the homotopies between those perturbed Kuranishi spaces for
which some of $x_{1},\cdots,x_{k}$ in (3.20) are the fundamental chains, and
those obtained from the forgetful maps.

We recall the notion of zero objects in $H{\mathbin{\cal F}}(X)$ and give a
corollary to Theorem 3.21.

######  Definition 3.23.

We call ${\mathbf{b}}\in H{\mathbin{\cal F}}_{\rm nc}(X)$ a zero object if the
unit of $HF^{*}({\mathbf{b}},{\mathbf{b}})$ is zero.

######  Corollary 3.24.

If ${\mathbf{b}}=(L,E,b)\in H{\mathbin{\cal F}}_{\rm nc}(X)$ is a zero object,
the Lagrangian $L\subset X$ has at least one self-intersection point of index
$-1.$

###### Proof.

Since $HF^{*}({\mathbf{b}},{\mathbf{b}})$ is ${\mathbin{\mathbb{Z}}}$-graded
it follows that the unit in $HF^{0}({\mathbf{b}},{\mathbf{b}})$ may be
cancelled out only by elements of $CF^{-1}(E,E),$ which come from self-
intersection points of $L$ of index $-1.$ ∎

From Corollary 3.24 we see immediately that:

######  Corollary 3.25.

Every object ${\mathbf{b}}\in H{\mathbin{\cal F}}_{\rm nc}(X)$ supported on an
embedded Lagrangian is nonzero.

Finally we consider special Lagrangians. We take such $\Omega^{2},J$ as in
Definition 3.4 so that we can speak of special Lagrangians in $X.$ We consider
in fact nearly-special Lagrangians which have phase sufficiently close (in the
$C^{0}$ sense) to a constant function. Combining Corollary 3.24 with Remark
3.7 we get:

######  Corollary 3.26.

Every object ${\mathbf{b}}\in H{\mathbin{\cal F}}_{\rm nc}(X)$ supported on a
nearly special Lagrangian is nonzero.

###  3.5 Duality

We introduce now the Fukaya category which we shall use most often:

######  Definition 3.27.

We denote by ${\mathbin{\cal F}}(X)\subseteq{\mathbin{\cal F}}_{\rm nc}(X)$
the full subcategory of objects supported on closed Lagrangians.

The following two results are Poincaré duality results for $H{\mathbin{\cal
F}}(X).$

######  Theorem 3.28.

For every ${\mathbf{b}}\in H{\mathbin{\cal F}}(X)$ there exists a
nondegenerate pairing

$HF^{*}({\mathbf{b}},{\mathbf{b}})\otimes
HF^{n-*}({\mathbf{b}},{\mathbf{b}})\to\Lambda.$ (3.27)

In particular:

######  Corollary 3.29.

For every nonzero object ${\mathbf{b}}\in H{\mathbin{\cal F}}(X)$ we have
$HF^{n}({\mathbf{b}},{\mathbf{b}})\neq 0.$ ∎

######  Remark 3.30.

This is also proved already for embedded Lagrangians by Fukaya, Oh, Ohta and
Ono [22, Theorem E].

We give a sketch proof of Theorem 3.28. The key notion we use is filtered
$A_{\infty}$ bimodules:

######  Definition 3.31.

Let $C$ be a ${\mathbin{\mathbb{Z}}}$-graded $\Lambda$-linear filtered curved
$A_{\infty}$ algebra. By a bimodule over $C$ we mean a
${\mathbin{\mathbb{Z}}}$-graded $\Lambda$ module ${\mathbin{\cal P}}$ equipped
with operations

${\mathfrak{m}}_{{\mathbin{\cal P}}}^{r|1|s}:\left(C[1]\right)^{\otimes
r}\otimes{\mathbin{\cal P}}\otimes\left(C[1]\right)^{\otimes
s}\to{\mathbin{\cal P}},$ (3.28)

$r,s=0,1,2,\cdots,$ which satisfy the bimodule version of the $A_{\infty}$
equation. An $A_{\infty}$ bimodule bimodule is filtered if it is equipped with
a valuation, so that $A_{\infty}$ bimodule operations do not decrease
valuation.

######  Example 3.32.

The $A_{\infty}$ algebra $C$ itself has the structure of a diagonal bimodule
with operations

${\mathfrak{m}}^{r|1|s}(x_{1},\ldots,x_{r},p,y_{1},\ldots,y_{s})=(-1)^{r+\sum\deg
x_{i}}{\mathfrak{m}}^{r+1+s}(x_{1},\ldots,x_{r},p,y_{1},\ldots,y_{s}),$ (3.29)

$r,s=0,1,2,\cdots;$ $x_{1},\cdots,x_{r},p,y_{1},\cdots,y_{r}\in C.$

######  Example 3.33.

If ${\mathbin{\cal P}}$ is an $A_{\infty}$ bimodule over $C$ then its dual
${\mathbin{\cal P}}^{\vee}:=\hom({\mathbin{\cal P}}^{\vee},\Lambda)$ has the
structure of a bimodule given by

${\mathfrak{m}}_{{\mathbin{\cal
P}}^{\vee}}(y_{1},\ldots,y_{s},\psi,x_{1},\ldots,x_{r})(p)=(-1)^{1+\deg\psi}\psi\left({\mathfrak{m}}_{{\mathbin{\cal
P}}}(x_{1},\ldots,x_{r},p,y_{1},\ldots,y_{s})\right),$ (3.30)

$r,s=0,1,2,\cdots;$ $x_{1},\cdots,x_{r},y_{1},\cdots,y_{r}\in C,$
$\psi\in{\mathbin{\cal P}}^{\vee}$ and $p\in{\mathbin{\cal P}}.$

Next, let ${\mathbf{b}}=(L,E,b)\in{\mathbin{\cal F}}(X)$ and apply to the
single brane $(L,E)$ the proof of Theorem 3.14. Then we get a filtered curved
$A_{\infty}$ algebra $C:={\mathbin{\cal C}}^{\rm sing}({\mathbin{\cal
L}}_{1})$ where ${\mathbin{\cal L}}_{1}$ is as in the proof of Theorem 3.14;
we call this an algebra, not a category, because there is only one object. We
use the same notation for the diagonal bimodule, and denote by $C^{\vee}$ its
dual bimodule. The ${\mathfrak{m}}^{1}$ of $C$ has leading term equal to the
usual differential, whose cohomology group is

$CF^{*}(E,E):=H^{*}(L\times_{X}L,\hom(E,E)[-\mu_{L}]).$ (3.31)

######  Proposition 3.34.

There exists a ${\mathbin{\mathbb{Z}}}$-graded $\Lambda$-linear natural
isomorphism of graded vector spaces

$CF^{*}(E,E)\cong CF^{*}(E,E)^{\vee}[-n].$ (3.32)

###### Proof.

If ${\mathbf{k}}$ has characteristic $\neq 2$ then the condition that $L$ be
relatively spin includes the data of an orientation of the domain $\hat{L}$ of
the immersed Lagrangian $L\subset X$, so there is a Poincaré duality
isomorphism

$H^{*}(\hat{L},\hom(E,E))\times H^{*}(\hat{L},\hom(E,E))\to\Lambda[-n]$ (3.33)

where we do not have to change the local-system term because
$\hom(E,E)^{\vee}\cong\hom(E,E).$ For an off-diagonal self-intersection pair
$(x,y)\in L\times_{X}L$ there is also a pairing

$(x,y)\otimes(y,x)\to\Lambda[-n]$ (3.34)

which completes the proof. ∎

Then we have:

######  Theorem 3.35.

There exists a bimodule homomorphism

$\Psi:C\to C^{\vee}[-n]$ (3.35)

such that the map $\Psi^{0|1|0}:C\to C^{\vee}[-n]$ has leading term equal to
(3.32).

###### Sketch Proof.

Recall that $C$ is by construction an inverse limit of $A_{K}$ algebras
$C_{K}:={\mathbin{\cal C}}^{\rm sing}_{K}({\mathbin{\cal L}}_{1})$ again in
the notation of the proof of Theorem 3.14. Regard this as a diagonal bimodule
with operations of the form

${\mathfrak{m}}^{r|1|s}=\sum{\mathfrak{m}}^{r|1|s}_{\gamma}T^{\gamma},$ (3.36)

$r,s\geqslant 0;r+s+\|\gamma\|\leq K.$ For each $K$ we construct a bimodule
map $\Psi_{K}:C_{K}\to C^{\vee}_{K}$ which consists of a collection of

$\Psi_{K}^{r|1|s}:C_{K}^{\times r}\times C_{K}\times C_{K}^{\times s}\to
C^{\vee}_{K}$ (3.37)

indexed by nonnegative integers $r$ and $s$. We regard the output as an input
by duality, so that we have to construct a compatible collection of maps

$C_{K}\times C_{K}^{\times r}\times C_{K}\times C_{K}^{\times s}\to\Lambda.$
(3.38)

In order to construct this map, we use the stable moduli spaces of
pseudoholomorphic discs with $r+s+2$ boundary points, ordered as in Figure
3.1: the marked point on the right corresponds to the input, and on the left
to the output.

$\Delta^{\vee}$$\Delta$$x_{r}$$x_{r-1}$$y_{1}$$y_{2}$$x_{1}$$y_{s}$ Figure
3.1: Moduli spaces of discs for the duality map

We choose inductively a family of perturbation data for the fibre products of
the form

${\mathbin{\cal M}}_{\gamma}^{r+s+2}\times_{\rm ev}(p\times
x_{1}\times\cdots\times x_{r}\times
p\times\tau_{1}\times\cdots\times\tau_{s})$ (3.39)

where ${\rm ev}$ denotes the evaluation maps at the boundary marked points
ordered along the boundary, the singular chains $x_{i}$ lies in
${\mathbin{\cal X}}_{j_{i}}$ while $y_{i}$ lies ${\mathbin{\cal
X}}_{j^{\prime}_{i}}$, and $p$ and $q$ respectively lie in ${\mathbin{\cal
X}}_{j}$ and ${\mathbin{\cal X}}_{j^{\prime}}$, and we assume that

$|\gamma|+\sum j_{i}+j+\sum j^{\prime}_{i}+j^{\prime}+r+s\leq K.$ (3.40)

We make these compatible with the previous perturbations of (3.20) and those
for the lower-order maps relative to $r,s.$ Using homological perturbation as
in the construction of the $A_{K}$ structure on ${\mathbin{\cal C}}_{K}^{\rm
sing}({\mathbin{\cal L}}_{N})$, we obtain the desired map of
$A_{K}$-bimodules.

From the $A_{K}$ bimodule maps of the form (3.37) we construct an $A_{\infty}$
bimodule by the same inductive procedure used to construct the Fukaya
category. The algebraic framework for this is given by Fukaya, Oh, Ohta and
Ono [22, §5.2], used originally for the deformation invariance of Floer
cohomology groups; which is in fact a homotopy equivalence of two bimodules
[22, Theorem D]. Our circumstances are easier because we only need to work
with Novikov field coefficients as in Definition 3.8; whereas Fukaya, Oh, Ohta
and Ono consider also Novikov ring coefficients.

Recall that we have constructed an $A_{K}$ homotopy equivalence between the
reduction of $C_{K+1}={\mathbin{\cal C}}^{\rm sing}_{K+1}({\mathbin{\cal
L}}_{1})$ to an $A_{K}$ algebra and $C_{K}.$ From this we get an $A_{K+1}$
algebra $C_{K,K+1}$ which is $A_{K+1}$ homotopy equivalent to $C_{K+1}$ and
whose reduction to an $A_{K}$ algebra is strictly isomorphic to $C_{K}.$ The
$A_{K+1}$ bimodule map $C_{K+1}\to C^{\vee}_{K+1}$ which induces an $A_{K+1}$
bimodule map (over the algebra $C_{K,K+1}$)

$C_{K,K+1}\to C^{\vee}_{K,K+1}.$ (3.41)

which we denote still by $\Psi_{K}.$ In the previous stage we have constructed
an $A_{K}$ bimodule map

$\Psi_{K-1}:C_{K-1,K}\to C^{\vee}_{K-1,K}.$ (3.42)

This is homotopic to the $A_{K}$ reduction of (3.41) by a choice of homotopy
between the defining perturbation data. Using homological perturbation, and
proceeding by induction, we get the desired system of bimodule maps. ∎

Applying to Theorem 3.35 the $A_{\infty}$ bimodule version of Whitehead’s
theorem [22, Theorem 5.2.35] we get:

######  Corollary 3.36.

$\Psi$ is a homotopy equivalence of bimodules.∎

We complete now the proof of Theorem 3.28. The bimodule homotopy equivalence
$\Psi:C\to C^{\vee}[-n]$ induces for every bounding cochain $b\in C^{1}$
another bimodule homotopy equivalence

$\Psi_{b}:C\to C^{\vee}[-n],$ (3.43)

relative to the $b$-deformed $A_{\infty}$ structure, such that for
$x_{1},\cdots,x_{r},p,y_{1},\cdots,y_{s}\in CF^{*}(E,E)$ the term

$\Psi^{r|1|s}_{b}(x_{1},\ldots,x_{r},p,y_{1},\ldots,y_{s})$ (3.44)

is given by

$\sum\Psi(b,\ldots,b,x_{1},b,\ldots,b,x_{r},b,\ldots,b,p,b,\ldots,b,y_{1},b,\ldots,b,y_{s},b,\ldots,b).$
(3.45)

Hence, passing to the cohomology groups, we get an isomorphism

$HF^{*}({\mathbf{b}},{\mathbf{b}})\cong
HF^{*}({\mathbf{b}},{\mathbf{b}})^{\vee}[-n]$ (3.46)

as asserted. ∎

###  3.6 Hamiltonian Continuations

We say that a symplectomorphism $\phi:X\to X$ is boundary-linear if it agrees
near ${\partial}X$ with the Hamiltonian diffeomorphism given by a constant
multiple of $r_{\lambda}.$

######  Lemma 3.37.

If $\phi:X\to X$ is a boundary-linear graded symplectomorphism which preserves
the object set of ${\mathbin{\cal C}}(X)$ then there exists an equivalence

$\phi_{*}:H{\mathbin{\cal F}}_{\rm nc}(X)\to H{\mathbin{\cal F}}_{\rm nc}(X)$
(3.47)

such that for every ${\mathbf{b}}=(L,E,b)\in{\mathbin{\cal F}}_{\rm nc}(X)$
the object $\phi_{*}{\mathbf{b}}$ is given by a certain bounding cochain
$\phi_{*}b$ for $(\phi_{*}L,\phi_{*}E)\in{\mathbin{\cal C}}(X).$ The functor
$\phi_{*}$ is compatible with composition in the sense that, for a pair $\phi$
and $\psi$ of boundary-linear symplectormorphisms, the composite
$\psi_{*}\circ\phi_{*}$ is naturally isomorphic to $(\psi\circ\phi)_{*}.$

###### Proof.

Note that a bounding cochain on $E\in{\mathbin{\cal C}}(X)$ relative to an
almost-complex structure $J$ induces a bounding cochain on $\phi_{*}E$
relative to $\phi_{*}J.$ Changing almost-complex structures in this way
induces an isomorphism of Floer cohomology groups, as in Theorem 11.2 (a) of
Akaho–Joyce’s paper [10]; so there is an isomorphism

$HF^{*}(E,E)\cong HF^{*}(\phi_{*}E,\phi_{*}E).$ (3.48)

More precisely, we can do this because $\phi$ is boundary linear and we can
use the maximum principle for pseudoholomorphic curves bounded by noncompact
Lagrangians. ∎

######  Remark 3.38.

We expect that the cohomology level functor (3.47) lifts to an $A_{\infty}$
functor ${\mathbin{\cal F}}(X)\to{\mathbin{\cal F}}(X).$ But the proof would
be substantially more involved, and we shall not need it for our purpose.

There is more to say if the symplectomorphism $\phi$ in Lemma 3.37 is a
Hamiltonian diffeomorphism:

######  Theorem 3.39.

If $\phi$ is a boundary-linear Hamiltonian diffeomorphism which preserves the
object set of ${\mathbin{\cal C}}(X),$ there exists a natural transformation
$\kappa_{\phi}$ from $\phi_{*}$ to the identity functor such that for any
${\mathbf{b}}_{1},{\mathbf{b}}_{2}\in{\mathbin{\cal F}}_{\rm nc}(X)$ there is
a commutative diagram

${HF^{*}({\mathbf{b}}_{1},{\mathbf{b}}_{2})}$${HF^{*}(\phi_{*}{\mathbf{b}}_{1},\phi_{*}{\mathbf{b}}_{2})}$${HF^{*}(\phi_{*}{\mathbf{b}}_{1},{\mathbf{b}}_{2}).}$$\scriptstyle{\circ\kappa_{\phi}{\mathbf{b}}_{1}}$$\scriptstyle{\phi_{*}}$$\scriptstyle{\kappa_{\phi}{\mathbf{b}}_{2}\circ}$
(3.49)

If either ${\mathbf{b}}_{1}$ or ${\mathbf{b}}_{2}$ are supported on a compact
Lagrangian, then these arrows are isomorphisms. If there is another such
Hamiltonian diffeomorphism $\psi_{*}:X\to X$ then

$\kappa_{\psi\phi}{\mathbf{b}}=\kappa_{\psi}(\phi_{*}{\mathbf{b}})\circ\kappa_{\phi}{\mathbf{b}},\;\;\;{\mathbf{b}}\in{\mathbin{\cal
F}}(X).$ (3.50)

###### Proof.

Such a natural transformation is obtained from Theorem 13.6 of Akaho–Joyce’s
paper [10]; the proof is not given in loc. cit. but it follows from the
arguments of Fukaya, Oh, Ohta and Ono [22].

The fact that the natural transformation for $\phi\circ\psi$ is given by the
composite may be proved by using Theorem 11.2 (b) of Akaho and Joyce. ∎

###  3.7 Exact Lagrangians

Suppose now that the Liouville $1$-form $\lambda$ is given globally on $X$ so
that we can speak of exact Lagrangians in $(X,\lambda).$ Denote by
${\mathbin{\cal F}}_{{\rm ex}}(X)\subseteq{\mathbin{\cal F}}(X)$ the full
subcategory of those objects $(L,E)$ for which:

  * •

the Lagrangian $L\subset X$ is embedded and exact; that is, $L$ has no self-
intersection point and $\lambda|_{L}$ is an exact $1$-form;

  * •

the local system $E$ is of the form ${\mathbin{\cal
E}}\otimes_{\mathbf{k}}\Lambda$ for some ${\mathbf{k}}$ local system
${\mathbin{\cal E}}$ on $L;$ and

  * •

$(L,E)$ is given the trivial bounding cochain, which exists because $L$ is
embedded and does not bound any noncontant pseudoholomorphic disc.

For $(L,E)\in{\mathbin{\cal F}}_{{\rm ex}}(X)$ take any primitive function
$h:L\to{\mathbin{\mathbb{R}}}$ with $dh=\lambda|_{L}.$ For
$(L_{0},E_{0}),(L_{1},E_{1})\in{\mathbin{\cal F}}_{{\rm ex}}(X)$ we define the
action function

${\mathbin{\cal A}}:L_{0}\times_{X}L_{1}\to{\mathbin{\mathbb{R}}}$ (3.51)

by ${\mathbin{\cal A}}(x):=h_{0}(x)-h_{1}(x)$ where $h_{0},h_{1}$ are the
primitives of $(L_{0},E_{0}),(L_{1},E_{1})$ respectively. We recall:

######  Proposition 3.40.

Let $(L_{0},E_{0}),\cdots,(L_{k},E_{k})\in{\mathbin{\cal F}}_{{\rm ex}}(X)$
and $h_{0},\cdots,h_{k}$ their primitives respectively. As in the proof of
Theorem 3.14 take on $X$ a compatible almost-complex structure $J,$ define the
moduli space ${\mathbin{\cal M}}(L_{1},\cdots,L_{k};L_{0})$ and define the
evaluation map

${\mathbin{\cal
M}}(L_{1},\cdots,L_{k};L_{0})\to\prod_{i=0}^{k-1}(L_{i}\times_{X}L_{i+1})\times(L_{0}\times_{X}L_{k}).$
(3.52)

Then the fibre of this map over a point $(x_{1},\cdots,x_{k};x_{0})$ consists
of $J$-holomorphic discs of area

$\left(\sum_{i=1}^{k}{\mathbin{\cal A}}(x_{i})\right)-{\mathbin{\cal
A}}(x_{0}).$ (3.53)

###### Proof.

In the cyclic notation, modulo $k+1$ for indices, the area is equal to

$\begin{split}\sum_{i=1}^{k+1}\int_{x_{i-1}}^{x_{i}}\lambda&=\sum_{i=1}^{k+1}h_{i-1}(x_{i})-h_{i-1}(x_{i-1})=\sum_{i=1}^{k+1}h_{i-1}(x_{i})-\sum_{i=0}^{k}h_{i}(x_{i})\\\
&=\left(\sum_{i=1}^{k}h_{i-1}(x_{i})-h_{i}(x_{i})\right)+h_{k}(x_{0})-h_{0}(x_{0})\end{split}$
(3.54)

which is equal to (3.53). ∎

This implies that the $A_{\infty}$ structure of ${\mathbin{\cal F}}_{{\rm
ex}}(X)$ is reducible to ${\mathbf{k}}$:

######  Corollary 3.41.

${\mathbin{\cal F}}_{{\rm ex}}(X)$ has the structure of a
${\mathbf{k}}$-linear $A_{\infty}$ category whose hom space between two
objects $(L_{0},E_{0}),(L_{1},E_{1})$ is given by

$CF_{\mathbf{k}}^{*}(E_{0},E_{1}):=H^{*}(L_{0}\times_{X}L_{1},\hom_{\mathbf{k}}(E_{0},E_{1})[-\mu_{L_{0},L_{1}}])$
(3.55)

and whose structure maps ${\mathfrak{m}}_{{\rm ex}}^{k}$ are defined by the
same disc counts but without the $T^{\gamma}$ factor. The $\Lambda$-linear
extension of this $A_{\infty}$ category is isomorphic to that of the
subcategory ${\mathbin{\cal F}}(X)$ with the same objects, under the maps

$T^{\mathbin{\cal
A}}:CF_{\mathbf{k}}^{*}(E_{0},E_{1})\otimes_{\mathbf{k}}\Lambda\to
CF^{*}(E_{0},E_{1}),\;\;\;E_{0},E_{1}\in{\mathbin{\cal F}}_{{\rm ex}}(X),$
(3.56)

defined as the multiplication by $T^{{\mathbin{\cal A}}(x)}$ on the component
of each $x\in L_{0}\times_{X}L_{1}.$ ∎

###### Proof.

This follows from the proof of Theorem 3.14; the point is that once we have
fixed the data $(x_{1},\cdots,x_{k};x_{0})$ relevant to the $A_{\infty}$
operation in Equation (3.22), the $\gamma$ in the structure map
${\mathfrak{m}}^{k}_{\gamma}$ will be equal to (3.53). ∎

We show now that $H{\mathbin{\cal F}}_{{\rm ex}}(X)$ is equivalent to the
exact Fukaya category which has appeared before in the literature; we need
only an equivalence of the two cohomology categories, not a quasi-equivalence
of the $A_{\infty}$ categories:

######  Proposition 3.42.

The cohomology category $H{\mathbin{\cal F}}_{{\rm ex}}(X)$ is equivalent to
that with the same objects but constructed in the more direct way by Seidel
[51].

###### Proof.

We show first that for $E_{0},E_{1}\in{\mathbin{\cal F}}_{{\rm ex}}(X)$ the
two cohomology groups $HF^{*}(E_{0},E_{1}),$ one given by the virtual method
of Fukaya, Oh, Ohta and Ono [22] and the other by the direct method of Seidel
[51], are isomorphic. If $E_{0}$ and $E_{1}$ have the same underlying
Lagrangian $L$, we make a Hamiltonian perturbation which intersects the
original one transversely. By this we may suppose that the two underlying
Lagrangians intersect each other transversely, which does not change the
isomorphism classes of both the cohomology groups given by the virtual and
direct methods. In the virtual method we give $X$ an almost-complex structure
which need not be regular, and in the direct method we give $X$ some generic
almost-complex structures which we can take to be regular because the two
Lagrangians are mutually transverse. Changing almost-complex structures
induces an isomorphism of the corresponding cohomology groups, as in Theorem
11.2 (a) of Akaho–Joyce’s paper [10], so we conclude that the morphism spaces
in the two categories are isomorphism.

We show next that for $E_{0},E_{1},E_{2}\in{\mathbin{\cal F}}_{{\rm ex}}(X)$
the two product maps of the form

$HF^{*}(E_{0},E_{1})\times HF^{*}(E_{1},E_{2})\to HF^{*}(E_{0},E_{2})$ (3.57)

are the same. By a Hamiltonian perturbation we may again suppose that the
three underlying Lagrangians are pairwise transverse and have no triple
intersection point. As before, in the virtual method we give $X$ an almost-
complex structure which need not be regular, and in the classical perturbation
method it is computed with respect to some generic almost-complex structures
which we can take to be regular. The invariance of the product at the level of
cohomology under the choice of almost complex structure completes the proof. ∎

###  3.8 Wrapped Fukaya Categories

We define the wrapped Fukaya category ${\mathbin{\cal W}}(X)$ by an
$A_{\infty}$ localisation of ${\mathbin{\cal F}}_{\rm nc}(X).$ This idea is
due to Abouzaid and Seidel, contained implicitly in their earlier work [8],
and has by now appeared in the literature [27].

In general, if ${\mathbin{\cal A}}$ is an $A_{\infty}$ category and
$Z\subseteq H^{0}{\mathbin{\cal A}}$ a set of morphisms, there exists a
universal pair consisting of an $A_{\infty}$ category ${\mathbin{\cal
A}}_{Z^{-1}}$ and an $A_{\infty}$-functor ${\mathbin{\cal A}}\to{\mathbin{\cal
A}}_{Z^{-1}}$ such that every element of $Z$ becomes invertible in
$H^{0}{\mathbin{\cal A}}_{Z^{-1}}.$ The key property that we shall use is the
fact that this construction is compatible with passage to triangulated
categories in the sense that there is a natural equivalence of triangulated
categories

$H^{0}\left({\mathbin{\cal A}}_{Z^{-1}}\right)\cong(H^{0}{\mathbin{\cal
A}})_{Z^{-1}}$ (3.58)

where the right-hand side is the localisation of a triangulated category.

The next result is a key computational tool for computing Fukaya categories
from localisations, and also implies the above equivalence of triangulated
categories in this specific context:

######  Proposition 3.43.

Let ${\mathbin{\cal A}}$ be an $A_{\infty}$ category, and $f:H{\mathbin{\cal
A}}\to H{\mathbin{\cal A}}$ an equivalence. Let $\kappa$ be a natural
transformation from $f$ to the identity functor; and ${\mathbin{\cal
A}}_{\kappa^{-1}}$ the localisation of ${\mathbin{\cal A}}$ by the constituent
morphisms $\\{\kappa_{a}\in\hom(fa,a)\\}_{a\in{\mathbin{\cal A}}}$. Then for
any two objects $a,b\in{\mathbin{\cal A}}$ there exist natural isomorphisms

$H{\mathbin{\cal
A}}_{\kappa^{-1}}(a,b)\cong\displaystyle\varinjlim_{i\to\infty}H{\mathbin{\cal
A}}(f^{i}a,b)\cong\\\ \displaystyle\varinjlim_{j\to\infty}H{\mathbin{\cal
A}}(a,f^{-j}b)\cong\displaystyle\varinjlim_{i,j\to\infty}H{\mathbin{\cal
A}}(f^{i}a,f^{-j}b)$ (3.59)

where $f^{-1}:H{\mathbin{\cal A}}\to H{\mathbin{\cal A}}$ is an inverse to
$f.$ Also for $a,b,c\in{\mathbin{\cal A}}$ there exists a commutative diagram

${\displaystyle\varinjlim_{i\to\infty}H{\mathbin{\cal
A}}(f^{i}a,b)\otimes\displaystyle\varinjlim_{j\to\infty}H{\mathbin{\cal
A}}(b,f^{-j}c)}$${\displaystyle\displaystyle\varinjlim_{i,j\to\infty}H{\mathbin{\cal
A}}(f^{i}a,f^{-j}c)}$${\displaystyle H{\mathbin{\cal
A}}_{\kappa^{-1}}(a,b)\otimes H{\mathbin{\cal
A}}_{\kappa^{-1}}(b,c)}$${H{\mathbin{\cal A}}_{\kappa^{-1}}(a,c)}$

$\scriptstyle\sim$

$\scriptstyle\sim$

(3.60)

where the horizontal arrows are products maps and the vertical isomorphisms
are obtained from those of (3.59). ∎

We shall also need the following observation:

######  Lemma 3.44.

Assume that $K\subseteq Z\subseteq H^{0}{\mathbin{\cal A}}$ are two sets of
morphisms such that for each element $\beta\in Z$, there are an element
$\alpha\in K$, an integer $d$ and a morphism $\gamma$ such that

$\beta\circ\gamma=\alpha^{d}.$ (3.61)

Then the natural functor $H{\mathbin{\cal A}}_{K^{-1}}\to H{\mathbin{\cal
A}}_{Z^{-1}}$ is an equivalence.

###### Proof.

It suffices to prove that ${\mathbin{\cal A}}_{K^{-1}}$ satisfies the
universal property of the localisation away from $Z$, i.e. that every morphism
in $Z$ is already invertible in ${\mathbin{\cal A}}_{K^{-1}}$. This is
immediate from Equation (3.61). ∎

We now arrive at the definition of the wrapped category in the contact-type
setting:

######  Definition 3.45.

Let $\Phi$ be a nonempty countable set of Hamiltonians
$\phi:X\to{\mathbin{\mathbb{R}}}$ with $\phi=cr,$ $c>0,$ near ${\partial}X.$
Identify these with the time-one maps they generate, and suppose that if
$\phi\in\Phi$ and $E\in{\mathbin{\cal C}}(X)$ then $\phi_{*}E\in{\mathbin{\cal
C}}(X).$ Then ${\mathbin{\cal W}}(X)$ is defined as the localisation of
${\mathbin{\cal F}}_{\rm nc}(X)$ by the set $\\{\kappa_{\phi}\in
HF^{0}(\phi_{*}{\mathbf{b}},{\mathbf{b}}):\phi\in\Phi,{\mathbf{b}}\in{\mathbin{\cal
F}}_{\rm nc}(X)\\}$ of continuation morphisms.

######  Remark 3.46.

The closed Fukaya category ${\mathbin{\cal F}}(X)$ is embedded naturally as a
full subcategory of ${\mathbin{\cal W}}(X)$ because the continuation morphisms
are invertible in $H^{0}{\mathbin{\cal F}}(X).$

There is also a subcategory $H{\mathbin{\cal W}}_{{\rm ex}}(X)\subseteq
H{\mathbin{\cal W}}(X)$ supported on exact Lagrangians, which is, by
Proposition 3.42, equivalent to that in the literature defined without the
virtual counts.

##  4 Remaining Proofs

In this section we prove Theorem 1.38 and Proposition 1.40. We fix a closed
connected manifold $Q$ and we use the material in the last section with
$X=T^{*}Q.$ To define the Fukaya categories ${\mathbin{\cal F}}(T^{*}Q),$
${\mathbin{\cal W}}(T^{*}Q)$ we do the following:

  * •

We take the background class on $T^{*}Q$ to be the pull-back by the projection
$T^{*}Q\to Q$ of the Stiefel–Whitney class $w_{2}(Q)\in
H^{2}(Q,{\mathbin{\mathbb{Z}}}_{2}).$ So the zero-section $Q\subset T^{*}Q$
has a relatively spin structure, and we give it a natural one which we shall
define shortly below.

  * •

Take the ${\mathbin{\mathbb{Z}}}$-grading of $T^{*}Q$ to be that by the
complexification of a squared real volume form on $Q.$ So there exist
${\mathbin{\mathbb{Z}}}$-gradings of $Q.$ We include in ${\mathbin{\cal
F}}(T^{*}Q)$ the object supported on $Q$ with the trivial rank-one local
system and the trivial bounding cochain.

  * •

We give $T^{*}Q$ the standard Liouville $1$-form $\lambda$ so that $Q$ is an
exact Lagrangian. We fix a point $q\in Q$ and include in ${\mathbin{\cal
W}}(T^{*}Q)$ the object supported on the fibre $T_{q}^{*}Q$ with the trivial
relative spin structure, an arbitrary ${\mathbin{\mathbb{Z}}}$-grading, the
trivial rank-one unitary local system and the trivial bounding cochain.

Here is the natural relative spin structure on $Q$:

######  Lemma 4.1.

The Lagrangian $Q\subset T^{*}Q$ has a relative spin structure such that the
$\pi_{1}Q$ representation on $HF^{*}(Q,T_{q}^{*}Q)$ is trivial. If
${\mathbf{k}}$ has characteristic $2$ this representation is automatically
trivial.

###### Proof.

If ${\mathbf{k}}$ has characteristic $2$ then $Q$ defines first an object of
the exact Fukaya category over ${\mathbin{\mathbb{Z}}}_{2}$ and the $\pi_{1}Q$
representation on $HF^{*}(Q,T_{q}^{*}Q)$ makes sense over
${\mathbin{\mathbb{Z}}}_{2}.$ But this is automatically trivial because
${\mathbin{\mathbb{Z}}}_{2}$ has only one unit.

Suppose now that ${\mathbf{k}}$ has characteristic $\neq 2$ and choose some
relative spin structure on $Q$. Then $Q$ defines an object of the exact Fukaya
category over ${\mathbin{\mathbb{Z}}}$, which then corresponds to a rank-$1$
representation of $\pi_{1}Q$. Since the set of relative spin structures is a
torsor over the set of representations, we can then choose a possibly
different relative spin structure which makes this representation trivial. ∎

We begin in §4.1 with the proof of Proposition 1.40, and then go to the proof
of Theorem 1.38, ending in §4.4. In §§4.2 and 4.3 we prove the following two
lemmata respectively (corresponding to (i), (ii) of Theorem 1.38):

######  Lemma 4.2.

For every closed $1$-form $\alpha$ on $Q$ and for every unitary local system
$E$ on the graph $Q^{\alpha},$ the $\pi_{1}Q$ representation on
$HW^{*}(E,T_{q}^{*}Q)$ is isomorphic to $T^{-[\alpha]}\otimes E$ where $E$ is
identified with the representation of $\pi_{1}Q^{\alpha}\cong\pi_{1}Q.$

With the same $\alpha$ the following holds as well: let $f:(P,p)\to(Q,q)$ be a
finite cover, $\beta:=f^{*}\alpha$ and $F$ a unitary local system on
$P^{\beta};$ then the $\pi_{1}Q$ representation on $HW^{*}(F,T_{p}^{*}P)$ is
isomorphic to $T^{-[\beta]}\otimes F.$

######  Lemma 4.3.

Let $\delta>0$ and let ${\mathbf{b}}\in{\mathbin{\cal F}}(T^{*}Q)$ be a
single-degree object supported within distance $\delta>0$ from the zero-
section $Q.$ Then every one-dimensional subrepresentation of
$HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$ has valuation $\leqslant\nobreak\delta$
(estimated as in Definition 4.7).

###  4.1 Proof of Proposition 1.40

We denote by $f$ either of the two finite covers $P\to Q$ and $T^{*}P\to
T^{*}Q.$ Every object of the Fukaya category ${\mathbin{\cal F}}(T^{*}Q)$ is
given by a pair $(L,E)$ consisting of a Lagrangian $L$ and a local system $E$,
together with a choice $b$ of bounding cochains in the curved filtered
$A_{\infty}$ algebra $C^{*}(L,E)$. The construction of this $A_{\infty}$
algebra itself entails many choices, and the main claim is that these choices
determine the choice to construct $C^{*}(f^{*}L,f^{*}E)$ in such a way that
the pullback map

$C^{*}(L,E)\to C^{*}(f^{*}L,f^{*}E)$ (4.1)

which assigns to a singular chain the sum of all possible inverse images in
$f^{*}L$ is a strict map of curved $A_{\infty}$ algebras (i.e. respects all
operations). From this, it immediately follows that the image of $b$ is a
bounding cochain in $C^{*}(f^{*}L,f^{*}E)$, which determines the functor.

To determine the choices for which (4.1) is a strict map of $A_{\infty}$
algebras, recall that the construction of the left hand side entails choosing
an increasing sequence $\\{{\mathbin{\cal C}}^{*}_{K}(L,E)\\}_{K=1}^{\infty}$
of subcomplexes of the singular chains of a Lagrangian $L$ with coefficients
in a local system $E$. We define $C^{*}_{K}(f^{*}L,f^{*}E)$ to be the complex
generated by all inverse images of chains supporting elements of
${\mathbin{\cal C}}^{*}_{K}(L,E)$. Using the fact that discs are simply
connected, we see that the moduli space of discs in $T^{*}P$ is a cover of the
moduli space of discs in $T^{*}Q$, and hence that a choice of triangulation of
a perturbation for moduli spaces of discs in $T^{*}Q$ determines the
corresponding choice for $T^{*}P$ by passing to a cover. This implies that the
choices on the cover can be chosen so that pullback map

${\mathbin{\cal C}}^{*}_{K}(L,E)\to{\mathbin{\cal C}}^{*}_{K}(f^{*}L,f^{*}E)$
(4.2)

is compatible with the operations constructed from moduli spaces of
holomorphic discs. The same argument shows that the homotopies required to
construct the inverse system are also strictly compatible with passing to
covers, from which the result follows. ∎

###  4.2 Proof of Lemma 4.2

We begin by explaining the notation used in Lemma 4.2:

######  Definition 4.4.

For a closed $1$-form $\alpha$ on $Q$ we denote by $Q^{\alpha}\subset T^{*}Q$
the graph of $\alpha.$ Because
$H_{2}(T^{*}Q,Q^{\alpha};{\mathbin{\mathbb{Q}}})=0$, any local local system
$E$ on ${\mathbin{\mathbb{Q}}}^{\alpha}$ determines an object of the Fukaya
category given by the trivial bounding cochain.

So we may treat $E$ as an object of ${\mathbin{\cal F}}(T^{*}Q)$ (for
technical reasons, our construction of the Fukaya category only allows
countably many Lagrangians as possible supports, so we cannot have all of them
as objects, but this does not affect our arguments, as they are always about a
specific choice of $\alpha$).

For $a\in H^{1}(Q,{\mathbin{\mathbb{R}}})$ we define

$T^{a}:\pi_{1}Q\to\Lambda^{*}$ (4.3)

to be the one-dimensional representation $x\mapsto T^{a\cdot x}.$ Every one-
dimensional representation $\pi_{1}Q\to\Lambda^{*}$ is of the form
$T^{a}\otimes E$ where $E$ is integral, that is, a representation on
$\Lambda^{0}.$

We come now to the proof of Lemma 4.2. Suppose first that $E$ is trivial of
rank one and regard $Q^{\alpha}$ as an object of ${\mathbin{\cal F}}(T^{*}Q).$
We compute the product map

${\mathfrak{m}}^{2}:HF^{*}(Q^{\alpha},T_{q}^{*}Q)\otimes
HW^{*}(T_{q}^{*}Q,T_{q}^{*}Q)\to HF^{*}(Q^{\alpha},T_{q}^{*}Q).$ (4.4)

Let $\phi:T^{*}Q\to T^{*}Q$ be the time-one map of a wrapping Hamiltonian such
that $\phi(T_{q}^{*}Q)$ intersects $T_{q}^{*}Q$ transversely and at least at
one point $\bar{x}\in\phi(T_{q}^{*}Q)\cap T_{q}^{*}Q$ corresponding to
$x\in\pi_{1}Q.$ Choose $\phi$ to be supported sufficiently far from the zero
section so that $\phi(T_{q}^{*}Q)$ intersects $Q^{t\cdot\alpha},$ for all
$t\in[0,1]$ transversely and exactly at one point. Now (4.4) comes from the
chain-level map

${\mathfrak{m}}^{2}:CF^{*}(Q^{\alpha},\phi(T_{q}^{*}Q))\otimes
CF^{*}(\phi(T_{q}^{*}Q),T_{q}^{*}Q)\to CF^{*}(Q^{\alpha},T_{q}^{*}Q)$ (4.5)

which is a count of pseudoholomorphic discs.

We show next that

${\mathfrak{m}}^{2}(\alpha|_{q},\bar{x})=T^{{\mathbin{\cal
A}}(\bar{x})-[\alpha]\cdot x}\alpha|_{q}.$ (4.6)

We deduce this from the following two propositions:

######  Proposition 4.5.

In (4.5) the number $($counted either modulo $2$ or with sign$)$ of
pseudoholomorphic discs for

$(\alpha|_{q},\bar{x},\alpha|_{q})\in
CF^{*}(Q^{\alpha},\phi(T_{q}^{*}Q))\times
CF^{*}(\phi(T_{q}^{*}Q),T_{q}^{*}Q)\times CF^{*}(Q^{\alpha},T_{q}^{*}Q)$ (4.7)

is equal to $1.$

###### Proof.

For $\alpha=0$ the exact Floer group $HW_{{\rm ex}}^{*}(Q,T_{q}^{*}Q)$ is
trivial as a $\pi_{1}Q$ representation so the number of pseudoholomorphic
discs for

$(q,\bar{x},q)\in CF^{*}(Q,\phi(T_{q}^{*}Q))\times
CF^{*}(\phi(T_{q}^{*}Q),T_{q}^{*}Q)\times CF^{*}(Q,T_{q}^{*}Q)$ (4.8)

equals $1.$ This is conserved under perturbations of $\alpha$ by the usual
cobordism argument associated to the family $t\cdot\alpha$. ∎

######  Proposition 4.6.

Every pseudoholomorphic disc for (4.7) has area ${\mathbin{\cal
A}}(\bar{x})-[\alpha]\cdot x.$

###### Proof.

This is the sum of the integrals of $\lambda$ over the three arcs on
$Q^{\alpha},\phi(T^{*}Q),T_{q}^{*}Q$ respectively. The integral for $Q$ is the
sum of those of $\lambda.$ The arc on $Q$ is actually a loop in $Q^{\alpha},$
say $\ell.$ As in Figure 4.1 we can fill in a triangle by flowing back the arc
on $\phi(T_{q}^{*}Q),$ which shows that $\ell$ is homotopic in $T^{*}Q$ to
$\bar{x}^{-1}.$

$q$$q$$\ell$$\bar{x}$ Figure 4.1: a pseudoholomorphic disc for $(q,\hat{x},q)$

So $\ell$ belongs to $x^{-1}\in\pi_{1}Q$ and $\int_{\ell}\alpha=-[\alpha]\cdot
x.$

The integral for $\phi(T_{q}^{*}Q)$ is exactly the action ${\mathbin{\cal
A}}(\bar{x})$ of $\bar{x}\in CF^{*}(\phi(T_{q}^{*}Q),T_{q}^{*}Q).$ The
integral for $T_{q}^{*}Q$ vanishes because $\lambda$ vanishes on $T_{q}^{*}Q.$
This completes the proof. ∎

These two propositions imply (4.6). Its action term will be cancelled out
under the isomorphism

$T^{\mathbin{\cal A}}:HW_{{\rm
ex}}^{0}(T_{q}^{*}Q,T_{q}^{*}Q)\otimes_{\mathbf{k}}\Lambda\cong
HW^{0}(T_{q}^{*}Q,T_{q}^{*}Q)$ (4.9)

so that the multiplication term becomes $T^{-[\alpha]\cdot x}.$ The
representation on $HW^{*}(Q^{\alpha},T_{q}^{*}Q)$ is thus isomorphic to
$T^{-[\alpha]}.$

If $Q^{\alpha}$ is given a local system $E$ then we get in (4.6) the parallel-
transport term and the representation on $HW^{*}(E,T_{q}^{*}Q)$ becomes
isomorphic to $T^{-[\alpha]}\otimes E.$

For a finite cover $f:(P,p)\to(Q,q)$ let $\beta:=f^{*}\alpha$ and take
$x\in\pi_{1}P.$ The Hamiltonian flow $\phi$ lifts to $T^{*}P$ which we denote
by the same symbol $\phi.$ Also the pseudoholomorphic discs for (4.7) lift to

$(\beta|_{p},\bar{x},\beta|_{p})\in CF^{*}(P^{\beta},\phi(T_{p}^{*}P))\times
CF^{*}(\phi(T_{p}^{*}P),T_{p}^{*}P)\times CF^{*}(P^{\beta},T_{p}^{*}P).$
(4.10)

So the same computation applies, which completes the proof of Lemma 4.2.∎

###  4.3 Proof of Lemma 4.3

We begin by explaining the meaning of the last part of Lemma 4.3:

######  Definition 4.7.

For a one-dimensional representation $\rho:\pi_{1}Q\to\Lambda^{*}$ define

${\mathbf{v}}\rho:={\mathbf{v}}\circ\rho\in\hom(\pi_{1}Q,{\mathbin{\mathbb{R}}})\cong
H^{1}(Q,{\mathbin{\mathbb{R}}}).$ (4.11)

We introduce now a norm on $\pi_{1}Q.$ For the unit $1\in\pi_{1}Q$ define
$|1|:=0$ and for $x\in\pi_{1}Q\setminus\\{1\\}$ define $|x|\in[0,\infty)$ to
be the least length of based geodesics in $x\in\pi_{1}Q.$ We introduce a norm
also on $H^{1}(Q,{\mathbin{\mathbb{R}}}).$ For $a\in
H^{1}(Q,{\mathbin{\mathbb{R}}})$ define

$|a|:=\sup_{x\in\pi_{1}Q\setminus\\{1\\}}\frac{a\cdot x}{|x|}.$ (4.12)

We come now to the proof of Lemma 4.3. The key notion we use is action-
completed Floer groups [52, 59, 60]. Consider the radius-$\delta$ disc
subbundle of $T^{*}Q$, and take on it any admissible Hamiltonians of the form
$H=H(|p|),$ $p\in T^{*}Q$ and $|p|$ the distance from $Q\subset T^{*}Q.$
Denote by ${\mathbin{\cal X}}$ the union of periodic orbits for these
Hamiltonians, and consider the action-complete wrapped complex from [52]:

$CW^{*}(T_{q}^{*}Q,T_{q}^{*}Q)^{\delta}:=\left\\{\sum_{i=1}^{\infty}c_{i}x_{i}:c_{i}\in\Lambda,x_{i}\in{\mathbin{\cal
X}};\lim_{i\to\infty}{\mathbf{v}}c_{i}-{\mathbin{\cal
A}}(x_{i})=\infty\right\\}.$ (4.13)

For $x\in{\mathbin{\cal X}}$ denote by $|x|$ the length of $x$ and recall that

${\mathbin{\cal
A}}(x)=\int_{0}^{1}x^{*}\lambda-x^{*}H(t)dt=r|x|-\int_{0}^{1}x^{*}H(t)dt$
(4.14)

where $H$ is the admissible Hamiltonian for $x.$ By the definition of
admissible Hamiltonians the last term on (4.14) is bounded, so the last
condition in (4.13) is equivalent to ${\mathbf{v}}c_{i}-\delta|x_{i}|$ tending
to infinity.

Consider now the singular chain complex $C_{-*}(\Omega_{q}Q,{\mathbf{k}})$ and
for a singular chain $x:\Delta^{n}\to\Omega_{q}Q$ define

$|x|:=\min_{t\in\Delta^{n}}\text{(the length of $x(t)$)}.$ (4.15)

Introduce