Lagrangian intersections and cuplength in generalised cohomology

Amanda Hirschi, Noah Porcelli

Abstract

We find lower bounds on the number of intersection points between two
relatively exact Hamiltonian isotopic Lagrangians. The bounds are given in
terms of the cuplength of the Lagrangian in various multiplicative generalised
cohomology theories. The intersection of the Lagrangians need not be
transverse, however, we require certain orientation assumptions. This gives
stronger bounds than previous estimates on the number of intersection points
for suitable Lagrangians diffeomorphic to $\operatorname{Sp}(2)$ or
$\operatorname{Sp}(3)$. Our proof uses Lusternik-Schnirelmann theory,
following and extending work by Hofer [Hof88].

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Main results
  2. 2 Homotopy and Fredholm theory
    1. 2.1 Generalised cohomology and Thom spectra
    2. 2.2 Index bundles
    3. 2.3 Proof of Theorem 2.12
  3. 3 Approximating pseudoholomorphic strips
  4. 4 Lusternik-Schnirelmann theory
  5. 5 Two examples

##  1 Introduction

###  1.1 Background

Let $(X,\omega)$ be a symplectic manifold and let $L$ and $L^{\prime}$ be two
Lagrangian submanifolds in $X$. A classical problem in symplectic topology is
to find lower bounds on the number of intersection points between $L$ and
$L^{\prime}$ when they are related by a Hamiltonian diffeomorphism.

This has been studied by many authors, under various different assumptions.
When the Lagrangians are assumed to be transverse, there are lower bounds
obtained from Floer homology, such as in [Flo88], [Oh93], and [FOOO09] among
many other references.

The classical Arnol’d conjecture concerns a special case of this question,
where $(X,\omega)$ is the product symplectic manifold $(Y\times
Y,\sigma\oplus-\sigma)$ for some compact symplectic manifold $(Y,\sigma)$, $L$
is the diagonal and $L^{\prime}$ is the graph of a Hamiltonian diffeomorphism
of $Y$. This case has been the subject of much study, both with and without
the additional assumption of transverse intersection. See [FO99], [Rud99],
[Par16] and [AB21] and the references therein.

We will not assume that $L$ and $L^{\prime}$ are transverse, but instead make
the following assumption. It excludes disk and sphere bubbling, guaranteeing
the compactness of certain moduli spaces of pseudoholomorphic curves.

######  Assumption 1.1.

Throughout this paper, we will assume that

  1. 1.

$X$ is either closed or a Liouville manifold.

  2. 2.

$L$ is connected, closed and _relatively exact_ , i.e.,

$\omega\cdot\pi_{2}(X,L)=0.$

Under these assumptions, Floer proved

######  Theorem 1.2 ([Flo88]).

If $L$ and $L^{\prime}$ intersect transversely, there is a lower bound

$\\#L\cap L^{\prime}\geq\sum\limits_{i}\mathrm{Rank}(H_{i}(L;\mathbb{Z}/2)).$

Without the transversality assumption, a version of the Arnol’d conjecture
states that

######  Conjecture 1.3.

Given Assumption 1.1, there is a lower bound

$\\#L\cap L^{\prime}\geq\mathrm{Crit}(L)$

where $\mathrm{Crit}(L)$ is the minimal number of critical points of any
smooth map $L\rightarrow\mathbb{R}$.

A standard application of the Weinstein neighbourhood theorem implies that if
true, this bound must be sharp.

Lusternik-Schnirelmann theory is a powerful tool for studying (numbers of)
critical points or intersection points without any transversality assumptions.
It has been used in many fields other than symplectic geometry. For example,
Klingenberg proved in [Kli78, Theorem 5.1.1] that any metric on $S^{2}$ admits
at least three closed geodesics. Another application is to show that any (not
necessarily Morse) function on a closed smooth manifold $M$ has at least
$c_{\mathbb{Z}}(M)$ critical points, where $c_{\mathbb{Z}}(M)$ is the
cuplength of $M$ in singular cohomology with integer coefficients. Lusternik-
Schnirelmann theory has also been used in contact topology, e.g. by Ginzburg
and Gürel in [GG20] to find lower bounds for numbers of Reeb orbits. For a
more comprehensive discussion and further applications we refer to [CLOT03] or
Chapter 11 in [MS17].

In particular, this technique can be applied to study Conjecture 1.3.

######  Theorem 1.4 ([Hof88, Theorem 3], [Flo89, Theorem 1], [Hof85, Theorem
1]).

Under Assumption 1.1, there is a lower bound

$\\#L\cap L^{\prime}\geq c_{\mathbb{Z}/2}(L).$ (1.1.1)

If $X=T^{*}L$ is a contangent bundle with $L$ the zero section, there is a
lower bound

$\\#L\cap L^{\prime}\geq c_{\mathbb{Z}}(L).$ (1.1.2)

Here $c_{R}(L)$ is the cuplength of $L$ in cohomology with coefficients in $R$
for a ring $R$, defined below.

This result was proved using Lusternik-Schnirelmann theory by Hofer.
Independently, Floer’s proof proceeds via Conley indices. In general, (1.1.1)
and (1.1.2) are weaker bounds than the one given by Conjecture 1.3. There are
examples in which it is a strictly weaker bound, such as [Rud99, Example 3.7].

Similar results have been obtained in the monotone setting (rather than the
relatively exact setting, in which we work) by Lê-Ono [LO96] and Gong
[Gon21a].

###  1.2 Main results

We will extend Theorem 1.4, as well as other results in [Hof88], to
generalised cohomology theories. The strategy is to prove the injectivity of a
certain pullback map relating the generalised cohomology of $L$ with the
generalised cohomology of a certain moduli space of pseudoholomorphic curves.

Fix a ring spectrum $R$, representing a multiplicative generalised cohomology
theory $R^{*}$. The main invariant we will use is the cuplength.

######  Definition 1.5.

Let $Y$ be a path-connected topological space. The _$R$ -cuplength_ of $Y$ is
the natural number (or $\infty$) c

$c_{R}(Y):=1+\sup\\{k\in\mathbb{N}:\exists\alpha_{1},\dots,\alpha_{k}\in\tilde{R}^{*}(Y):\alpha_{1}\cdot\dots\cdot\alpha_{k}\neq
0\\}.$ (1.2.1)

The proof of Theorem 1.4 relies on various moduli spaces of pseudoholomorphic
curves, which might not be cut out transversely. Thus they do not admit an a
tangent bundle; instead one has to work with a virtual vector bundle, induced
by the linearisation of the Cauchy-Riemann operator. We make the following
assumption throughout the paper, which will allow us to apply a version of
Atiyah duality later on.

######  Assumption 1.6.

This virtual vector bundle is $R$-orientable.

From [Por22] we have the following criteria for Assumption 1.6 to be satisfied
for some generalised cohomology theories.

######  Proposition 1.7 ([Por22, Proposition 1.13]).

Assumption 1.6 holds when

  1. 1.

$R$ is the Eilenberg-MacLane spectrum $H\mathbb{Z}/2$.

  2. 2.

$R$ is the Eilenberg-MacLane spectrum $H\mathbb{Z}$, and $L$ is (relatively)
spin.

  3. 3.

$R^{*}$ is complex $K$-theory, and $L$ is spin.

  4. 4.

$R^{*}$ is real $K$-theory, and $TL$ admits a stable trivialisation over a
3-skeleton of $L$ which extends (after complexification) to a stable
trivialisation of $TX$ over a 2-skeleton of $X$ (as a complex vector bundle).

Fix an $\omega$-compatible almost complex structure $J$. If $X$ is Liouville,
we assume $J$ to be convex at infinity. Let $Z$ be the infinite strip
$\mathbb{R}+[0,1]i$ in the complex plane. By pseudoholomorphic, we will always
mean with respect to $J$. We are interested in the following standard moduli
space of pseudoholomorphic strips with Lagrangian boundary conditions

$\mathcal{M}_{L,L^{\prime}}:=\left\\{u\in
C^{\infty}(Z,X):\overline{\partial}_{J}u=0,\;E(u)<\infty,\;u(\mathbb{R})\subseteq
L,\;u(\mathbb{R}+i)\subseteq L^{\prime}\right\\}.$

Evaluation at 0 defines a map $\pi:\mathcal{M}_{L,L^{\prime}}\rightarrow L$.
Following Hofer’s strategy, we can approximate $\mathcal{M}_{L,L^{\prime}}$
with spaces of pseudoholomorphic discs and use Theorem 2.12 to prove the
following.

######  Proposition 1.8.

The map $\pi^{*}\colon R^{*}(L)\rightarrow R^{*}(\mathcal{M}_{L,L^{\prime}})$
is injective.

The main work of the paper, done in Section 2, lies in the proof of the
auxiliary result Theorem 2.12. In Section 3 we approximate
$\mathcal{M}_{L,L^{\prime}}$ by moduli spaces of pseudoholomorphic discs. The
continuity axiom, which is satisfied by generalised cohomology theories as
defined in Section 2.1, allows us to deduce Proposition 1.8.

As in [Hof88], we combine this result in section 4 with standard Lusternik-
Schnirelmann theory to obtain the following lower bound.

######  Theorem 1.9.

Suppose $L$ satisfies Assumption 1.1 and Assumption 1.6 for a ring spectrum
$R$. Then the number of intersection points between $L$ and $L^{\prime}$
satisfies

$\\#L\cap L^{\prime}\geq c_{R}(L).$

######  Remark 1.10.

If $L=L_{1}\sqcup\dots\sqcup L_{k}$ is a disjoint union of Lagrangians
satisfying Assumption 1.1, we may write
$L^{\prime}=L^{\prime}_{1}\sqcup\dots\sqcup L^{\prime}_{k}$ with
$L^{\prime}_{i}$ Hamiltonian isotopic to $L_{i}$. Theorem 1.9 applied to each
pair $L_{i}$ and $L^{\prime}_{i}$ shows that

$\\#L\cap L^{\prime}\geq\sum_{i=1}^{k}c_{R}(L_{i}).$

In Section 5 we give two examples where Theorem 1.9 represents a stronger
bound than what was previously known. This uses computations in [IM04] of the
cuplengths of the compact symplectic groups $\operatorname{Sp}(2)$ and
$\operatorname{Sp}(3)$ with respect to certain generalised cohomology
theories.

As we do not assume the transversality of $L$ and $L^{\prime}$, to our
knowledge there is no analogue of our strategy of proof using the setup in
[CJS95, Coh09]. However, it may be possible to use their setup to prove
Theorem 1.9 using the strategy of [Gon21b] instead.

Suppose $X$ is compact and symplectically aspherical. If $\psi$ is a (possibly
degenerate) Hamiltonian diffeomorphism of $X$, we can apply Theorem 1.9 to the
graph of $\psi$ in $X\times X$ to deduce the Hamiltonian version of this
inequality.

######  Corollary 1.11.

The number of fixed points of $\psi$ satisfies

$\\#\operatorname{Fix}(\psi)\geq c_{R}(X).$

In this setting, Conjecture 1.3 (which implies Corollary 1.11) has already
been verified; see [Rud99, Theorem A], [OR99, Corollary 4.2] and [CLOT03,
Theorem 8.28].

### Acknowledgements

Both authors are grateful to Ailsa Keating and Ivan Smith for valuable
discussions and for suggesting this project in the case of the latter. Oscar
Randal-Williams pointed out the example of $\operatorname{Sp}(2)$ and gave
helpful feedback, as did Jonny Evans. The first author thanks Matija Sreckovic
for comments on an earlier draft. The second author would also like to thank
Jack Smith, Nick Nho and Sam Frengley for helpful discussions.
Both authors were funded by the EPSRC during their graduate studies. The
second author is supported by the Engineering and Physical Sciences Research
Council [EP/W015889/1]. This work was finished while both authors were
resident at the Simons Laufer Mathematical Sciences Institute, supported by
the NSF grant No. DMS-1928930.

##  2 Homotopy and Fredholm theory

###  2.1 Generalised cohomology and Thom spectra

For our purposes, it suffices to work with classical spectra as defined in
[Rud98] or [Ada95]. However, our definitions require some care, as we will
take the generalised (co)homology of spaces which are not necessarily homotopy
equivalent to a CW complex. A generalised cohomology theory defined on CW
complexes can always be extended to all spaces, but this extension may not be
unique. We need an extension that satisfies a certain continuity property,
namely the statement of [AMS21, Lemma 5.2].

Unless otherwise specified, we work with spectra whose level spaces are
homotopy equivalent to CW complexes. We denote by $\mathbb{S}$ the sphere
spectrum. All our spaces are assumed to be compactly generated and Hausdorff.

Given an $\Omega$-spectrum $R$, we define, for a pointed space $X$, the
_$n^{\mathrm{th}}$ $R$-homology and $R$-cohomology groups_ to be

$R_{n}(X):=\pi_{n}(X\wedge R_{n})\quad\quad\quad\quad R^{n}(X):=[X,R_{n}]_{*}$

respectively, for $n$ in $\mathbb{Z}$, where $[\cdot,\cdot]_{*}$ denotes the
set of pointed homotopy classes of maps. As $R_{n}\simeq\Omega^{2}R_{n+2}$,
the sets $R_{n}(X)$ and $R^{n}(X)$ carry a natural abelian group structure for
all $n$.

We recall the definition of relative $R$-(co)homology.
Given an inclusion $j\colon A\hookrightarrow X$ of pointed spaces, we denote
by

$C_{X}A:=C_{A}\cup_{j}X$

the _(reduced) mapping cone_ of $j$, where $C_{A}$ is the cone over $A$. We
view this as a pointed space, with basepoint the vertex of $C_{A}$ (or
equivalently the basepoint of $X$).

The inclusion $X\hookrightarrow C_{X}A$ is a cofibration and collapsing $X$
induces a natural map $C_{X}A\rightarrow\Sigma A$. By [tD08, Theorem 4.6.4]
these maps fit into an $h$-coexact sequence

$A\rightarrow X\rightarrow C_{X}A\rightarrow\Sigma A\rightarrow\Sigma
X\rightarrow\dots$

In particular, there exists for any pointed space $W$ a long exact sequence of
pointed sets

$\dots\rightarrow[\Sigma X,W]_{*}\rightarrow[\Sigma
A,W]_{*}\rightarrow[C_{X}A,W]_{*}\rightarrow[X,W]_{*}\rightarrow[A,W]_{*}$
(2.1.1)

The _relative $R$-(co)homology_ of $(X,A)$ is defined by

$R_{*}(X,A):=R_{*}(C_{X}A)\qquad\qquad R^{*}(X,A):=R^{*}(C_{X}A).$

We recover the usual long exact sequence of a pair in $R$-cohomology due to
(2.1.1). Similarly, there is a long exact sequence of a pair in $R$-homology.

In the case of a pair of unpointed spaces $(X,A)$, one simply considers the
pair $(X_{+},A_{+})$, where $\cdot_{+}$ denotes the addition of a disjoint
basepoint.

A _ring spectrum_ consists of an $\Omega$-spectrum $R$ endowed with both a
multiplication map $\mu\colon R\wedge R\to R$ and a unit
$\iota\colon\mathbb{S}\to R$ such that the usual associativity and unit
diagrams commute up to homotopy. In this case we can define a cup product

$\cdot\;\colon R^{*}(X)\otimes R^{*}(X)\rightarrow R^{*}(X)$

by letting $\alpha\cdot\beta$ be the composite

$X\xrightarrow{\Delta}X\wedge X\xrightarrow{\alpha\wedge\beta}R_{n}\wedge
R_{k}\rightarrow(R\wedge R)_{n+k}\xrightarrow{\mu}R_{n+k}$ (2.1.2)

for $\alpha\in R^{n}(X)$ and $\beta\in R^{k}(X)$, where the map $R_{n}\wedge
R_{k}\rightarrow(R\wedge R)_{n+k}$ comes from the construction of the smash
product.

When one works outside the setting of CW complexes, the cup product does not
necessarily descend to a map

$R^{*}(X,A)\otimes R^{*}(X,B)\rightarrow R^{*}(X,A\cup B).$

However, we can make the following two observations, which will be useful in
Section 4.

######  Remark 2.1.

Suppose $\alpha\in R^{n}(X)$ and $\beta\in R^{m}(X)$ admit representatives
$\tilde{\alpha}:X\rightarrow R_{n}$ and $\tilde{\beta}:X\rightarrow R_{m}$
which send subspaces $A,B\subseteq X$ to the basepoints $*$ respectively.
Then, by construction, $\alpha\cdot\beta$ admits a representative
$X\rightarrow R_{n+m}$ sending $A\cup B$ to $*$. Induction shows the same
result for classes $\alpha_{1},\ldots,\alpha_{k}\in R^{*}(X)$ that admit
representatives $\tilde{\alpha_{i}}$ mapping subspaces $A_{i}\subseteq X$ to
$*$ respectively. In particular,

$\alpha_{1}\cdot\ldots\cdot\alpha_{k}=0\quad\text{in}\quad
R^{*}(A_{1}\cup\ldots\cup A_{k}).$

######  Lemma 2.2.

Let $(X,d)$ be a compact metric space, with an open cover
$U_{1},\ldots,U_{k}$. Let $\alpha_{i}\in R^{n_{i}}(X)$ be cohomology classes
such that $\alpha_{i}|_{{U_{i}}}=0$ in $R^{n_{i}}({U}_{i})$ for all $i$. Then
the product $\alpha_{1}\cdot\ldots\cdot\alpha_{k}$ vanishes in $R^{*}(X)$.

###### Proof.

Let $V_{1},\dots,V_{k}$ be an open cover of $X$ so that
$\overline{V_{i}}\subseteq U_{i}$ for each $i$. Set
$d_{i}:=d(\cdot,\overline{V_{i}})$. As $A_{i}:=X\setminus U_{i}$ is disjoint
from $\overline{V_{i}}$ and compact, there exists $\epsilon_{i}>0$ so that
$d_{i}^{-1}([0,\varepsilon_{i}])\subseteq U_{i}$.

Pick maps $\tilde{\alpha}_{i}:X\rightarrow R_{n_{i}}$ representing each
$\alpha_{i}\in R^{n_{i}}(X)$. Since $\alpha_{i}|_{U_{i}}=0$, we can choose
nullhomotopies

$H_{i}:U_{i}\times[0,1]\rightarrow R_{n_{i}}$

such that $H_{i}(\cdot,0)\equiv*$ and
$H_{i}(\cdot,1)=\tilde{\alpha}_{i}|_{U_{i}}$.

Define maps $\beta_{i}:X\rightarrow R_{n_{i}}$ by

$\beta_{i}(x):=\begin{cases}\tilde{\alpha}_{i}(x)&\mathrm{if}\;\varepsilon_{i}\leq
d_{i}(x)\\\ H_{i}(x,\varepsilon_{i}^{-1}d_{i}(x))&\mathrm{if}\;0\leq
d_{i}(x)\leq\varepsilon_{i}.\end{cases}$

Then $\beta_{i}\simeq\tilde{\alpha}_{i}$, via the homotopy
$G_{i}:X\times[0,1]\rightarrow R_{n_{i}}$ given by

$G_{i}(x,s):=\begin{cases}\tilde{\alpha}_{i}(x)&\mathrm{if}\;\varepsilon_{i}\leq
d_{i}(x)\\\ H_{i}(x,s+(1-s)\varepsilon_{i}^{-1}d_{i}(x))&\mathrm{if}\;0\leq
d_{i}(x)\leq\varepsilon_{i}.\end{cases}$

Hence $\beta_{i}$ is a representative of $\alpha_{i}\in R^{n_{i}}(X)$.
$\beta_{i}$ sends $V_{i}$ to the basepoint $*$ so
$\alpha_{i}\cdot\ldots\cdot\alpha_{k}=0$ in $R^{*}(X)$ by Remark 2.1. ∎

Suppose now that $X$ is a compact space and that $\xi\colon F\rightarrow X$ is
a vector bundle of rank $k$. Let $F_{\infty}$ be its fibrewise one-point
compactification. This is a sphere bundle over $X$ with a canonical section
$s_{\infty}$ given by the point at infinity in every fibre. The _Thom space_
of $\xi$ is defined to be the pointed space

$\text{Th}(\xi):=C_{F_{\infty}}\operatorname{im}(s_{\infty})$

and its _Thom spectrum_ , written $X^{\xi}$ or $X^{F}$, to be the spectrum
$\Sigma^{\infty}\text{Th}(\xi)$. In particular, if $\xi$ is the trivial bundle
of rank $k$, then $\text{Th}(\xi)=\Sigma^{k}X_{+}$ and
$X^{\xi}=\Sigma^{\infty+k}X_{+}$. Note that if $X$ is not a CW complex,
$\text{Th}(\xi)$ might not be either. These are the only spectra whose level
spaces are not CW complexes that we will encounter.

For a virtual vector bundle of the form $\eta-\mathbb{R}^{N}$, we let
$\operatorname{rank}(\eta)-N$ be its _rank_ and define its _Thom spectrum_ to
be

$X^{\eta-\mathbb{R}^{N}}:=\Sigma^{-N}X^{\eta}.$

All our virtual bundles will be of this form, so this definition is sufficient
for our purposes.111The homotopy type of the Thom spectrum only depends on the
stable isomorphism class of the virtual vector bundle. Due to compactness, any
virtual bundle on $X$ is stably isomorphic to one of the considered form, so
it suffices for our applications.

Let $R$ be a ring spectrum and $\xi$ a virtual vector bundle over $X$ of rank
$k$. An _$R$ -orientation_ of $\xi$ is an element $u\in R^{k}(X^{\xi})$ such
that for any map $j:\Sigma^{k}\mathbb{S}\rightarrow X^{\xi}$ which is a
(stable) homotopy equivalence to a fibre, we have

$j^{*}u=\pm[\iota]\in R^{k}(\Sigma^{k}\mathbb{S})\cong R^{0}(\mathbb{S}).$

where $[\iota]$ is the homotopy class of the unit map. Any trivial bundle is
$R$-orientable, and if two out of $\xi$, $\eta$ and $\xi\oplus\eta$ are
$R$-oriented, then so is the third.

By the Thom isomorphism theorem, [Rud98, Theorem V.1.3] any $R$-orientation on
a virtual vector bundle $\xi$ over a compact CW complex $X$ induces a natural
isomorphism

$R^{*+k}(X^{\xi})\cong R^{*}(X).$

By [AMS21, Lemma 5.2], this also holds when $X$ is a compact subset of a
manifold $M$, and both $\xi$ and its $R$-orientation are pulled back from $M$.

We will need the following form of Atiyah duality, which can be viewed as a
form of Poincaré duality for generalised cohomology theories.

######  Theorem 2.3 ([AMS21, Theorem 5.2]).

Let $M$ be a smooth (not necessarily compact) manifold, possibly with
boundary, and suppose $Z\subseteq M$ is any compact subset. Then there is a
canonical isomorphism

$R_{-*}(M,M\setminus Z)\;\cong\;R^{*}\Bigl{(}Z^{-TM|_{Z}}\Bigr{)}$

compatible with restriction to smaller closed subsets $Z^{\prime}\subseteq Z$.

If $M$ is $R$-oriented on a neighbourhood of $Z$ and of dimension $n$, the
Thom isomorphism theorem then gives an isomorphism

$R^{*+n}(Z)\;\cong\;R_{-*}(M,M\setminus Z).$

for compact subsets $Z\subseteq M$. In this case, we define the _fundamental
class of $M$ along $Z$_ $[M]_{Z}\in R_{n}(M,M\setminus Z)$ to be the image of
the unit in $R^{0}(Z)$ under this isomorphism. The class $[M]_{Z}$ depends
only on the choice of orientation of which there can be more than two.

We will need the following version of the fact that, given an (oriented)
compact manifold with boundary $M$, the fundamental class of $\partial M$ has
vanishing image in the homology of $M$.

######  Lemma 2.4.

Let $W^{n+1}$ be an $R$-oriented smooth manifold with boundary and $K\subseteq
W$ a compact subset. Then the image of $[\partial W]_{K\cap\partial W}$ in
$R_{n}(W,W\setminus K)$ is $0$.

###### Proof.

Suppose first that $W$ is compact. The map $\partial$ in the exact sequence of
a pair

$R_{n+1}(W,\partial W)\xrightarrow{\partial}R_{n}(\partial W)\rightarrow
R_{n}(W)$

sends $[W]$ to $[\partial W]$ \- see [Rud98, Remark V.2.14.a)]. The claim with
$K=W$ then follows from the exactness of this sequence.

Now assume $W$ is non-compact and set $C=K\cap\partial W$. By excision, we may
modify $W$ away from $K$, and replace $W$ with a compact smooth neighbourhood
of $K$. Then $[\partial W]_{C}$ is the restriction of the fundamental class
$[\partial W]\in R_{n}(\partial W)$. We may deduce the claim now from the
first step and the commutativity of the following diagram.

${R_{n+1}(W,\partial W)}$${R_{n}(\partial W)}$${R_{n}(W)}$${R_{n}(\partial
W,\partial W\setminus C)}$${R_{n}(W,W\setminus K)}$$\scriptstyle{\partial}$

∎

###  2.2 Index bundles

Let $s\colon\mathcal{B}\rightarrow\mathcal{E}$ be a smooth Fredholm section of
a Banach bundle over a Banach manifold intersecting the zero section of
$\mathcal{E}$ transversely. By the infinite-dimensional implicit function
theorem [MS04, Theorem A.3.3], the zero locus $s^{-1}(0)$ is a smooth manifold
of dimension $\operatorname{index}(Ds)=\dim(\ker(Ds))$ with tangent bundle
$\ker(Ds)\rightarrow s^{-1}(0)$. If $Ds$ is not fibrewise surjective, the zero
locus is not necessarily smooth or may have excess dimension. The natural
replacement of $\ker(Ds)$ in this case is the _index bundle_ , a virtual
vector bundle constructed below. It relies on the notion of the stabilisation
of a Fredholm operator.

######  Definition 2.5.

Let $D\colon X\rightarrow Y$ be a Fredholm operator between two Banach spaces.
We call an operator $T\colon\mathbb{R}^{N}\rightarrow Y$ a _stabilisation of
$D$_ if $D+T\colon X\oplus\mathbb{R}^{N}\rightarrow Y$ is surjective.

As $T$ is compact, $D+T$ is still a Fredholm operator and

$\operatorname{index}(D+T)=\operatorname{index}(D)+N$

by [MS04, Theorem A.1.5(i)]. Given a smoothly varying family of Fredholm
operators we will show the existence of a smoothly varying family of
stabilisations near a compact subset in Lemma 2.6.

Let us fix our setting for the rest of this subsection. Let $Y$ be a separable
Hilbert manifold, $\mathcal{H}$ a separable Hilbert space and $\Lambda$ a
compact finite-dimensional manifold with boundary. We assume that $\psi\colon
V\rightarrow\mathcal{H}$ is a smooth Fredholm map with $V\subseteq
Y\times\Lambda$ an open subset. Define the open subset

$V^{\operatorname{reg}}:=\\{(x,\lambda)\in V:d_{1}\psi(x,\lambda)\text{ is
surjective}\\}$

where $d_{1}$ is the derivative with respect to the first argument.

######  Lemma 2.6.

For any closed subset $A\subseteq V^{\operatorname{reg}}$ and any
neighbourhood $U\subseteq V$ of a compact subset $K\subseteq V$, there exists
a smooth map $T\colon V\times\mathbb{R}^{k}\rightarrow\mathcal{H}$ such that

  1. 1.

$T(z,\cdot)$ is linear for each $z\in V$;

  2. 2.

$T(z,\cdot)=0$ for $z\in A$;

  3. 3.

$T(z,\cdot)=0$ for $z\in V\setminus U$;

  4. 4.

$d_{1}\psi(x,\lambda)+T(x,\lambda,\cdot)\colon
T_{(x,\lambda)}V\oplus\mathbb{R}^{k}\rightarrow\mathcal{H}$ is surjective for
$(x,\lambda)\in U$.

###### Proof.

For each $z\in K$ there exists an open neighbourhood $U_{z}\subseteq V$ of
$z$, an integer $k_{z}\geq 0$, and an operator
$T_{z}\colon\mathbb{R}^{k_{z}}\rightarrow\mathcal{H}$ such that

$d_{1}\psi(y,\mu)+T_{z}\colon
T_{y}Y\oplus\mathbb{R}^{k_{z}}\rightarrow\mathcal{H}$

is surjective for $(y,\mu)\in U_{z}$. Let $Z\subseteq K$ be a finite subset
such that $U:=\bigcup\limits_{z\in Z}{U_{z}}$ contains $K$ and set
$k:=\sum\limits_{z\in Z}{k_{z}}$. Using a smooth partition of unity
subordinate to $\\{U_{z}\\}_{z\in Z}\cup\\{V\setminus K\\}$, we obtain an
operator $T^{\prime}\colon V\times\mathbb{R}^{k}\rightarrow\mathcal{H}$
satisfying all conditions save for the second one. Multiplying $T^{\prime}$
with a smooth bump function which is identically one on $V\setminus
V^{\operatorname{reg}}$ and vanishes on $A$, we obtain the desired map. ∎

######  Definition 2.7.

A family of operators $T$ as in Lemma 2.6 is said to be a _stabilisation of
$\psi$ along $K$ relative to $A$, of rank $k$_. We call

$\text{Ind}_{K}(\psi;T):=\ker(d_{1}\psi+T)|_{U}-\mathbb{R}_{U}^{k}$

the _index bundle of $\psi$ along $K$ (with respect to $T$)_, defined over a
neighbourhood $U$ of $K$.

######  Lemma 2.8.

Any two index bundles of $\psi$ along $K$ are stably equivalent as germs near
$K$.

###### Proof.

Suppose $T$ and $S$ are two stabilisations along $K$. We may assume without
loss of generality that $d_{1}\psi+T$ and $d_{1}\psi+S$ are surjective over
the same subset. As we may add factors of $\mathbb{R}$ to the domain of $T$,
respectively $S$, without changing the index bundle, we may assume that $T$
and $S$ are smooth maps $V\times\mathbb{R}^{k+\ell}\rightarrow\mathcal{H}$
with $T$ vanishing on $V\times\mathbb{R}^{k}\times\\{0\\}$ and $S$ vanishing
on $V\times\\{0\\}\times\mathbb{R}^{\ell}$. Now we can linearly interpolate
between them and apply [tD08, Theorem 14.3.2]. ∎

######  Definition 2.9.

We let $\text{Ind}_{K}(\psi)$ denote the stable equivalence class of any
$\text{Ind}_{K}(\psi;T)$ and call it the _index bundle of $\psi$ along $K$_.

######  Definition 2.10.

Given a ring spectrum $R$, the map $\psi$ is _$R$ -orientable along $K$_ if
$\text{Ind}_{K}(\psi)$ is $R$-orientable on a neighbourhood of $K$. We say
that $\psi$ is _$R$ -orientable_ if it is $R$-orientable along any compact
subset.

###  2.3 Proof of Theorem 2.12

The following result generalises [Hof88, Theorem 5] to arbitrary ring spectra.

######  Proposition 2.11.

Let $\mathcal{Y}$ be a smooth separable Hilbert manifold and $\mathcal{H}$ be
a separable Hilbert space. Let
$\psi:\mathcal{Y}\times[0,1]\rightarrow\mathcal{H}$ be a smooth Fredholm map
of index $n+1$, and write $\psi_{t}$ for its restriction to
$\mathcal{Y}\times\\{t\\}$. Given a ring spectrum $R$, assume

  1. 1.

$\psi$ is proper with respect to a neighbourhood of $0$ in $\mathcal{H}$ and
$R$-orientable along $\psi^{-1}(0)$,

  2. 2.

$\psi_{0}$ is submersive near $\psi_{0}^{-1}(0)$,

  3. 3.

there exists a smooth map $\pi\colon\mathcal{Y}\rightarrow N$ to a connected
closed manifold $N$ such that

$\pi|_{\psi_{0}^{-1}(0)}\colon\psi_{0}^{-1}(0)\rightarrow N$

is a diffeomorphism.

If $N$ is $R$-oriented, then $\pi^{*}\colon R^{*}(N)\rightarrow
R^{*}(\psi_{1}^{-1}(0))$ is injective.

###### Proof.

Set $K:=\psi^{-1}(0)$ and $I:=[0,1]$. By (1), $K$ is compact. Given any subset
$W\subseteq\mathcal{Y}\times I$ we denote by $W_{t}$ its fibre over $t\in I$.
Let $T$ be a stabilisation of $\psi$ along $K$ relative to
$K_{0}\times\\{0\\}$ of rank $k$. Set

$S:=(\psi+T)^{-1}(0)\subseteq\mathcal{Y}\times I\times\mathbb{R}^{k}.$

Then $S$ is a smooth (non-compact) cobordism from $S_{0}$ to $S_{1}$ with
$TS=\ker(d\psi+T)$. Assumption (1) on $\psi$ implies that $S$ is $R$-oriented
on a neighbourhood of $K$. By the compactness of $T(v,t,\cdot)$ for
$(v,t)\in\mathcal{Y}\times I$ and [MS04, Theorem A.1.5.i],

$\dim(S)=n+k+1.$

Note that $K=\\{(x,t,z)\in S:z=0\\}$ and $S_{0}=K_{0}\times\mathbb{R}^{k}$.
Set

$\tilde{\pi}:=\pi\times\operatorname{id}_{I}\times\operatorname{id}_{\mathbb{R}^{k}}:S\rightarrow
N\times I\times\mathbb{R}^{k}$

and let $\tilde{\pi}_{t}$ be the restriction to
$\mathcal{Y}\times\\{t\\}\times\mathbb{R}^{k}$. This fits into a commutative
diagram of pairs

${(S_{0},S_{0}\setminus
K_{0})}$${(N\times\mathbb{R}^{k},N\times(\mathbb{R}^{k}\setminus
0))}$${(S,S\setminus K)}$${(N\times I\times\mathbb{R}^{k},N\times
I\times(\mathbb{R}^{k}\setminus 0))}$${(S_{1},S_{1}\setminus
K_{1})}$${(N\times\mathbb{R}^{k},N\times(\mathbb{R}^{k}\setminus
0))}$$\scriptstyle{\tilde{\pi}_{0}}$$\scriptstyle{i_{0}}$$\scriptstyle{\iota_{0}}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{i_{1}}$$\scriptstyle{\tilde{\pi}_{1}}$$\scriptstyle{\iota_{1}}$

Consider the composition

$R^{*}(N)\xrightarrow{\pi_{1}^{*}}R^{*}(K_{1})\xrightarrow[\cong]{\mathrm{AD}}R_{n+k-*}(S_{1},S_{1}\setminus
K_{1})\xrightarrow{(\pi_{1})_{*}}R_{n+k-*}(N\times\mathbb{R}^{k},N\times(\mathbb{R}^{k}\setminus
0))\xrightarrow[\cong]{\mathrm{AD}}R^{*}(N)$ (2.3.1)

where $\mathrm{AD}$ denotes the Atiyah duality isomorphism. Note that in the
second map we use the $R$-orientability assumption in (1).

By construction, this map is given by multiplication by
$\mathrm{AD}((\pi_{1})_{*}[S_{1}]_{K_{1}})\in R^{0}(N)$, which is equal to
$\mathrm{AD}((\pi_{0})_{*}[S_{0}]_{K_{0}})$ by Lemma 2.4. As $\pi_{0}$ is a
diffeomorphism, this cohomology class is a unit. Hence (2.3.1) is an
isomorphism. Because it factors through the pullback
$\pi_{1}^{*}:R^{*}(N)\rightarrow R^{*}(K_{1})$, the latter must be injective.∎

We apply this to our situation. Let $G\subseteq\mathbb{C}$ be a simply-
connected compact submanifold with smooth boundary, and suppose
$\\{L_{z}\\}_{z\in\partial G}$ is a _Hamiltonian family_ of Lagrangians in
$X$. That is, there exists a (relatively exact) Lagrangian $L\subseteq X$ and
a smooth family $\\{\phi^{t}_{z}\\}_{z\in\partial G,t\in[0,1]}$ of Hamiltonian
isotopies (which we can assume to be compactly supported) of $X$ such that
$L_{z}=\phi_{z}^{1}(L)$ for all $z$. We can assume that $L=L_{z_{0}}$ for some
$z_{0}\in\partial G$.

Consider the following moduli space of pseudoholomorphic discs with moving
Lagrangian boundary conditions:

$\mathcal{P}:=\left\\{u\in
C^{\infty}(G,X):\overline{\partial}_{J}u=0,\;E(u)<\infty,\;\forall
z\in\partial G:u(z)\in L_{z}\right\\}$ (2.3.2)

where $\overline{\partial}_{J}$ is the Cauchy-Riemann operator associated to
$J$ and $E$ is the symplectic energy. Let $\pi:\mathcal{P}\rightarrow L$ be
evaluation at $z_{0}$.

######  Theorem 2.12.

The pullback $\pi^{*}:R^{*}(L)\rightarrow R^{*}(\mathcal{P})$ is injective.

######  Remark 2.13.

If the moduli space $\mathcal{P}$ were cut out transversely, this could be
proved using a cobordism argument as in [Por22]. On the other hand, following
[Por22, Remark 4.6], Theorem 2.12 can be used to give a slightly different
proof of [Por22, Corollary 1.9], without using any transversality results.

######  Remark 2.14.

Hofer [Hof88] proves Theorem 2.12 as well as Theorem 1.8, 1.9 and Corollary
1.11 in the case where $R^{*}$ is Čech cohomology with coefficients in
$\mathbb{Z}/2$.

Using an extension of an associated family of Hamiltonians we may extend the
family of Hamiltonian isotopies $\\{\phi_{z}\\}_{z\in\partial G}$ to a smooth
family $\\{\phi_{z}\\}_{z\in G}$ of Hamiltonian isotopies, parametrised by
$G$.

Fix $k\geq 3$. Given $t\in[0,1]$, we define $\psi_{t}\colon
W^{k,2}(G,X)\rightarrow W^{k,2}(G,X)$ by setting

$\psi_{t}(u)(z):=\phi^{t}_{z}(u(z))$

for $z\in G$. By assumption, $\psi_{0}$ is the identity. Let

$\mathcal{A}:=\left\\{u\in W^{k,2}(G,X):u(\partial G)\subseteq L\right\\}.$

The smooth Banach bundle $\mathcal{E}\rightarrow\mathcal{A}$ with fibre

$\mathcal{E}_{u}:=W^{k-1,2}(G,u^{*}TX).$

admits a smooth Fredholm section
$\overline{\partial}_{J}:\mathcal{A}\rightarrow\mathcal{E}$ given by

$\overline{\partial}_{J}u=\partial_{s}u+J(u)\partial_{t}u.$

The canonical evaluation map defines a map of pairs
$\text{ev}\colon\mathcal{A}\times(G,\partial G)\rightarrow(X,L)$. By pulling
back, this defines a bundle pair

$(F,F^{\prime}):=\text{ev}^{*}(TX,TL)\rightarrow\mathcal{A}\times(G,\partial
G).$

Using a connection on $TX$, the linearisation of $\overline{\partial}_{J}$
defines a real Cauchy-Riemann operator on $(F,F^{\prime})$ by [MS04,
Proposition 3.1.1]. Then Assumption 1.6 states exactly that its index bundle
is $R$-oriented. For $u\in\overline{\partial}_{J}^{-1}(0)$ we have, by the
Riemann-Roch theorem, [MS04, Theorem C.1.10], that

$\operatorname{index}(D_{u}\overline{\partial}_{J})=\dim(L)+\mu(F|_{u},F^{\prime}|_{u}),$
(2.3.3)

where $\mu(F|_{u},F^{\prime}|_{u})$ is the boundary Maslov index of the
pullback of $(F,F^{\prime})$ to $\\{u\\}\times G$.

######  Remark 2.15.

If $u\in\mathcal{A}$ is pseudoholomorphic, then
$\mu(F|_{u},F^{\prime}|_{u})=0$ as $u$ must be constant due to relative
exactness. However, if $u$ instead satisfies that $\psi_{t}(u)$ is
pseudoholomorphic for some $t>0$, $u$ need not be constant and may lie in a
non-trivial relative homotopy class of discs. In this case,
$\mu(F|_{u},F^{\prime}|_{u})$ might not vanish.

By [Kui65, Theorem (3)] we can fix a smooth isometric trivialisation
$\Psi\colon\mathcal{E}\rightarrow\mathcal{A}\times\mathcal{H}$, where
$\mathcal{H}$ is some separable Hilbert space. Define
$\mathcal{F}\colon\mathcal{A}\times[0,1]\rightarrow\mathcal{H}$ by

$\mathcal{F}_{t}(u):=\operatorname{pr}_{2}\Psi(\overline{\partial}_{J}\psi_{t}(u))$
(2.3.4)

letting $\operatorname{pr}_{2}$ denote the projection to the second factor.
Note that $\mathcal{F}_{1}^{-1}(0)$ is diffeomorphic via $\psi_{1}$ to the
space $\mathcal{P}$ of pseudoholomorphic maps from $G$ to $M$ which have
finite energy and map $z\in\partial G$ to $L_{z}$.

###### Proof of Theorem 2.12.

Let

$\mathcal{W}:=\left\\{(u,t)\in\mathcal{A}\times[0,1]:\mu(F|_{\psi_{t}(u)},F^{\prime}|_{\psi_{t}(u)})=0\right\\}.$

This is an open subset of $\mathcal{A}\times[0,1]$. We restrict to the subset
where the Maslov index is 0 in order to have control over the index of
$\mathcal{F}$, due to Remark 2.15. However, with a little more care the entire
argument could also be applied without this restriction. Let
$\pi:\mathcal{W}\rightarrow L$ be the evaluation map at $z_{0}$.

By [Hof88, Proposition 6] there exists a neighbourhood $U\subseteq\mathcal{W}$
of $\mathcal{F}^{-1}(0)$ such that $\mathcal{F}|_{U}:U\rightarrow\mathcal{H}$
is a Fredholm map of index $\dim(L)+1$ and such that $\mathcal{F}|_{U}$ is
proper with respect to a neighbourhood of $0\in\mathcal{H}$. We note that $U$
and $\mathcal{H}$ are separable Hilbert manifolds. Since pseudoholomorphic
discs with boundary on $L$ are constant, $\pi$ defines a diffeomorphism
$\mathcal{F}_{0}^{-1}(0)\rightarrow L$. Moreover, $\mathcal{F}_{0}$ is
submersive by the proof of [Hof88, Lemma 5], and $\mathcal{F}$ is
$R$-orientable by Assumption 1.6. Thus the claim follows from Proposition
2.11. ∎

##  3 Approximating pseudoholomorphic strips

The key idea in the proof of Theorem 1.8 is to study a one-parameter family of
moduli spaces of pseudoholomorphic discs $\mathcal{P}_{\ell}$ with moving
boundary conditions. They approximate the moduli space of pseudoholomorphic
strips $\mathcal{M}_{L,L^{\prime}}$ as the parameter $\ell$ tends to $\infty$.
Together with [AMS21, Lemma 5.2], this allows us to infer Proposition 1.8 from
Theorem 2.12.

Throughout this section we fix a convex domain $G$ in $\mathbb{C}$ with smooth
boundary, such that both $(-\eta,\eta)$ and $(-\eta,\eta)+i$ are contained in
$\partial G$ for some $\eta>0$. For $\ell>0$, define
$Z_{\ell}:=[-\ell,\ell]+[0,1]i$, and let

$G_{\ell}:={Z_{\ell}\cup(G+\ell)\cup(G-\ell)}$

be a smoothing as shown below.

$0$ $i$ $\ell$ $-\ell$ $Z_{\ell}\mathchar 24634\relax$ $0$ $i$ $\ell$ $-\ell$
$G_{\ell}\mathchar 24634\relax$

######  Definition 3.1.

For a domain $W$ in $\mathbb{C}$ and a smooth map $u:W\rightarrow X$, we
define the _symplectic energy_ to be

$E(u):=\frac{1}{2}\int_{W}u^{*}\omega$

whenever this integral is defined.

When $u$ is pseudoholomorphic, the symplectic energy of $u$ is defined and
non-negative, although not necessarily finite.

We consider the following moduli spaces. Recall from the introduction that $L$
is a closed, relatively exact Lagrangian in $X$ and $L^{\prime}$ is
Hamiltonian isotopic to $L$. We denote by $Z:=\mathbb{R}+[0,1]i$ the infinite
strip.

######  Definition 3.2.

We define

$\mathcal{D}_{L,L^{\prime}}:=\left\\{u\in
C^{\infty}(Z,X):|E(u)|<\infty,\;u(\mathbb{R})\subseteq
L,\;u(\mathbb{R}+i)\subseteq L^{\prime}\right\\}.$

It contains
$\mathcal{M}_{L,L^{\prime}}:=\\{u\in\mathcal{D}_{L,L^{\prime}}:\overline{\partial}_{J}u=0\\}$
as the subspace of pseudoholomorphic maps.

Given $\ell>0$ and $A\geq 0$, we set

$\mathcal{F}_{\ell,A}:=\left\\{u\in C^{\infty}(Z_{\ell},X):E(u)\leq
A,\;\overline{\partial}_{J}u=0,\;u([-\ell,\ell])\subseteq
L,\;u([-\ell,\ell]+i)\subseteq L^{\prime}\right\\}.$

Given $\ell\geq 0$ and a Hamiltonian family $\\{L_{t}\\}_{t\in[0,1]}$ of
Lagrangians in $X$ with $L_{0}=L$ and $L_{1}=L^{\prime}$, we define the moduli
space

$\mathcal{P}_{\ell}:=\left\\{u\in
C^{\infty}(G_{\ell},X):\overline{\partial}_{J}u=0,\;u(s+it)\in L_{t}\text{ for
}s+it\in\partial G_{\ell}\right\\}.$

All of these spaces are endowed with the weak $C^{\infty}$ Whitney topology.
By the Nash Embedding Theorem applied to the metric
$g_{J}=\omega(\cdot,J\cdot)$, this topology is metrisable. Hofer showed in
[Hof88, Theorems 1 and 2] that the moduli spaces $\mathcal{P}_{\ell}$ and
$\mathcal{M}_{L,L^{\prime}}$ are compact.

Evaluation at $0\in\mathbb{C}$ defines a continuous map, denoted by $\pi$,
from each of these spaces to $L$.

######  Remark 3.3.

Pick some Hamiltonian isotopy $\\{\psi^{t}\\}_{t\in[0,1]}$ such that
$\psi^{t}(L)=L_{t}$ for all $t$. Setting

$L_{x+iy}:=L_{y}\qquad\quad\text{and}\qquad\quad\psi^{t}_{x+iy}:=\psi^{ty}$

for $x+iy$ in $\partial G$ shows that $\mathcal{P}_{\ell}$ is the space of
pseudoholomorphic maps $G_{\ell}\to M$ of finite energy which map
$z\in\partial G_{\ell}$ to $L_{z}$, i.e, of the form (2.3.2). This allows us
to apply Theorem 2.12 with $\mathcal{P}=\mathcal{P}_{\ell}$.

We require the following uniform energy bound.

######  Lemma 3.4 ([Hof88, Lemma 2]).

The symplectic energy is uniformly bounded on all $\mathcal{P}_{\ell}$. More
precisely, there exists a constant $C\geq 0$ such that for all $\ell>0$ and
all $u$ in $\mathcal{P}_{\ell}$, we have $E(u)\leq C$.

Fix a smooth cutoff function $\rho:\mathbb{R}\rightarrow[0,1]$ with

$\rho(t)=\begin{cases}1\quad&t\leq\frac{1}{2}\\\
0\quad&t\geq\frac{3}{2}\end{cases}$

and define for $\ell>0$ the function
$r_{\ell}:\mathcal{F}_{\ell,A}\rightarrow\mathcal{D}_{L,L^{\prime}}$ by

$r_{\ell}(u)(x+iy):=u(\rho(\ell^{-1}|x|)x+iy).$

By construction, $r_{\ell}(u)$ agrees with $u$ on $Z_{\frac{\ell}{2}}$.

We require the following result which one should consider as a variant of
Gromov compactness.

######  Proposition 3.5 ([Hof88, Proposition 3]).

For any neighbourhood $U$ of $\mathcal{M}_{L,L^{\prime}}$ in
$\mathcal{D}_{L,L^{\prime}}$ and any $A\geq 0$, there exists $\ell_{0}>0$ such
that $r_{\ell}(\mathcal{F}_{\ell,A})\subseteq U$ for all $\ell\geq\ell_{0}$.

###### Proof of Proposition 1.8.

Let $C$ be the uniform energy bound from Lemma 3.4. Any $u$ in any
$\mathcal{P}_{\ell}$ clearly satisfies $E(u|_{Z_{\ell}})\leq C$. Picking $U$
an open neighbourhood of $\mathcal{M}_{L,L^{\prime}}$ in
$\mathcal{D}_{L,L^{\prime}}$, and taking $\ell_{0}$ as in Proposition 3.5 with
$A=C$, we obtain a commutative diagram

${\mathcal{P}_{\ell_{0}}}$${\mathcal{F}_{\ell_{0},C}}$${U}$${L}$$\scriptstyle{\cdot|_{Z_{\ell_{0}}}}$$\scriptstyle{\pi}$$\scriptstyle{r_{\ell_{0}}}$$\scriptstyle{\pi}$

By Theorem 2.12, the pullback $\pi^{*}:R^{*}(L)\rightarrow R^{*}(U)$ must thus
be injective. Using the isomorphism

$R^{*}(\mathcal{M}_{L,L^{\prime}})\cong\varinjlim R^{*}(U)$

(taking a direct limit over open neighbourhoods of
$\mathcal{M}_{L,L^{\prime}}$ in $\mathcal{D}_{L,L^{\prime}}$) given by [AMS21,
Lemma 5.2] and the exactness of the direct limit functor, we may conclude. ∎

######  Remark 3.6.

[AMS21, Lemma 5.2] states that generalised cohomology theories as defined in
Section 2.1 satisfy the continuity axiom. This is one key ingredient for our
generalisation of the results in [Hof88].

##  4 Lusternik-Schnirelmann theory

Our goal in this section is the proof of Theorem 1.9. Observe that there is a
natural $\mathbb{R}$-action on $\mathcal{M}_{L,L^{\prime}}$, by setting
$t\cdot u:=u(\cdot-t)$. The fixed points of this action are exactly the
constant maps to points in $L\cap L^{\prime}$. Hence there is a bijection
between these fixed points and $L\cap L^{\prime}$.

######  Lemma 4.1 ([Hof88, Lemma 4]).

There exists a continuous map
$\sigma:\mathcal{M}_{L,L^{\prime}}\rightarrow\mathbb{R}$ such that for any $u$
which is not a fixed point of the $\mathbb{R}$-action, the function
$t\mapsto\sigma(t\cdot u)$ is strictly decreasing.

###### Sketch of the construction.

One should think of $\sigma$ as something akin to the Floer action functional.
Indeed, if $X$ is Liouville and $L$ is an exact Lagrangian, we can take
$\sigma$ to be the usual Floer action functional.

If not, for each path component $Q$ in $\mathcal{M}_{L,L^{\prime}}$, we fix a
basepoint $u_{0}$ in $Q$, and define $\sigma(u_{0}):=0$. Then for some other
$u_{1}$ in $Q$, we pick a path $\\{u_{t}\\}_{t\in[0,1]}$ from $u_{0}$ to
$u_{1}$, and define

$\sigma(u_{1})=\int_{[0,1]^{2}}v^{*}\omega$

where $v:[0,1]^{2}\rightarrow X$ is a smoothing (rel endpoints) of the map
sending $(s,t)$ to $u_{s}(ti)$. This is well-defined due to relative
exactness. ∎

Fix some basepoint $x_{0}\in L$. For any subset $S$ of
$\mathcal{M}_{L,L^{\prime}}$ or $\mathcal{D}_{L,L^{\prime}}$, we consider the
map of pairs

$\pi_{S}:(S,\emptyset)\rightarrow(L,x_{0}):u\mapsto u(0),$

as well as the pullback

$\pi_{S}^{*}:R^{*}(L,x_{0})\rightarrow R^{*}(S).$

######  Definition 4.2.

To each subset $S$ of $\mathcal{M}_{L,L^{\prime}}$, we assign the non-negative
integer

$I(S):=\min\left\\{k\geq
1:\exists\;U_{1},\ldots,U_{k}\subseteq\mathcal{M}_{L,L^{\prime}}\text{ open
}:S\subseteq U_{1}\cup\dots\cup U_{k}\text{ and }\pi^{*}_{U_{i}}=0\right\\}.$

Note $I$ has a uniform upper bound. Indeed, let $N$ be the minimal number of
contractible open subsets of $L$ required to cover $L$. Then $I(S)\leq N$ for
any $S\subseteq\mathcal{M}_{L,L^{\prime}}$.

######  Lemma 4.3.

Fix subsets $S$ and $T$ of $\mathcal{M}_{L,L^{\prime}}$.

  1. 1.

If $S\subseteq T$, then $I(S)\leq I(T)$.

  2. 2.

There is some open neighbourhood $U$ of $S$ such that $I(S)=I(U)$.

  3. 3.

$I(S\cup T)\leq I(S)+I(T)$.

  4. 4.

If $\\{\varphi_{t}\\}_{t\in\mathbb{R}}$ is a flow on
$\mathcal{M}_{L,L^{\prime}}$, then $I(S)=I(\varphi_{t}(S))$ for all
$t\in\mathbb{R}$.

  5. 5.

$I(\\{u_{1},\dots,u_{n}\\})=1$ for any
$u_{1},\dots,u_{n}\in\mathcal{M}_{L,L^{\prime}}.$

Thus $I$ is an _index function_ in the sense of [Rud99, Definition 4.2].

###### Proof.

If $S\leq T$, we take the minimum over a larger set, so the inequality is
immediate. If $U_{1},\ldots,U_{I(S)}$ are open subsets of
$\mathcal{M}_{L,L^{\prime}}$ covering $S$ with $\pi^{*}_{U_{i}}=0$ for all
$i$, set

$U=U_{1}\cup\ldots\cup U_{I(S)}$

Then $I(U)\leq I(S)$, so equality holds by (1). The union of two suitable open
covers for $S$, respectively $T$ defines a suitable open cover for $S\cup T$
which must have cardinality at least $I(S\cup T)$. This shows 3. As
$\varphi_{t}$ is homotopic to the identity, it takes a suitable cover for $S$
to a suitable cover for $\varphi_{t}(S)$. Thus $I(\varphi_{t}(S))\leq I(S)$,
and equality follows from applying the same argument to
$\varphi_{-t}(\varphi_{t}(S))$. To see the last claim, denote
$\\{p_{1},\dots,p_{k}\\}=\\{u_{1}(0),\dotsc,u_{n}(0)\\}$. For each $j\leq k$
choose a contractible open neighbourhood $V_{j}$ of $p_{j}$ in $L$, such that
$\overline{V_{i}}\cap\overline{V_{j}}=\emptyset$ for $i\neq j$. Then the
preimage $U$ = $\pi^{-1}(V_{1}\cup\dots\cup V_{k})$ defines a suitable cover
for $\\{u_{1},\dots,u_{n}\\}$. ∎

######  Lemma 4.4.

$I(\mathcal{M}_{L,L^{\prime}})\geq c_{R}(L)$.

###### Proof.

Fix an open cover $U_{1},\ldots,U_{k}$ of $\mathcal{M}_{L,L^{\prime}}$ such
that $\pi^{*}_{U_{i}}=0$ for $i\leq k$ and let $\alpha_{1},\dots,\alpha_{k}\in
R^{*}(L,x_{0})$ be arbitrary. By Lemma 2.2 the product
$\pi_{\mathcal{M}_{L,L^{\prime}}}^{*}\alpha_{1}\cdot\ldots\cdot\pi_{\mathcal{M}_{L,L^{\prime}}}^{*}\alpha_{k}$
vanishes in $R^{*}(\mathcal{M}_{L,L^{\prime}})$. By Theorem 2.12,
$\pi_{\mathcal{M}_{L,L^{\prime}}}^{*}$ is injective, so
$\alpha_{1}\cdot\ldots\cdot\alpha_{k}=0$ in $R^{*}(L,x_{0})$ and $c_{R}(L)\leq
k$. Taking the infimum over all such open covers completes the proof. ∎

Given Lemma 4.4, the proof of Theorem 1.9 reduces to showing that

$\\#L\cap L^{\prime}\geq I(\mathcal{M}_{L,L^{\prime}})$ (4.0.1)

Since $I$ is an index function, this follows from [Rud99, Theorem 4.2]. For
the sake of exposition, we give a proof here, using standard Lusternik-
Schnirelman theory as in [Hof88, Section V].

######  Definition 4.5.

For $1\leq i\leq I(\mathcal{M}_{L,L^{\prime}})$, we define

$d_{i}:=\inf_{I(S)\geq i}\sup\sigma(S)$

where the infimum is taken over subsets of $\mathcal{M}_{L,L^{\prime}}$.

For any $d\in\mathbb{R}$, we denote

$Cr(d):=\left\\{u\in\mathcal{M}_{L,L^{\prime}}:\sigma(u)=d,\;\mathbb{R}\cdot
u=u\right\\}.$

It follows that

$\sum_{d}\\#Cr(d)=\\#L\cap L^{\prime}.$

######  Lemma 4.6.

$-\infty<d_{1}\leq\ldots\leq d_{I(\mathcal{M}_{L,L^{\prime}})}<\infty.$

###### Proof.

First observe that $d_{j}\leq d_{j+1}$ for all $j$ since we take the infimum
over a smaller set. The compactness of $\mathcal{M}_{L,L^{\prime}}$ implies
that $-\infty<d_{1}$ and $d_{I(\mathcal{M}_{L,L^{\prime}})}<\infty$. ∎

######  Lemma 4.7.

For any neighbourhood $U$ of $Cr(d)$, there exists some $\varepsilon>0$ such
that

$u\in\sigma^{-1}((-\infty,d+\varepsilon])\setminus U\quad\Rightarrow\quad
1\cdot u\in\sigma^{-1}((-\infty,d-\varepsilon]).$

###### Proof.

This follows from the compactness of $\sigma^{-1}((-\infty,d])\setminus U$
along with the continuity of the $\mathbb{R}$-action. ∎

######  Lemma 4.8.

$Cr(d_{j})$ is non-empty for all $j$.

###### Proof.

Suppose $Cr(d_{j})$ is empty. Applying Lemma 4.7 to $U=\emptyset$ we obtain
some $\varepsilon>0$ such that

$1\cdot\sigma^{-1}((-\infty,d_{j}+\varepsilon])\subseteq\sigma^{-1}((-\infty,d_{j}-\varepsilon]).$

By definition of $d_{j}$, there exists $S\subseteq\mathcal{M}_{L,L^{\prime}}$
such that $I(S)\geq j$ and

$d_{j}\leq\sup\sigma(S)\leq d_{j}+\varepsilon.$

But then $I(1\cdot S)\geq j$ and $\sup\sigma(1\cdot S)<d_{j}$, which is a
contradiction. ∎

######  Lemma 4.9.

If $d_{j}=d_{j+1}$ for any $j$, then $Cr(d_{j})$ is infinite.

###### Proof.

If $Cr(d_{j})$ is finite, then $I(Cr(d_{j}))=1$ by Lemma 4.3.(5). So it
suffices to show that $I(Cr(d_{j}))\geq 2$. Suppose by contradiction
$I(Cr(d_{j}))\leq 1$. Since $Cr(d_{j})$ is non-empty, we must have equality.
Then there is some open neighbourhood $U$ of $Cr(d_{j})$ such that
$\pi^{*}_{U}=0$. Given this $U$, fix $\varepsilon>0$ as in the statement of
Lemma 4.7.

Choose $S\subseteq\mathcal{M}_{L,L^{\prime}}$ such that $I(S)\geq j+1$ and

$d_{j}\leq\sup\sigma(S)\leq d_{j}+\varepsilon.$

Then $I(1\cdot(S\setminus U))\geq j$ but $\sigma(1\cdot(S\setminus U))\leq
d_{j}-\varepsilon$, a contradiction. ∎

The inequality in (4.0.1), and hence Theorem 1.9, follows from Lemmas 4.8 and
4.9.

##  5 Two examples

Let

$\operatorname{Sp}(n):=\text{Sp}(2n;\mathbb{C})\cap U(2n)$

be the _compact symplectic group_. It is a compact simply-connected Lie group
of dimension $n(2n+1)$. The zero section defines a Lagrangian embedding
$\operatorname{Sp}(n)\hookrightarrow T^{*}\operatorname{Sp}(n)$, where we
endow $T^{*}\operatorname{Sp}(n)$ with the canonical symplectic structure. As
this embedding is a homotopy equivalence, $\operatorname{Sp}(n)$ is relatively
exact. We will consider $\operatorname{Sp}(2)$ and $\operatorname{Sp}(3)$
since their cuplength with respect to a certain generalised cohomology theory
was computed in [IM04] (see also [Kis07]) and is strictly greater than their
cuplength with respect to integral cohomology.

######  Proposition 5.1 ([IMN03, IM04]).

The mod-2 and integral cuplengths of $\operatorname{Sp}(2)$ are

$c_{\mathbb{Z}/2}(\operatorname{Sp}(2))=c_{\mathbb{Z}}(\operatorname{Sp}(2))=3,$

while

$c_{h^{*}}(\operatorname{Sp}(2))=4$

where $h^{*}$ is the cohomology theory associated to the truncated sphere
spectrum $\mathbb{S}[0,2]$. In particular, [IM04] shows that its cuplength in
real $K$-theory is

$c_{KO}(\operatorname{Sp}(2))=4.$

Similarly the cuplengths of $\operatorname{Sp}(3)$ in the same cohomology
theories are given by

$c_{\mathbb{Z}/2}(\operatorname{Sp}(3))=c_{\mathbb{Z}}(\operatorname{Sp}(3))=4$

and

$c_{h^{*}}(\operatorname{Sp}(3))=c_{KO}(\operatorname{Sp}(3))=5.$

######  Remark 5.2.

We use Hofer’s convention in [Hof88] for cuplengths which differs by one
compared to that of [IM04].

Since $\operatorname{Sp}(2)$ and $\operatorname{Sp}(3)$ are Lie groups, they
are parallelisable. By Proposition 1.7 we can therefore apply Theorem 1.9 with
real $K$-theory to either one as the zero section lying inside its cotangent
bundle. This gives a (strictly) stronger bound on the Arnol’d number than
Hofer’s cup length estimate, though this estimate was already known due to
work of Laudenbach and Sikorav [LS85], using finite-dimensional
approximations.

######  Corollary 5.3.

The minimum number of intersection points between a relatively exact
Lagrangian embedding of $\operatorname{Sp}(2)$ (satisfying Proposition 1.7.4)
and its image under any Hamiltonian diffeomorphism is at least $4$. The same
is true for $\operatorname{Sp}(3)$ with $5$ instead of $4$.

######  Proposition 5.4.

The critical number of $\operatorname{Sp}(2)$ is 4.

###### Proof.

The critical number of $\operatorname{Sp}(2)$ is bounded below by 4 by
Proposition 5.1. On the other hand, [Sma62, Theorem 6.1] combined with the
computation of its integral homology in Lemma 5.5 implies that the Morse
number of $\operatorname{Sp}(2)$ is $4$, which is an upper bound for the
critical number. ∎

The bound for $\operatorname{Sp}(2)$ (and a stronger bound for
$\operatorname{Sp}(3)$) were shown by [LS85] for the respective zero section
in the cotangent bundle. However, their methods are specifically geared
towards cotangent bundles while our bounds persists under Weinstein handle
attachments. As an example, one can plumb a copy of
$T^{*}\operatorname{Sp}(2)$ containing $\operatorname{Sp}(2)$ as the zero
section, with the cotangent bundle of any other $2$-connected manifold of the
same dimension to obtain a new Weinstein manifold $X$. Since
$\operatorname{Sp}(2)$ is $2$-connected, the resulting manifold admits the the
same $2$-skeleton (up to homotopy) as $T^{*}\operatorname{Sp}(2)$, which is
trivial. Therefore Proposition 1.74 still holds for the embedding
$\operatorname{Sp}(2)\hookrightarrow X$. This gives a stronger bound than
Hofer’s estimate, in a case for which the estimate of [LS85] does not apply.

We recap the computation of the integral cohomology rings of
$\operatorname{Sp}(2)$ and $\operatorname{Sp}(3)$ here.

######  Lemma 5.5 ([IM04]).

The integral cuplengths of $\operatorname{Sp}(2)$ and $\operatorname{Sp}(3)$
are given by

$c_{\mathbb{Z}}(\operatorname{Sp}(2))=3\quad\text{and}\quad
c_{\mathbb{Z}}(\operatorname{Sp}(3))=4.$

Furthermore, $H^{*}(\operatorname{Sp}(2))$ and $H^{*}(\operatorname{Sp}(3))$
are free of rank 4 and 8 respectively, over both $\mathbb{Z}$ and
$\mathbb{Z}/2$.

###### Proof.

Given $n$ we can identify $\mathbb{H}^{n}$ with $\mathbb{R}^{4n}$ to see the
existence of a principal $\operatorname{Sp}(n-1)$-bundle
$\operatorname{Sp}(n)\to S^{4n-1}$ induced by the canonical action on the unit
quaternions. Thus we can apply the Leray-Serre spectral sequence to compute
the cohomology of $\operatorname{Sp}(2)$ with coefficients in $A=\mathbb{Z}$
or $A=\mathbb{Z}/2$. The $E_{2}$-page is given by

$E^{p,q}_{2}=H^{p}(S^{7},H^{q}(\operatorname{Sp}(1);A))$

which vanishes for $p\notin\\{0,7\\}$ and $q\notin\\{0,3\\}$. Hence the
spectral sequence collapses for degree reasons at the second page. As
$\operatorname{Sp}(1)\cong S^{3}$, we obtain

$H^{*}(\operatorname{Sp}(2);A)\cong H^{*}(S^{7};A)\otimes_{A}H^{*}(S^{3};A).$
(5.0.1)

By the multiplicativity of the spectral sequence, (5.0.1) is an isomorphism of
graded rings. Therefore,

$H^{n}(\operatorname{Sp}(2);A)=\begin{cases}A\quad&n\in\\{0,3,7,10\\},\\\
0\quad&\text{otherwise}\end{cases}$

and we can deduce the values of $c_{\mathbb{Z}/2}(\operatorname{Sp}(2))$ and
$c_{\mathbb{Z}}(\operatorname{Sp}(2))$.

The same argument gives an isomorphism of graded rings

$H^{*}(\operatorname{Sp}(3);A)\cong
H^{*}(S^{11};A)\otimes_{A}H^{*}(\operatorname{Sp}(2);A).$

Therefore,

$H^{n}(\operatorname{Sp}(2);A)=\begin{cases}A\quad&n\in\\{0,3,7,10,11,14,18,21\\},\\\
0\quad&\text{otherwise}\end{cases}$

from which we deduce the corresponding statements for $\operatorname{Sp}(3)$.
∎

## References

  * [AB21] Mohammed Abouzaid and Andrew J. Blumberg.  Arnold conjecture and Morava K-theory, 2021.
  * [Ada95] Frank Adams.  Stable homotopy and generalised homology.  Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1995.  Reprint of the 1974 original.
  * [AMS21] Mohammed Abouzaid, Mark McLean, and Ivan Smith.  Complex cobordism, Hamiltonian loops and global Kuranishi charts, 2021\.  arXiv:2103.01507.
  * [CJS95] Ralph L. Cohen, John D. S. Jones, and Graeme B. Segal.  Floer’s infinite-dimensional Morse theory and homotopy theory.  In The Floer memorial volume, volume 133 of Progr. Math., pages 297–325. Birkhäuser, Basel, 1995.
  * [CLOT03] Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanré.  Lusternik-Schnirelmann category, volume 103 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2003.
  * [Coh09] Ralph L. Cohen.  Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners.  In Algebraic topology, volume 4 of Abel Symp., pages 39–59. Springer, Berlin, 2009.
  * [Flo88] Andreas Floer.  Morse theory for Lagrangian intersections.  J. Differential Geom., 28(3):513–547, 1988.
  * [Flo89] Andreas Floer.  Cuplength estimates on Lagrangian intersections.  Comm. Pure Appl. Math., 42(4):335–356, 1989.
  * [FO99] Kenji Fukaya and Kaoru Ono.  Arnold conjecture and Gromov-Witten invariant.  Topology, 38(5):933–1048, 1999.
  * [FOOO09] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.  Lagrangian intersection Floer theory: anomaly and obstruction. Part I, volume 46 of AMS/IP Studies in Advanced Mathematics.  American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009.
  * [GG20] Viktor L. Ginzburg and Başak Z. Gürel.  Lusternik-Schnirelmann theory and closed Reeb orbits.  Math. Z., 295(1-2):515–582, 2020.
  * [Gon21a] Wenmin Gong.  Lagrangian Ljusternik-Schnirelman theory and Lagrangian intersections, 2021.
  * [Gon21b] Wenmin Gong.  A short proof of cuplength estimates on Lagrangian intersections, 2021\.
  * [Hof85] Helmut Hofer.  Lagrangian embeddings and critical point theory.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 2(6):407–462, 1985.
  * [Hof88] Helmut Hofer.  Lusternik-Schnirelman-theory for Lagrangian intersections.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 5(5):465–499, 1988.
  * [IM04] Norio Iwase and Mamoru Mimura.  L-S Categories of Simply-connected Compact Simple Lie Groups of Low Rank, pages 199–212.  Birkhäuser Basel, Basel, 2004.
  * [IMN03] Norio Iwase, Mamoru Mimura, and Tetsu Nishimoto.  Lusternik-Schnirelmann categories of non-simply connected compact simple Lie groups.  arXiv Mathematics e-prints, page math/0303085, March 2003.
  * [Kis07] Daisuke Kishimoto.  L-S category of quaternionic Stiefel manifolds.  Topology and its Applications, 154(7):1465–1469, 2007.  Special Issue: The Third Joint Meeting Japan-Mexico in Topology and its Applications.
  * [Kli78] Wilhelm Klingenberg.  Lectures on closed geodesics.  Grundlehren der Mathematischen Wissenschaften, Vol. 230. Springer-Verlag, Berlin-New York, 1978.
  * [Kui65] Nicolaas H. Kuiper.  The homotopy type of the unitary group of Hilbert space.  Topology, 3:19–30, 1965.
  * [LO96] Hông Vân Lê and Kaoru Ono.  Cup-length estimates for symplectic fixed points.  In Contact and symplectic geometry (Cambridge, 1994), volume 8 of Publ. Newton Inst., pages 268–295. Cambridge Univ. Press, Cambridge, 1996.
  * [LS85] François Laudenbach and Jean-Claude Sikorav.  Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibré cotangent.  Invent. Math., 82(2):349–357, 1985.
  * [MS04] Dusa McDuff and Dietmar Salamon.  $J$-holomorphic curves and symplectic topology, volume 52 of American Mathematical Society Colloquium Publications.  American Mathematical Society, Providence, RI, 2004.
  * [MS17] Dusa McDuff and Dietmar Salamon.  Introduction to symplectic topology, volume 27 of Oxford Graduate Texts in Mathematics.  Ofxford University Press, third edition, 2017.
  * [Oh93] Yong-Geun Oh.  Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.  Comm. Pure Appl. Math., 46(7):949–993, 1993.
  * [OR99] John Oprea and Yuli B. Rudyak.  On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture.  Math. Z., 230(4):673–678, 1999.
  * [Par16] John Pardon.  An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves.  Geom. Topol., 20(2):779–1034, 2016.
  * [Por22] Noah Porcelli.  Families of relatively exact Lagrangians, free loop spaces and generalised homology, 2022.  arXiv:2202.09677.
  * [Rud98] Yuli B. Rudyak.  On Thom spectra, orientability, and cobordism.  Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.  With a foreword by Haynes Miller.
  * [Rud99] Yuli B. Rudyak.  On analytical applications of stable homotopy (the Arnold conjecture, critical points).  Math. Z., 230(4):659–672, 1999.
  * [Sma62] S. Smale.  On the structure of manifolds.  Amer. J. Math., 84:387–399, 1962.
  * [tD08] Tammo tom Dieck.  Algebraic topology.  EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008.

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