# Unobstructed Lagrangian cobordism groups of surfaces

Dominique Rathel-Fournier<EMAIL_ADDRESS>
###### Abstract.

We study Lagrangian cobordism groups of closed symplectic surfaces of genus
$g\geq 2$ whose relations are given by unobstructed, immersed Lagrangian
cobordisms. Building upon work of Abouzaid [Abo08] and Perrier [Per19], we
compute these cobordism groups and show that they are isomorphic to the
Grothendieck group of the derived Fukaya category of the surface.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Unobstructed cobordisms and the proof of Theorem A
  4. 4 Topologically unobstructed cobordisms
  5. 5 Topological obstruction of surgery cobordisms in dimension 2
  6. 6 Lagrangian cobordism invariants of curves
  7. 7 Relations between isotopic curves
  8. 8 Action of the mapping class group
  9. 9 Computation of $\Omega_{\operatorname{unob}}(\Sigma)$
  10. A Action-type invariants of Lagrangians in monotone manifolds
  11. B Proof of Proposition 7.6

##  1\. Introduction

Lagrangian cobordism is a relation between Lagrangian submanifolds of a
symplectic manifold $(M,\omega)$ that was introduced by Arnold [Arn80]. A
consequence of the Gromov-Lees h-principle is that the study of general
immersed Lagrangian cobordisms essentially reduces to algebraic topology, see
[Eli84, Aud87]. In contrast, cobordisms satisfying suitable geometric
constraints display remarkable rigidity phenomena. A fundamental example of
this is the proof by Biran and Cornea [BC13, BC14] that monotone, embedded
Lagrangian cobordisms give rise to cone decompositions in the derived Fukaya
category $\operatorname{DFuk}(M)$. See also [BC17, BCS21, BC21] for further
developments, as well as the work of Nadler and Tanaka [NT20], who obtained
related results from a different perspective.

A natural problem is to establish to what extent Lagrangian cobordisms
determine the structure of $\operatorname{DFuk}(M)$. In this paper, we
consider the following formalization of this question. Fix classes of
Lagrangians in $M$ and of Lagrangian cobordisms in $\mathbb{C}\times M$,
possibly equipped with extra structures (the precise classes considered in
this paper will be specified shortly). The associated _Lagrangian cobordism
group_ $\Omega(M)$ is the abelian group whose elements are formal sums of
Lagrangians in $M$, modulo relations coming from Lagrangian cobordisms. On the
side of $\operatorname{DFuk}(M)$, part of the information about cone
decompositions is encoded in the _Grothendieck group_
$K_{0}\operatorname{DFuk}(M)$. (See Section 2 for precise definitions of
$\Omega(M)$ and $K_{0}\operatorname{DFuk}(M)$.)

Whenever the cone decompositions results of [BC14] can be extended to the
classes of Lagrangians under consideration, there is an induced surjective
morphism

$\Theta:\Omega(M)\to K_{0}\operatorname{DFuk}(M),$

see [BC14, Corollary 1.2.1]. Biran and Cornea posed the following question.

######  Question 1.

When is the map $\Theta$ an isomorphism?

There are only two cases for which $\Theta$ is known to be an isomorphism (for
appropriate classes of Lagrangians): the torus $T^{2}$ by work of Haug [Hau15]
(see also [Hen20] for a refinement of this result), and Liouville surfaces by
work of Bosshard [Bos21].

In this paper, we consider this problem in the case of a closed surface
$\Sigma$ of genus $g\geq 2$. In this case, the group
$K_{0}\operatorname{DFuk}(\Sigma)$ was computed by Abouzaid [Abo08], who
showed that there is an isomorphism

$K_{0}\operatorname{DFuk}(\Sigma)\cong
H_{1}(S\Sigma;\mathbb{Z})\oplus\mathbb{R},$ (1)

where $S\Sigma$ is the unit tangent bundle of $\Sigma$. The main goal of this
paper is to answer Question 1 by exhibiting a Lagrangian cobordism group
$\Omega_{\operatorname{unob}}(\Sigma)$ whose relations are given by
_unobstructed_ immersed cobordisms, such that the associated map
$\Theta:\Omega_{\operatorname{unob}}(\Sigma)\to
K_{0}\operatorname{DFuk}(\Sigma)$ is an isomorphism.

###  1.1. Main results

####  1.1.1. Unobstructed cobordisms and cone decompositions

Our first main result is an extension of the cone decomposition results of
[BC14] to a class of immersed Lagrangian cobordisms that are _unobstructed_ ,
in the sense that they have a well-behaved Floer theory. There exist many
different mechanisms to achieve unobstructedness in the litterature, at
varying levels of generality and technical complexity; see Remark 1.3 for a
discussion. The precise notion of unobstructedness considered in this work is
the following: we say that an immersed Lagrangian cobordism is _quasi-exact_
if it does not bound $J$-holomorphic disks and teardrops for suitable almost
complex structures $J$ (see Section 3.2 for the precise definition). In this
work, we will use the terms _quasi-exact_ and _unobstructed_ interchangeably.

To state the first theorem, we let $(M,\omega)$ be a closed symplectic
manifold which is symplectically aspherical, in the sense that
$\omega|_{\pi_{2}(M)}=0$. Recall that a Lagrangian submanifold $L\subset M$ is
called weakly exact if $\omega|_{\pi_{2}(M,L)}=0$. In this paper, we assume
that all Lagrangians and Lagrangian cobordisms are oriented and equipped with
a spin structure.

######  Theorem A.

Let $V:L\rightsquigarrow(L_{1},\ldots,L_{k})$ be a quasi-exact cobordism
between weakly exact, embedded Lagrangian submanifolds of $M$. Then $L$ admits
a cone decomposition in $\operatorname{DFuk}(M)$ with linearization
$(L_{1},\ldots,L_{k})$.

We make a fews remarks on Theorem A.

######  Remark 1.1.

In this work, we will only apply Theorem A to the case of curves on surfaces
of genus $g\geq 2$. However, since the weakly exact case requires no
additional ingredients, we include it for the sake of generality.

######  Remark 1.2.

The notion of quasi-exactness was introduced in [BCS21] in the case of
embedded cobordisms. The same notion also appears in the work of Sheridan-
Smith [SS21] under the name of _unobstructed Lagrangian branes_. The
definition given here is a straightforward generalization of this notion to
the case of immersed Lagrangians, where one must also have control over
holomorphic disks with corners at self-intersection points, of which teardrops
are a special case. One of the results of [BCS21] is a cone decomposition
theorem for embedded quasi-exact cobordisms; Theorem A is thus an extension of
this result to the immersed case.

######  Remark 1.3.

  1. (i)

It is expected that Theorem A should generalize to Lagrangian cobordisms that
may bound disks and teardrops, but are nevertheless _Floer-theoretically
unobstructed_ in a suitable sense. See the work of Biran-Cornea [BC21] for
such a result in the case of exact Lagrangians equipped with an extra
structure called a _marking_. See also the work of Hicks [Hic22b][Hic22a] for
partial results concerning Lagrangian cobordisms that are _unobstructed by
bounding cochains_ in the sense of [FOOO09] and [AJ10].

The advantage of the class of unobstructed cobordisms considered in this paper
is that it requires much less technical machinery to set up Floer theory. The
tradeoff is that cobordisms that are unobstructed in this more restrictive
sense are rarer. For instance, most surgery cobordisms between curves on
surfaces bound teardrops, which makes the computation of
$\Omega_{\operatorname{unob}}(\Sigma)$ more delicate.

  2. (ii)

For the purpose of answering Question 1, one seeks a class of Lagrangian
cobordisms which is

    1. (a)

constrained enough that one can prove a result analogous to Theorem A, ideally
with a minimal amount of technical machinery;

    2. (b)

large enough that one can construct enough cobordisms – for example Lagrangian
surgeries – to recover all the relations in $K_{0}\operatorname{DFuk}(M)$.

There is an obvious conflict between these two requirements, and the class of
unobstructed Lagrangian cobordisms that we consider is in some sense a
compromise between them. There are simpler classes of cobordisms that are
known to satisfy condition (a), but it is unknown whether they also satisfy
(b). For example, in the case of a surface $\Sigma$, it is unknown to the
author whether the class of oriented, monotone, embedded Lagrangian cobordisms
is large enough to recover $K_{0}\operatorname{DFuk}(\Sigma)$.

####  1.1.2. Computation of $\Omega_{\operatorname{unob}}(\Sigma)$ and
comparison with $K$-theory

In the remainder of this work, $\Sigma$ denotes a closed symplectic surface of
genus $g\geq 2$. We denote by $\Omega_{\operatorname{unob}}(\Sigma)$ the
Lagrangian cobordism group whose generators are non-contractible, oriented,
embedded closed curves in $\Sigma$, and whose relations are given by quasi-
exact cobordisms. Our second main result is a computation of
$\Omega_{\operatorname{unob}}(\Sigma)$.

######  Theorem B.

There is an isomorphism

$\Omega_{\operatorname{unob}}(\Sigma)\cong
H_{1}(S\Sigma;\mathbb{Z})\oplus\mathbb{R},$

where $S\Sigma$ is the unit tangent bundle of $\Sigma$.

In Theorem B, the map to $H_{1}(S\Sigma;\mathbb{Z})$ is given by the homology
class of lifts of curves to $S\Sigma$. The map to $\mathbb{R}$ is a
generalization to closed surfaces of the symplectic area of plane curves. At
the level of generators, these invariants are the same as those considered by
Abouzaid in their computation of $K_{0}\operatorname{DFuk}(\Sigma)$.

It follows from Theorem A that the Yoneda embedding induces a well-defined
morphism $\Theta:\Omega_{\operatorname{unob}}(\Sigma)\to
K_{0}\operatorname{DFuk}(\Sigma)$. By combining Theorem B and Abouzaid’s
result (1), we immediately obtain the following corollary, which gives an
answer to Question 1 for higher genus surfaces.

######  Corollary C.

The map $\Theta:\Omega_{\operatorname{unob}}(\Sigma)\longrightarrow
K_{0}\operatorname{DFuk}(\Sigma)$ is an isomorphism.

####  1.1.3. The immersed cobordism group

Let $\Omega_{\operatorname{imm}}(\Sigma)$ be the Lagrangian cobordism group
whose generators are oriented immersed curves in $\Sigma$, and whose relations
are given by all oriented, immersed Lagrangian cobordisms. There is an obvious
forgetful morphism
$\Omega_{\operatorname{unob}}(\Sigma)\to\Omega_{\operatorname{imm}}(\Sigma)$.
As a byproduct of our computation of $\Omega_{\operatorname{unob}}(\Sigma)$,
we also obtain a computation of $\Omega_{\operatorname{imm}}(\Sigma)$.

######  Corollary 1.4.

The natural map
$\Omega_{\operatorname{unob}}(\Sigma)\to\Omega_{\operatorname{imm}}(\Sigma)$
is an isomorphism.

######  Remark 1.5.

Corollary 1.4 corrects an error in Theorem 1.5 of [Per19], where it is claimed
that $\Omega_{\operatorname{imm}}(\Sigma)\cong
H_{1}(\Sigma;\mathbb{Z})\oplus\mathbb{Z}_{\chi(\Sigma)}$, the latter group
being isomorphic to $H_{1}(S\Sigma;\mathbb{Z})$. The error originates from a
mistaken application of the h-principle for Lagrangian immersions in the proof
of Lemma 2.1 of [Per19].

######  Remark 1.6.

Using the Gromov-Lees h-principle for Lagrangian immersions, Eliashberg
[Eli84] computed the groups $\Omega_{\operatorname{imm}}(M)$ in the case of an
exact symplectic manifold $M$ in terms of stable homotopy groups of certain
Thom spectra. The group $\Omega_{\operatorname{imm}}(\Sigma)$ for a surface of
genus $g\geq 2$ can also be computed using these methods.

###  1.2. Outline of the proofs of the main results

####  1.2.1. Outline of the proof of Theorem A

The proof of Theorem A follows the same scheme as the proof of the cone
decomposition results of [BC14] and the extension to embedded quasi-exact
cobordisms developed in [BCS21]. The main technical argument is the
construction of a Fukaya category
$\operatorname{Fuk}_{\operatorname{cob}}(\mathbb{C}\times M)$ whose objects
are quasi-exact cobordisms. The main difference between our setup and that of
[BC14] and [BCS21] is that we consider Lagrangian cobordisms that may have
self-intersections. The Floer theory of Lagrangian immersions was developed in
its most general form by Akaho-Joyce [AJ10]. Restricting to Lagrangians that
do not bound holomorphic disks and teardrops allows us to use the
transversality and compactness arguments of [BC14] and [BCS21] with minimal
modifications, rather than the more complicated virtual perturbation
techniques of [AJ10]. The extension of the results of [BC14] to this case thus
poses no new technical challenges.

####  1.2.2. Outline of the proof of Theorem B

The general structure of the computation of
$\Omega_{\operatorname{unob}}(\Sigma)$ is similar to the computation of
$K_{0}\operatorname{DFuk}(\Sigma)$ in [Abo08]. In fact, many of our arguments
can be seen as translations to the language of Lagrangian cobordisms of
arguments appearing in [Abo08]. The main idea of the proof is to use the
action of the mapping class group $\operatorname{Mod}(\Sigma)$ of $\Sigma$ on
(a quotient of) $\Omega_{\operatorname{unob}}(\Sigma)$ to obtain a simple set
of generators of $\Omega_{\operatorname{unob}}(\Sigma)$. The computation of
the action of $\operatorname{Mod}(\Sigma)$ is done by realizing Dehn twists as
iterated Lagrangian surgeries. The key technical difficulty in doing this is
to perform these surgeries in such a way that the associated surgery
cobordisms are unobstructed.

In Section 5, we give topological criteria that ensure that such cobordisms
are unobstructed, which rely on techniques from surface topology. Roughly
speaking, these criteria state that surgery cobordisms are unobstructed
provided that the curves being surgered do not bound certain types of
polygons. These conditions may be hard to ensure in practice for curves
intersecting in many points. To bypass this difficulty, we use an inductive
argument, based on a technique of Lickorish [Lic64], that reduces the
computation of the action of Dehn twists to the simplest cases of curves which
have only one or two intersections points, where unobstructedness can be
achieved easily. This method may have interesting applications to the study of
$\operatorname{DFuk}(\Sigma)$.

###  1.3. Relation to previous work

Lagrangian cobordism groups of higher genus surfaces were studied previously
by Perrier in their thesis [Per19]. Here, we make some remarks to clarify the
relationship between the present work and [Per19].

The present work is heavily inspired by the results and methods of Perrier. In
particular, the general structure of the computation of
$\Omega_{\operatorname{unob}}(\Sigma)$ and many of the surgery procedures that
we use are the same as in [Per19]. However, it appears that the proofs of the
key unobstructedness results in [Per19] contain significant gaps, which the
present author was unable to fix in a straightforward way. It is therefore one
of the main purposes of this paper to clarify and correct some of the results
and proofs of [Per19].

The main novel contributions of the present work are as follows. First, we
provide a proof that unobstructed cobordisms give rise to cone decompositions,
filling a gap in [Per19]. It should be noted that the class of unobstructed
cobordisms studied in the present work differs from the class considered in
[Per19]. This allows us to deal with some technical issues with the definition
used in [Per19]; see Remark 3.5.

Our second main contribution concerns the proofs that the cobordisms
constructed in [Per19] are unobstructed. In Section 5, we formulate a precise
criterion for the existence of teardrops on $2$-dimensional surgery
cobordisms, fixing a gap in the proof of the main obstruction criterion used
in [Per19] (see Proposition 5.11 therein). We also treat the case of disks
with boundary on surgery cobordisms, which is overlooked in [Per19]. We then
apply these obstruction criteria to verify that the cobordisms appearing in
the computation of $\Omega_{\operatorname{unob}}(\Sigma)$ are unobstructed. As
explained in Section 1.2.2, we use an inductive argument that allows us to
circumvent the delicate combinatorial arguments used in [Per19], greatly
simplifying the proofs.

Finally, we give a direct proof that holonomy is an invariant of oriented
Lagrangian cobordism, and use this to correct the computation of
$\Omega_{\operatorname{imm}}(\Sigma)$ appearing in [Per19], see Remark 1.5.

###  1.4. Organization of the paper

This paper is organized as follows. In Section 2, we recall some well-known
facts about Lagrangian cobordisms and Fukaya categories. In Section 3, we
construct the Fukaya category of unobstructed cobordisms
$\operatorname{Fuk}_{\operatorname{cob}}(\mathbb{C}\times M)$ and use it to
prove Theorem A.

The rest of the paper is devoted to the proof of Theorem B. In Sections 4 and
5, we establish the unobstructedness criteria that we will use repeatedly
throughout the computation of $\Omega_{\operatorname{unob}}(\Sigma)$. In
Section 6, we define the morphism $\Omega_{\operatorname{unob}}(\Sigma)\to
H_{1}(S\Sigma;\mathbb{Z})\oplus\mathbb{R}$ appearing in Theorem B. In Section
7, we determine relations between isotopic curves and reduce the computation
of $\Omega_{\operatorname{unob}}(\Sigma)$ to that of a quotient, the reduced
group $\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$. In Section 8, we
compute the action of Dehn twists on
$\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$. Finally, in Section 9, we
combine the results of Sections 4 – 8 to complete the computation of
$\Omega_{\operatorname{unob}}(\Sigma)$.

This paper has two appendices. In Appendix A, we define cobordism invariants
of Lagrangians in monotone symplectic manifolds, which generalize the holonomy
of curves on higher genus surfaces (as defined by Abouzaid [Abo08]). In
Appendix B, we prove a minor modification of a result appearing in [Per19],
which we state as Proposition 7.6.

###  1.5. Acknowledgements

This work is part of the author’s PhD thesis at the University of Montreal
under the supervision of Octav Cornea and François Lalonde. The author would
like to thank Octav Cornea for his invaluable support and patience throughout
this project. The author would also like to thank Jordan Payette for helpful
discussions in the early stages of this project. The author acknowledges the
financial support of NSERC Grant #504711 and FRQNT Grant #300576.

##  2\. Preliminaries

###  2.1. Conventions and terminology

By a _curve_ in a surface $\Sigma$, we mean an equivalence class of smooth
immersions $S^{1}\to\Sigma$, where two immersions are equivalent if they
differ by an orientation-preserving diffeomorphism of $S^{1}$. By convention,
the circle $S^{1}$ is always equipped with its standard orientation as the
boundary of the disk. A curve is _simple_ if it is an embedding. We will often
identify simple curves with their image in $\Sigma$.

We say that a Lagrangian immersion has _generic self-intersections_ if its
only self-intersections are transverse double points. Likewise, two Lagrangian
immersions $\alpha$ and $\beta$ are in _general position_ if the union
$\alpha\cup\beta$ has generic self-intersections. As in the case of curves, we
will generally not distinguish between two Lagrangian immersions that differ
by a diffeomorphism of the domain isotopic to the identity.

###  2.2. Lagrangian cobordisms

Let $(M,\omega)$ be a symplectic manifold. We equip the product
$\widetilde{M}=\mathbb{C}\times M$ with the symplectic form
$\widetilde{\omega}=\omega_{\mathbb{C}}\oplus\omega$, where
$\omega_{\mathbb{C}}$ is the standard symplectic form on $\mathbb{C}$. We
denote by $\pi_{\mathbb{C}}$ and $\pi_{M}$ the projections on the first and
the second factor, respectively.

Let $(\alpha_{i}:L_{i}\to M)_{i=1}^{r}$ and $(\beta_{j}:L_{j}^{\prime}\to
M)_{j=1}^{s}$ be two finite collections of Lagrangian immersions of compact
manifolds into $M$. We recall the definition of a Lagrangian cobordism between
these collections, following Biran and Cornea [BC13].

######  Definition 2.1.

A Lagrangian cobordism from $(\alpha_{i})_{i=1}^{r}$ to
$(\beta_{j})_{j=1}^{s}$ is a compact cobordism $V$ with positive end $\coprod
L_{i}$ and negative end $\coprod L_{j}^{\prime}$, along with a proper
Lagrangian immersion $\iota_{V}:V\to[0,1]\times\mathbb{R}\times M$ which is
cylindrical near $\partial V$ in the following sense.

For each positive end $L_{i}$, there is a collar embedding
$(1-\varepsilon,1]\times L_{i}\to V$ over which the immersion $\iota_{V}$ is
given by $(t,p)\mapsto(t,i,\alpha_{i}(p))$. Likewise, for each negative end
$L_{j}^{\prime}$ there is a collar embedding $[0,\varepsilon)\times
L_{j}^{\prime}\to V$ over which $\iota_{V}$ is given by
$(t,p)\mapsto(t,j,\beta_{j}(p))$.

We will use the notation $\iota_{V}:(\alpha_{i})\rightsquigarrow(\beta_{j})$
to denote a Lagrangian cobordism, or more succintly
$V:(L_{i})\rightsquigarrow(L_{j}^{\prime})$ when the immersions are clear from
the context.

In this work, we will always assume that the Lagrangians are equipped with an
orientation and spin structure. Likewise, we always assume that cobordisms are
equipped with an orientation and spin structure that restrict compatibly to
the ends, in the sense that the collar embeddings in Definition 2.1 can be
chosen to preserve the orientations and spin structures.

The basic examples of Lagrangian cobordisms are Lagrangian suspensions and
Lagrangian surgeries, which we recall below.

Figure 1. The projection to $\mathbb{C}$ of a cobordism
$V:(L_{i})\rightsquigarrow(L_{j}^{\prime})$.

####  2.2.1. Lagrangian suspension

Recall that a Lagrangian regular homotopy $\phi:L\times I\to M$ is called
_exact_ if for each $t\in I$ the form
$\beta_{t}=\phi_{t}^{*}\iota_{\xi_{t}}\omega$ is exact, where $\xi_{t}$ is the
vector field along $\phi$ that generates the homotopy. A function $H:L\times
I\to\mathbb{R}$ such that $dH_{t}=\beta_{t}$ is called a Hamiltonian
generating $\phi$. We call $\phi$ a _Hamiltonian isotopy_ if each map
$\phi_{t}:L\to M$ is an embedding. Note that in this case $\phi$ extends to an
isotopy of Hamiltonian diffeomorphisms of $M$.

To an exact homotopy $\phi$ generated by a Hamiltonian $H$, we can associate a
Lagrangian cobordism $\eta:L\times I\to\mathbb{C}\times M$ called the
_Lagrangian suspension_ of $\phi$. This cobordism is defined by the following
formula:

$\eta(x,t)=(t,H_{t}(\phi_{t}(x)),\phi_{t}(x)).$

It is easily checked that this is a Lagrangian immersion, and that it is
cylindrical provided that $\phi$ is stationary near the endpoints.

####  2.2.2. Lagrangian surgery

Let $L$ and $L^{\prime}$ be immersed Lagrangians in general position that
intersect at $s\in M$. One can perform the Lagrangian surgery of $L$ and
$L^{\prime}$ at $s$ to eliminate this intersection point, obtaining a new
Lagrangian $L\\#_{s}L^{\prime}$ (which depends on additional choices).
Moreover, there is a Lagrangian cobordism with ends $L$, $L^{\prime}$ and
$L\\#_{s}L^{\prime}$. This operation was described by several authors at
various levels of generality: see for example the works of Arnold [Arn80],
Audin [Aud90], Lalonde-Sikorav [LS91], Polterovich [Pol91] and Biran-Cornea
[BC13].

The Lagrangian surgery construction will be our main source of Lagrangian
cobordisms between curves. We describe it in more details in Section 5.1.

####  2.2.3. Lagrangian cobordism groups

We recall the general definition of the Lagrangian cobordism groups of a
symplectic manifold $M$, following [BC14, Section 1.2]. Let $\mathsf{Lag}$ be
a class of Lagrangian submanifolds of $M$ and $\mathsf{Cob}$ be a class of
Lagrangian cobordisms. The associated cobordism group $\Omega(M)$ is the
quotient of the free abelian group generated by $\mathsf{Lag}$ by the
relations

$L_{1}+\ldots+L_{r}=0$

whenever there is a cobordism $(L_{1},\ldots,L_{r})\rightsquigarrow\emptyset$
that belongs to $\mathsf{Cob}$.

In this paper, we are interested in the Lagrangian cobordism groups of a
closed symplectic surface $\Sigma$. In this case, we take the class
$\mathsf{Lag}$ to consist of all oriented, non-contractible, simple curves in
$\Sigma$. Following the setting of [Abo08], we assume that all curves are
equipped with the bounding spin structure on the circle. Note that the class
$\mathsf{Lag}$ coincides with the objects of $\operatorname{Fuk}(\Sigma)$, as
defined in Section 2.3 below.

There are several interesting choices for the class $\mathsf{Cob}$. In this
paper, we focus on the case where $\mathsf{Cob}$ consists of all _quasi-exact_
cobordisms, which are defined in Section 3.2. We denote by
$\Omega_{\operatorname{unob}}(\Sigma)$ the associated cobordism group.

The inverse in $\Omega_{\operatorname{unob}}(\Sigma)$ is given by reversing
the orientation and spin structure of curves. Note also that any oriented
cobordism between oriented curves admits a spin structure that restricts to
the bounding spin structure on its ends (this is easily seen in the case of
the cylinder and the pair of pants; in the general case, consider a pants
decomposition of the cobordism). This means that dropping the assumption that
cobordisms carry compatible spin structures leads to the same cobordism group,
so that we may ignore spin structures in the computation of
$\Omega_{\operatorname{unob}}(\Sigma)$.111Note that in the context of Theorem
A the cone decomposition induced by a cobordism may depend on the choice of
spin structure, but this dependence is not seen at the level of K-theory.

###  2.3. Fukaya categories

In this section, we define the version of the Fukaya category that will be
considered in this paper. The purpose of this section is mainly to fix our
conventions, hence most details will be omitted. A comprehensive exposition of
the material presented here can be found in the book of Seidel [Sei08]. See
also [Sei11] and [Abo08] for the specific case of surfaces.

In the following, we assume that $(M,\omega)$ is a closed symplectic manifold
which is _symplectically aspherical_ , in the sense that
$\omega|_{\pi_{2}(M)}=0$.

####  2.3.1. Coefficients.

The version of $\operatorname{Fuk}(M)$ that we will consider is a
$\mathbb{Z}_{2}$-graded $A_{\infty}$-category which is linear over the
universal Novikov ring over $\mathbb{Z}$

$\Lambda=\Big{\\{}\sum_{i\geq
0}a_{i}\,q^{t_{i}}:a_{i}\in\mathbb{Z},t_{i}\in\mathbb{R},t_{i}\nearrow\infty\Big{\\}}.$

####  2.3.2. Objects.

Recall that a Lagrangian submanifold $L\subset M$ is _weakly exact_ if
$\omega|_{\pi_{2}(M,L)}=0$. The objects of $\operatorname{Fuk}(M)$ are weakly
exact, spin, connected, closed Lagrangian submanifolds $L\subset M$, equipped
with an orientation and spin structure.

######  Remark 2.2.

In the case where $M$ is a surface, we will follow the setting of [Abo08] and
restrict our attention to the subcategory of $\operatorname{Fuk}(M)$ whose
objects are equipped with the _bounding_ spin structure on $S^{1}$. Note that
there is no obstruction to incorporating curves equipped with the trivial spin
structure. However, this would require minor adjustments to some of the
statements and proofs from [Abo08], and we shall not do so here.

####  2.3.3. Morphisms.

Given two objects $L_{0}$ and $L_{1}$, the morphism space
$\operatorname{Hom}(L_{0},L_{1})$ is the Floer cochain complex
$CF(L_{0},L_{1};H,J)$, which is defined as follows.

The complex depends on a choice of Floer datum
$\mathcal{D}(L_{0},L_{1})=(H,J)$, which consists of a Hamiltonian
$H:[0,1]\times M\to\mathbb{R}$ and a time-dependent compatible almost complex
structure $(J_{t})_{t\in[0,1]}$ on $M$. We require that $\phi_{H}^{1}(L_{0})$
be transverse to $L_{1}$, where $(\phi_{H}^{t})_{t\in[0,1]}$ denotes the
isotopy generated by the Hamiltonian vector field $X_{H}$. Moreover, we assume
that the pair $(H,J)$ is _regular_ in the sense of [Sei08, Section (8i)].

Let $\mathcal{O}(L_{0},L_{1};H)$ be the set of orbits $y:[0,1]\to M$ of the
Hamiltonian flow $\phi_{H}^{t}$ that satisfy $y(0)\in L_{0}$ and $y(1)\in
L_{1}$. Recall that here is a bijection
$\mathcal{O}(L_{0},L_{1};H)\to\phi_{H}^{1}(L_{0})\cap L_{1}$, which associates
to $y$ the endpoint $y(1)$. The _Floer complex_ $CF(L_{0},L_{1};H,J)$ is
defined as the free $\Lambda$-module generated by
$\mathcal{O}(L_{0},L_{1};H)$.

####  2.3.4. Grading.

The Floer complex has a $\mathbb{Z}_{2}$-grading, which is defined as follows.
Take $y\in\mathcal{O}(L_{0},L_{1};H)$ and see it as an intersection point
$p\in\phi_{H}^{1}(L_{0})\cap L_{1}$. Let $\lambda_{\operatorname{can}}$ be a
canonical short path from $T_{p}\phi_{H}^{1}(L_{0})$ to $T_{p}L_{1}$ in the
Lagrangian Grassmannian of $T_{p}M$. Then we set $|y|=0$ if
$\lambda_{\operatorname{can}}$ lifts to a path in the oriented Lagrangian
Grassmannian of $T_{p}M$ from $T_{p}\phi_{H}^{1}(L_{0})$ to $T_{p}L_{1}$
(equipped with their given orientations). Otherwise, we set $|y|=1$.

Equivalently, we have $|y|=0$ if the contribution of $p$ to the intersection
number $\phi^{1}_{H}(L_{0})\cdot L_{1}$ is $(-1)^{\frac{n(n+1)}{2}}$.
Otherwise, $|y|=1$.

####  2.3.5. Differential.

The Floer differential is defined by counting rigid solutions
$u:\mathbb{R}\times[0,1]\to M$ to the _Floer equation_

$\begin{cases}&\partial_{s}u+J_{t}(u)(\partial_{t}u-X_{H}(t,u))=0\\\
&u(s,0)\in L_{0},\;u(s,1)\in L_{1}.\end{cases}$ (2)

For $x,y\in\mathcal{O}(L_{0},L_{1};H)$, let $\widetilde{\mathcal{M}}(y,x)$ be
the set of solutions to (2) with asymptotics $\lim_{s\to-\infty}u(s,-)=y$,
$\lim_{s\to\infty}u(s,-)=x$. For a regular Floer datum $(H,J)$,
$\widetilde{\mathcal{M}}(y,x)$ is a manifold, which may have several connected
components of different dimensions. The dimension of the component containing
$u$ is given by the Maslov index $\mu(u)$. Let $\mathcal{M}(y,x)$ be the
quotient of $\widetilde{\mathcal{M}}(y,x)$ by the action of $\mathbb{R}$ by
translation in the $s$ variable.

The Floer differential $\mu^{1}:CF^{*}(L_{0},L_{1};H,J)\to
CF^{*+1}(L_{0},L_{1};H,J)$ is defined by setting

$\mu^{1}(x)=(-1)^{|x|}\,\sum_{y\in\mathcal{O}(L_{0},L_{1};H)}\;\sum_{\begin{subarray}{c}u\in\mathcal{M}(y,x)\\\
\mu(u)=1\end{subarray}}\varepsilon(u)\,q^{\omega(u)}\,y.$ (3)

Here, $\varepsilon(u)=\pm 1$ is a sign which is described in Section 2.3.7.

####  2.3.6. $A_{\infty}$-operations.

More generally, the higher $A_{\infty}$-operations $(\mu^{d})_{d=2}^{\infty}$
are defined by counting solutions to inhomogeneous Cauchy-Riemann equations
for maps $u:S\to M$, where $S$ is a disk with $d+1$ punctures on the boundary.

More precisely, one considers the Deligne-Mumford moduli space
$\mathcal{R}^{d+1}$ of disks with $d+1$ boundary punctures
$\zeta_{0},\ldots,\zeta_{d}$, with the convention that $\zeta_{0}$ is
incoming, $\zeta_{1},\ldots,\zeta_{d}$ are outgoing, and the punctures are
ordered anti-clockwise along the boundary. Let
$\mathcal{S}^{d+1}\to\mathcal{R}^{d+1}$ be the universal curve over
$\mathcal{R}^{d+1}$. For $r\in\mathcal{R}^{d+1}$, denote by
$S_{r}\subset\mathcal{S}^{d+1}$ the fiber over $r$. Let $C_{0},\ldots,C_{d}$
be the boundary components of $S_{r}$ ordered anti-clockwise, with the
convention that $\zeta_{0}$ is adjacent to $C_{0}$ and $C_{d}$.

Fix a consistent universal choice of strip-like ends as in [Sei08, Section
(9g)]. For each family of Lagrangians $(L_{0},\ldots,L_{d})$, choose a
_perturbation datum_
$\mathcal{D}(L_{0},\ldots,L_{d})=(\mathbf{\Theta},\mathbf{J})$. Here,
$\mathbf{\Theta}=(\Theta^{r})_{r\in\mathcal{R}^{d+1}}$ is a smooth family of
$1$-forms $\Theta^{r}\in\Omega^{1}(S^{r};C^{\infty}(M))$. Moreover,
$\mathbf{J}=(J_{z})_{z\in\mathcal{S}^{d+1}}$ is a smooth family of compatible
almost complex structures on $M$ parametrized by $\mathcal{S}^{d+1}$. The
perturbation data is required to restrict to the previously chosen Floer data
over the strip-like ends. Moreover, it is required to be _consistent_ with
respect to breaking and gluing of disks (see [Sei08, Section (9i)] for the
precise definitions).

The perturbation datum $\mathcal{D}(L_{0},\ldots,L_{d})$ gives rise to the
following inhomogeneous Cauchy-Riemann equation for maps $u:S_{r}\to M$

$\begin{cases}Du+J(z,u)\circ Du\circ j=Y(z,u)+J(z,u)\circ Y(z,u)\circ j\\\
u(C_{k})\subset L_{k}.\\\ \end{cases}$ (4)

Here, $Y\in\Omega^{1}(S_{r};C^{\infty}(TM))$ is the $1$-form with values in
Hamiltonian vector fields of $M$ induced by $\Theta^{r}$. Moreover, $j$
denotes the complex structure of $S_{r}$.

Fix orbits $y_{0}\in\mathcal{O}(L_{0},L_{d})$ and
$y_{k}\in\mathcal{O}(L_{k-1},L_{k})$ for $k=1,\ldots,d$. Let
$\mathcal{M}^{d+1}(y_{0},\ldots,y_{d})$ be the moduli space of pairs $(r,u)$,
where $r\in\mathcal{R}^{d+1}$ and $u:S_{r}\to M$ is a solution of (4) which is
asymptotic to $y_{k}$ at the puncture $\zeta_{k}$ in strip-like coordinates.

Under suitable regularity assumptions, $\mathcal{M}^{d+1}(y_{0},\ldots,y_{d})$
is a manifold whose local dimension near $u$ is $\mu(u)+d-2$, where $\mu$ is
the Maslov index. Write $\mathcal{M}_{0}^{d+1}(y_{0},\ldots,y_{d})$ for the
$0$-dimensional part of $\mathcal{M}^{d+1}(y_{0},\ldots,y_{d})$. The operation

$\mu^{d}:\operatorname{Hom}(L_{d-1},L_{d})\otimes\cdots\otimes\operatorname{Hom}(L_{0},L_{1})\to\operatorname{Hom}(L_{0},L_{d})[2-d]$

is now defined by setting

$\mu^{d}(y_{d},\ldots,y_{1})=(-1)^{\dagger}\;\sum_{y_{0}\in\mathcal{O}(L_{0},L_{d})}\;\sum_{u\in\mathcal{M}_{0}^{d+1}(y_{0},y_{1},\ldots,y_{d})}\varepsilon(u)\,q^{\omega(u)}\,y_{0}.$
(5)

Here, $\varepsilon(u)=\pm 1$ is a sign which is described in Section 2.3.7.
Moreover, $(-1)^{\dagger}$ is an additional sign given by
$\dagger=\sum_{k=1}^{d}k|y_{k}|$.

####  2.3.7. Orientations and signs.

The signs appearing in the definition of the operations $\mu^{d}$ depend on
choices of orientations of the moduli spaces of inhomogeneous polygons
$\mathcal{M}^{d+1}(y_{0},\ldots,y_{d})$. We orient these moduli spaces using
the arguments of [FOOO09, Chapter 8] (see also [Sei08, Section 11] for a
closely related approach).

The orientations of the moduli spaces depend on the orientation and spin
structures of the Lagrangians involved, as well as some extra data which we
now describe. Fix a pair of objects $(L_{0},L_{1})$ and a generator of the
Floer complex $y\in\mathcal{O}(L_{0},L_{1};H)$. Up to replacing $L_{0}$ by
$\phi^{1}_{H}(L_{0})$ in what follows, we may assume that $y$ is a constant
trajectory at an intersection point $p\in L_{0}\cap L_{1}$.

We introduce the following notations. Let
$\mathcal{L}^{\operatorname{or}}(T_{p}M)$ be the oriented Lagrangian
Grassmannian of $T_{p}M$. Let $\lambda$ be a path in
$\mathcal{L}^{\operatorname{or}}(T_{p}M)$ with $\lambda(0)=T_{p}L_{0}$ and
$\lambda(1)=T_{p}L_{1}$. The path $\lambda$ defines an _orientation operator_
$D_{\lambda}$ as follows. Let $S_{+}$ be the unit disk with one incoming
boundary puncture. Let $E\to S_{+}$ be the trivial vector bundle with fiber
$T_{p}M$. The path $\lambda$ defines a totally real subbundle $F\subset
E|_{\partial S_{+}}$. Define $D_{\lambda}$ as the standard Cauchy-Riemann
operator on $E$ with boundary conditions given by $F$. Denote by
$\widetilde{\lambda}$ the vector bundle over $[0,1]$ given by

$\widetilde{\lambda}=\bigcup_{t\in[0,1]}\\{t\\}\times\lambda(t).$

For each pair $(L_{0},L_{1})$ and for each $y\in\mathcal{O}(L_{0},L_{1};H)$,
we fix the following _orientation data_ :

  1. (1)

A path $\lambda$ in $\mathcal{L}^{\operatorname{or}}(T_{p}M)$ from
$T_{p}L_{0}$ to $T_{p}L_{1}$.

  2. (2)

An orientation of the determinant line $\det(D_{\lambda})$.

  3. (3)

A spin structure on $\widetilde{\lambda}$ that extends the given spin
structures of $T_{p}L_{0}$ and $T_{p}L_{1}$.

By standard gluing arguments (see for example Chapter 8 of [FOOO09] or Section
12 of [Sei08]), the above data determine orientations of the moduli spaces
$\mathcal{M}^{d+1}(y_{0},\ldots,y_{d})$. In particular, each isolated element
$u\in\mathcal{M}^{d+1}_{0}(y_{0},\ldots,y_{d})$ carries a sign
$\varepsilon(u)=\pm 1$ which determines its contribution to $\mu^{d}$ (see
Equation (3) and Equation (5)).

######  Remark 2.3.

In the case where $M$ is a surface, the signs appearing in the definition of
$\mu^{d}$ can also be defined in a purely combinatorial way. This is described
in [Sei11, Section 7] and is also the definition used in [Abo08].

Figure 2. Choices of orientation paths for intersection points between curves.

The combinatorial definition of the signs in [Sei11] is related to the one
described above in the following way. In [Sei11], each Lagrangian is equipped
with a marked point and a trivialization of the spin structure on the
complement of the marked point. These extra choices determine canonical
choices of the orientation data (1)–(3) as follows. For an intersection point
$p\in L_{0}\cap L_{1}$, one takes the path $\lambda$ to be as in Figure 2.
More precisely, if $|p|=0$ then $\lambda$ is the canonical short path from
$L_{0}$ to $L_{1}$. If $|p|=1$, one takes the reverse of the canonical short
path from $L_{1}$ to $L_{0}$ and perturbs it to add a positive crossing. The
spin structure on $\widetilde{\lambda}$ is uniquely determined (up to
isomorphism relative to the boundary) by the trivializations of the spin
structures of $T_{p}L_{0}$ and $T_{p}L_{1}$. Finally, $\det(D_{\lambda})$ is
oriented as follows. If $|p|=0$, then $D_{\lambda}$ is invertible, so that
$\det(D_{\lambda})$ is canonically oriented. If $|p|=1$, there is an
isomorphism $\det(D_{\lambda})\cong T_{p}L_{1}$ given by the spectral flow
(see [Sei08, Lemma 11.11]). As $T_{p}L_{1}$ is oriented, this determines an
orientation of $\det(D_{\lambda})$.

With these compatible choices of data, it can be checked that the
combinatorial definition of the signs given in [Sei11] agrees with the one
obtained from gluing arguments, at least up to a global sign (see the
discussion following Example 13.5 in [Sei08]).

####  2.3.8. Invariance.

The Fukaya category depends on several auxiliary choices, such as choices of
strip-like ends, Floer data, perturbation data and orientation data. It
follows from the constructions in [Sei08, Section (10a)] that the resulting
category is independent of these choices up to quasi-isomorphism. With this in
mind, we will omit the choice of auxiliary data from the notation and simply
write $\operatorname{Fuk}(M)$ for any of these categories.

####  2.3.9. Derived Fukaya categories and $K$-theory.

We consider the following model for the derived Fukaya category
$\operatorname{DFuk}(M)$. First, consider the Yoneda embedding
$\mathcal{Y}:\operatorname{Fuk}(M)\to\operatorname{mod}(\operatorname{Fuk}(M))$.
Then, let $\operatorname{Fuk}(M)^{\wedge}$ be the triangulated closure of the
image of $\mathcal{Y}$ inside $\operatorname{mod}(\operatorname{Fuk}(M))$.
Finally, define
$\operatorname{DFuk}(M)=H^{0}(\operatorname{Fuk}(M)^{\wedge})$. Note that we
do not complete with respect to idempotents.

The derived category $\operatorname{DFuk}(M)$ is a triangulated category (in
the usual sense). Geometrically, the shift functor is realized by the
operation of reversing the orientation and spin structure of objects of
$\operatorname{Fuk}(M)$.

We let $K_{0}\operatorname{DFuk}(M)$ be the Grothendieck group of
$\operatorname{DFuk}(M)$. Recall that for a triangulated category
$\mathcal{C}$, the Grothendieck group $K_{0}\mathcal{C}$ is the free abelian
group generated by the objects of $\mathcal{C}$, quotiented by the relations
$A+B-C=0$ whenever there is an exact triangle $A\to B\to C$ in $\mathcal{C}$.

##  3\. Unobstructed cobordisms and the proof of Theorem A

In this section, we construct a Fukaya category
$\operatorname{Fuk}_{\operatorname{cob}}(\mathbb{C}\times M)$ whose objects
are quasi-exact cobordisms, and use it to prove Theorem A. The proof closely
follows the scheme formulated in [BC14], and its extension to embedded quasi-
exact cobordisms in [BCS21]. As a complete proof is outside of the scope of
this paper, we will content ourselves with describing the small modifications
that are needed to adapt the framework developed in [BC14] and [BCS21] to the
present case. The main differences are as follows:

  1. (i)

We consider Lagrangian cobordisms that may have self-intersections. In this
case, holomorphic curves that appear through bubbling may have corners at
self-intersection points. We show that ruling out holomorphic curves with at
most $1$ corner is sufficient to ensure the compactness of the moduli spaces
relevant to the definition of
$\operatorname{Fuk}_{\operatorname{cob}}(\mathbb{C}\times M)$.

  2. (ii)

The Fukaya category defined in [BC14] is linear over $\mathbb{Z}_{2}$ and
ungraded. The version of the Fukaya category that we consider is linear over
the Novikov ring over $\mathbb{Z}$ and is $\mathbb{Z}_{2}$-graded. In our
setting, the adjustments needed to deal with signs and gradings were described
by Haug [Hau15, Section 4].

Following the setting of [Abo08] and [Hau15], we use cohomological conventions
for complexes, which leads to some superficial differences with the
homological conventions used in [BC14].

###  3.1. Holomorphic maps with boundary on immersed Lagrangians

The Floer theory of Lagrangian immersions was originally developed by Akaho
[Aka05] and Akaho-Joyce [AJ10]. The main difference with the embedded case is
that holomorphic curves with boundary on an immersed Lagrangian may have
branch jumps at self-intersection points of the Lagrangian. To describe this
behaviour, holomorphic curves are equipped with the data of _boundary lifts_
that record the branch jump type.

We recall the following definitions from [AJ10]. Let $S$ be a nodal disk, that
is a compact nodal Riemann surface of genus $0$ with $1$ boundary component.
In order to define boundary lifts, $S$ is equipped with a boundary
parametrization, i.e. a continuous orientation-preserving map
$\ell:S^{1}\to\partial S$ such that

  * •

the preimage of a boundary node consists of two points,

  * •

the preimage of a smooth point of $\partial S$ consists of one point.

Note that $\ell$ is unique up to reparametrization.

Let $N$ be a symplectic manifold and let $\iota:L\looparrowright N$ be a
Lagrangian immersion of a manifold $L$ (which we do not assume to be connected
or compact). Fix a compatible almost complex structure $J$ on $N$.

######  Definition 3.1.

A (genus $0$) $J$-holomorphic map with corners with boundary on
$\iota:L\looparrowright N$ is a tuple $(S,\ell,u,\Delta,\overline{u})$, where

  1. (1)

$S$ is a nodal disk with boundary parametrization $\ell$,

  2. (2)

$u:(S,\partial S)\to(N,\iota(L))$ is a continuous map,

  3. (3)

$\Delta\subset\partial S$ is a finite set of marked points distinct from the
nodes,

  4. (4)

$\overline{u}:S^{1}\setminus\ell^{-1}(\Delta)\to L$ is a continuous map,

which satisfies the following conditions:

  1. (i)

$u$ is $J$-holomorphic on $S\setminus\Delta$,

  2. (ii)

$u$ has finite energy, i.e. $\int_{S\setminus\Delta}u^{*}\omega<\infty$,

  3. (iii)

$u\circ\ell=\iota\circ\overline{u}$ on $S^{1}\setminus\ell^{-1}(\Delta)$,

  4. (iv)

for each $\zeta\in\ell^{-1}(\Delta)$, the one-sided limits of $\overline{u}$
at $\zeta$ are distinct.

The elements of $\Delta$ are called the _corner points_ of $u$ and the map
$\overline{u}$ is called the _boundary lift_ of $u$. Condition (iv) in
Definition 3.1 means that the boundary of $u$ has a branch jump at each corner
point. Note that $u$ cannot have branch jumps at the nodes of $S$. However,
the individual components of $u$, seen as maps defined on a disk, may have
branch jumps at the nodal points.

We introduce the following terminology. We denote by $\mathbb{D}$ the closed
unit disk in $\mathbb{C}$.

######  Definition 3.2.

  1. (i)

A $J$-holomorphic disk is a $J$-holomorphic map with domain $\mathbb{D}$ and
no corners, i.e. $\Delta=\emptyset$.

  2. (ii)

A $J$-holomorphic teardrop is a $J$-holomorphic map with domain $\mathbb{D}$
and $1$ corner.

A $J$-holomorphic map $u$ is stable if its automorphism group is finite.
Equivalently, if $S_{\alpha}$ is a component of $S$ such that
$u|_{S_{\alpha}}$ is constant, then $S_{\alpha}$ carries at least $3$ special
points (with the convention that interior nodes count twice).

###  3.2. Definition of quasi-exact cobordisms

Given a Lagrangian cobordism $V\subset[0,1]\times\mathbb{R}\times M$, we can
extend its ends towards $\pm\infty$ by gluing appropriate cylinders of the
form $(-\infty,0]\times\\{k\\}\times L$ or $[1,\infty)\times\\{k\\}\times L$.
This defines a non-compact Lagrangian $\overline{V}$, called the extension of
$V$. When discussing the Floer theory of $V$, we make the convention that $V$
is always to be replaced by its extension.

Recall that we write $\widetilde{M}=\mathbb{C}\times M$. We say that a
compatible almost complex structure $J$ on $\widetilde{M}$ is admissible if
there is a compact set $K\subset(0,1)\times\mathbb{R}$ such that the
projection $\pi_{\mathbb{C}}:\widetilde{M}\to\mathbb{C}$ is
$(J,i)$-holomorphic outside $K\times M$.

We now define quasi-exact cobordisms, which generalize to the immersed case
Definition 4.2 of [BCS21] and the unobstructed Lagrangian brane cobordisms of
[SS21].

######  Definition 3.3 (Quasi-exact cobordisms).

Let $V\subset\widetilde{M}$ be an immersed Lagrangian cobordism with generic
self-intersections and $J$ be an admissible almost complex structure. We say
that the pair $(V,J)$ is quasi-exact if $V$ does not bound non-constant
$J$-holomorphic disks and teardrops.

######  Remark 3.4.

  1. (i)

By definition, quasi-exact cobordisms have embedded ends. It is possible in
some situations to work with cobordisms having immersed ends; see for example
[BC21] for an implementation. However, doing so would involve additional
technical difficulties and is not necessary for our purpose.

One drawback of this restriction is that surgeries that produce immersed
Lagrangians are _never_ unobstructed in the sense of Definition 3.3. However,
concatenations of such surgeries may give rise to unobstructed cobordisms
after perturbing their self-intersection locus; a special case of this
procedure is explained in Section 4.4.

  2. (ii)

It follows from the open mapping theorem that the concatenation of two quasi-
exact cobordisms along two matching ends is quasi-exact (for a suitable choice
of $J$); see Proposition 6.2 of [BCS21] for a detailed proof. In contrast,
classes of cobordisms satisfying topological constraints such as weak
exactness or monotonicity are typically not closed under concatenations.

######  Remark 3.5 (Comparison with the definition in [Per19]).

The class of unobstructed cobordisms considered in [Per19] consists of
cobordisms that do not bound any _continuous_ teardrops. This definition is
thus closer to the definition of a topologically unobstructed cobordism which
we introduce in Section 4.

There are two problems with the definition used in [Per19]. The first is that
there is no condition on disks with boundary on the cobordism, which poses
technical issues in the setup of Floer theory and the proofs of cone
decompositions. These issues are not addressed in [Per19]. The second problem
is that the non-existence of continuous teardrops is generally not preserved
under concatenations of cobordisms. The use of holomorphic maps in Definition
3.3 aims to fix this issue. Indeed, the class of quasi-exact cobordisms is
well-behaved with respect to concatenations; see Remark 3.4 and Proposition
4.11.

######  Lemma 3.6.

Let $(V,J)$ be a quasi-exact cobordism. Then $V$ does not bound stable
$J$-holomorphic maps with at most $1$ corner.

###### Proof.

Suppose that $V$ bounds a stable $J$-holomorphic map $u$ with at most $1$
corner. If $u$ has one component, then this contradicts that $(V,J)$ is quasi-
exact. If $u$ has more than one component, then there is a component
$\mathbb{D}_{\alpha}$ that has $1$ boundary node and no marked point. By
stability, the restriction $u_{\alpha}=u|_{\mathbb{D}_{\alpha}}$ is non-
constant. Moreover, it can only have a corner at the unique nodal point of
$\mathbb{D}_{\alpha}$. Hence $u_{\alpha}$ is a non-constant $J$-holomorphic
disk with at most $1$ corner, which is again a contradiction. ∎

For the purpose of defining suitable classes of Floer and perturbation data,
we will need a version of Lemma 3.6 that holds for a larger class of almost
complex structures, which we now define.

######  Definition 3.7.

Let $(V,J_{V})$ be a quasi-exact cobordism. A compatible almost complex
structure $J$ on $\widetilde{M}$ is adapted to $(V,J_{V})$ if

  1. (i)

$J=J_{V}$ on $[0,1]\times\mathbb{R}\times M$.

  2. (ii)

The projection $\pi_{\mathbb{C}}:\widetilde{M}\to\mathbb{C}$ is
$(J,i)$-holomorphic on $U\times M$, where $U$ is a neighborhood of
$\pi_{\mathbb{C}}(V)$.

######  Lemma 3.8.

Suppose that $(V,J_{V})$ is a quasi-exact cobordism and that $J$ is adapted to
$(V,J_{V})$. Then $V$ does not bound stable $J$-holomorphic maps with at most
$1$ corner.

###### Proof.

We prove the any $J$-holomorphic map $u:\mathbb{D}\to\widetilde{M}$ with
boundary on $V$ is actually $J_{V}$-holomorphic. Let $v=\pi_{\mathbb{C}}\circ
u$. By the open mapping theorem for holomorphic functions (see Proposition
3.3.1 of [BC14] for the precise version used here), $v$ cannot meet the
unbounded components of $\mathbb{C}\setminus\pi_{\mathbb{C}}(V)$. Therefore,
either $v(\mathbb{D})\subset[0,1]\times\mathbb{R}$, or
$v(\operatorname{int}\mathbb{D})$ meets
$\pi_{\mathbb{C}}(V)\cap([0,1]\times\mathbb{R})^{c}$. In the first case, it
follows from condition (i) in Definition 3.7 that $u$ is $J_{V}$-holomorphic.
In the second case, it follows from condition (ii) and the open mapping
theorem that $v$ is constant, so that $u$ is also constant. ∎

###  3.3. Definition of
$\operatorname{Fuk}_{\operatorname{cob}}(\widetilde{M})$

We now define the Fukaya category of quasi-exact cobordisms
$\operatorname{Fuk}_{\operatorname{cob}}(\widetilde{M})$, closely following
the construction described in [BC14]. We shall only describe the modifications
required to adapt this construction to the present case, and refer to [BC14]
for further details.

####  3.3.1. Objects

An object of $\operatorname{Fuk}_{\operatorname{cob}}(\widetilde{M})$ is a
quasi-exact Lagrangian cobordism $(V,J)$ with weakly exact ends, equipped with
an orientation and spin structure on $V$. For notational convenience, we will
most of the time write such an object by only specifying the cobordism $V$ and
keeping the other parts of the data implicit.

######  Remark 3.9.

For a given cobordism $V$, each choice of $J$ such that $(V,J)$ is quasi-exact
defines a different object of
$\operatorname{Fuk}_{\operatorname{cob}}(\widetilde{M})$. A priori, these
objects need not be quasi-isomorphic. As a consequence, the cone decomposition
of Theorem A may depend on the choice of $J$. Note, however, that the induced
relation on the level of $K$-theory does not depend on $J$.

####  3.3.2. Floer data

Recall the following notion from [BC14]. To define the class of admissible
Floer data, we fix a _profile function_ $h:\mathbb{R}^{2}\to\mathbb{R}$, whose
role is to specify the form of the Hamiltonian perturbations near infinity.
The definition is the same as that of [BC14, Section 3.2], except that the
sign of $h$ is switched, i.e. $h$ is such that $-h$ satisfies the properties
i-iv on p.1761 of [BC14]. The switch in the sign of $h$ is due to our use of
cohomology rather than homology.

For each pair of objects $(V_{0},V_{1})$, we fix a choice of Floer datum
$\mathcal{D}(V_{0},V_{1})=(\widetilde{H},\widetilde{J})$, where
$\widetilde{H}:[0,1]\times\widetilde{M}\to\mathbb{R}$ is a Hamiltonian and
$\widetilde{J}(t)$ is a time-dependent compatible almost complex structure on
$\widetilde{M}$. The Floer data are required to satisfy the following
conditions:

  1. (i)

$\phi^{1}_{\tilde{H}}(V_{0})$ is transverse to $V_{1}$ and their intersections
are not double points.

  2. (ii)

There exists a compact set
$K\subset(-\frac{5}{4},\frac{9}{4})\times\mathbb{R}\subset\mathbb{C}$ and a
Hamiltonian $H:[0,1]\times M\to\mathbb{R}$ such that
$\widetilde{H}(t,x+iy,p)=h(x,y)+H(t,p)$ for $x+iy\in\mathbb{C}\setminus K$.

  3. (iii)

For all $t\in[0,1]$, the projection
$\pi_{\mathbb{C}}:\widetilde{M}\to\mathbb{C}$ is
$(\widetilde{J}(t),(\phi_{h}^{t})_{*}i)$-holomorphic outside $K\times M$.

  4. (iv*)

For $k\in\\{0,1\\}$, $\widetilde{J}(k)$ is adapted to $J_{k}$ in the sense of
Definition 3.7.

Note that conditions (ii)–(iii) are the same as in [BC14] (see p.1792), and
that condition (i) is also the same except for the restriction concerning the
double points. The purpose of condition (iv*) is to ensure that there is no
bubbling of holomorphic disks with boundary on $V_{k}$. Note that condition
(iv*) does not interfere with condition (iii) since the profile function
satisfies $(\phi_{h}^{t})_{*}i=i$ over $[0,1]\times\mathbb{R}$ and over the
projections of the ends of the cobordisms. We remark however that it is
generally not possible to impose that $\widetilde{J}(k)$ satisfies condition
(i) of Definition 3.3.

It follows from condition (ii) and the definition of the profile function $h$
that $\phi^{1}_{\tilde{H}}(V_{0})$ and $V_{1}$ are distinct at infinity.
Combined with condition (i), this implies that the set of Hamiltonian chords
$\mathcal{O}(V_{0},V_{1};\widetilde{H})$ is finite.

####  3.3.3. Perturbation data

To define the admissible class of perturbation data, we fix the following
choices. First, fix a consistent universal choice of strip-like ends as in
[Sei08, Section (9g)]. Secondly, fix a family of _transition functions_
$\mathbf{a}:\mathcal{S}^{d+1}\to[0,1]$ for $d\geq 1$ as in Section 3.1 of
[BC14]. These transition functions are required to satisfy several conditions;
we refer to [BC14] for the precise definitions.

For each collection of objects $(V_{0},\ldots,V_{d})$, we fix a perturbation
datum $\mathcal{D}(V_{0},\ldots,V_{d})=(\mathbf{\Theta},\mathbf{J})$. Here,
$\mathbf{\Theta}=(\Theta^{r})_{r\in\mathcal{R}^{d+1}}$ is a smooth family of
$1$-forms $\Theta^{r}\in\Omega^{1}(S_{r};C^{\infty}(\widetilde{M}))$ and
$\mathbf{J}=(J_{z})_{z\in\mathcal{S}^{d+1}}$ is a smooth family of compatible
almost complex structures on $\widetilde{M}$.

The perturbation data are required to satisfy the conditions (i)–(iii) stated
on pp.1763-1764 of [BC14]. Moreover, in order to rule out bubbling off of
holomorphic disks, we impose the following additional condition:

  1. (iv*)

For $0\leq k\leq d$ and $z\in C_{k}$, $J_{z}$ is adapted to $J_{V_{k}}$ in the
sense of Definition 3.7.

As usual, the perturbation data are required to be consistent with breaking
and gluing as in [Sei08, Section (9i)].

####  3.3.4. Transversality and compactness

Having made choices of Floer and perturbation data as described above, the
moduli spaces of Floer polygons are defined as in Section 2.3.5 and Section
2.3.6. The main point here is that Floer polygons are assumed to have _no
corners_ , i.e. they are required to be smooth up to the boundary.

Before going further, we need to justify that the Floer and perturbation data
can be chosen to achieve the transversality and compactness up to breaking of
the moduli spaces of Floer polygons.

For transversality, the argument is the same as in Section 4.2 of [BCS21]. The
main point is that the additional constraints imposed on the Floer and
perturbation data (i.e. condition (iv*) above) only concern their restriction
to the boundary of $S_{r}$, so that arbitrary perturbations are allowed in the
interior of $S_{r}$.

We now address the compactness issues. The first point is that we must have
$C^{0}$ bounds for Floer polygons of bounded energy. In our setting, the proof
is the same as in [BC14, Section 3.3]. The only difference is that there is no
uniform bound on the energy of Floer polygons; instead the areas of curves are
encoded using Novikov coefficients.

The second point is that we must show that no bubbling of holomorphic disks
can occur for sequences of Floer polygons with bounded energy. This is a
consequence of Lemma 3.8. Indeed, since Floer polygons are assumed to have no
corners, it follows from the Gromov compactness theorem for curves with
boundary lifts (see [IS02]) that a sequence of such polygons with bounded
energy converges to a stable map with no corners. If the limit stable map
contains a bubble tree $u$ with boundary on a cobordism $V_{k}$, then by
condition (iv*) in the definition of the Floer and perturbation data $u$ must
be $J$-holomorphic for an almost complex structure $J$ which is adapted to the
base structure $J_{k}$ associated to $V_{k}$. Moreover, since the limit map
has no corners, $u$ has at most $1$ corner. By restricting to a subtree if
necessary, we can assume that $u$ is stable. Hence, $V_{k}$ bounds a stable
$J$-holomorphic map with at most $1$ corner, which contradicts Lemma 3.8.

####  3.3.5. Summing it up

Having made the choices described above, the definition of
$\operatorname{Fuk}_{\operatorname{cob}}(\widetilde{M})$ now follows the same
recipe as the definition of $\operatorname{Fuk}(M)$ outlined in Section 2.3.
For objects $V_{0}$ and $V_{1}$, the morphism space
$\operatorname{Hom}(V_{0},V_{1})$ is the complex generated by Hamiltonian
orbits for the chosen Floer datum. The grading of
$\operatorname{Hom}(V_{0},V_{1})$ is defined as in Section 2.3.4. The
$A_{\infty}$-operations $\mu^{d}$ are defined by counting rigid Floer polygons
asymptotic to Hamiltonian orbits. The signs appearing in the definition of
$\mu^{d}$ are defined by fixing orientation data as in Section 2.3.7.

###  3.4. End of the proof of Theorem A

In the previous sections, we described the technical adjustments that are
needed to extend the theory developed in [BC14, BCS21] to immersed quasi-exact
cobordisms. Having made these modifications, the proof of Theorem A now
follows the same arguments as the proof Theorem A of [BC14]. The only missing
ingredient is the verification that the $A_{\infty}$-functors defined in
[BC14] are compatible with gradings and signs. In the present setting, these
verifications were carried out by Haug, see [Hau15, Section 4].

##  4\. Topologically unobstructed cobordisms

In this section, we introduce a class of cobordisms that are _topologically
unobstructed_ , in the sense that they do not bound any homotopically non-
trivial continuous disks and teardrops. This condition will play an important
role in this paper because it is easier to check than unobstructedness in the
sense of Section 3. Moreover, as we will see in subsequent sections, all the
cobordisms appearing in the computation of
$\Omega_{\operatorname{unob}}(\Sigma)$ can be chosen to either satisfy this
stronger property, or at least be concatenations of topologically unobstructed
cobordisms.

In Section 4.1, we define topologically unobstructed cobordisms. In Section
4.2, we characterize topological obstruction for Lagrangian suspensions. In
Section 4.3, we show that topological unobstructedness is an open condition in
the $C^{1}$ topology.

In Section 4.4, we consider concatenations of cobordisms along possibly
immersed ends. There are two issues with such concatenations: the first is
that their self-intersections are not generic and therefore must be perturbed;
the second is that topological unobstructedness is generally not preserved
under concatenations. The main result of Section 4.4 is that appropriate
concatenations of topologically unobstructed cobordisms become unobstructed in
the sense of Section 3 after a suitable perturbation.

###  4.1. Definitions

In Section 3.1, we considered holomorphic disks with corners with boundary on
Lagrangian immersions. In this section, we consider _continuous_ polygons with
boundary on immersions, which are defined in a similar way.

######  Definition 4.1.

Let $\iota:L\to M$ be an immersion. A continuous polygon with boundary on
$\iota$ consists of a continuous map
$u:(\mathbb{D},\partial\mathbb{D})\to(M,\iota(L))$, a finite set
$\Delta\subset\partial\mathbb{D}$ and a continuous map
$\overline{u}:\partial\mathbb{D}\setminus\Delta\to L$ such that

  1. (i)

$\iota\circ\overline{u}=u$ over $\partial\mathbb{D}\setminus\Delta$,

  2. (ii)

for each $\zeta\in\Delta$, the one-sided limits of $\overline{u}$ at $\zeta$
are distinct.

In the rest of this paper, by a _polygon_ we will always mean a _continuous
polygon_ (not necessarily holomorphic). As before, we call a polygon a _disk_
if it has $0$ corners, and a _teardrop_ if it has $1$ corner.

Note that it follows from the definition that an immersion $\iota:L\to M$
bounds a teardrop if and only if there is a path $\gamma$ in $L$ with distinct
endpoints such that $\iota\circ\gamma$ is a contractible loop in $M$.

We will need to consider homotopy classes of disks with boundary on an
immersion. For this purpose, recall that to a based map of based spaces
$f:X\to Y$, we can associate relative homotopy groups $\pi_{n}(f)$, which are
a straightforward generalization of the relative homotopy groups of a pair.
The elements of $\pi_{n}(f)$ are homotopy classes of diagrams of based maps

${\mathbb{D}^{n}}$${Y}$${\partial\mathbb{D}^{n}}$${X}$inc$\scriptstyle{f}$

where the homotopies are also required to factor through $f$ over the boundary
of the disk. Equivalently, we may define $\pi_{n}(f)$ as the relative homotopy
group $\pi_{n}(M_{f},X)$, where $M_{f}$ is the mapping cylinder of $f$ and $X$
is embedded in $M_{f}$ in the usual way. There is then a long exact sequence
of homotopy groups

${\ldots}$${\pi_{n}(X)}$${\pi_{n}(Y)}$${\pi_{n}(f)}$${\pi_{n-1}(X)}$${\ldots}$$\scriptstyle{f_{*}}$
(6)

######  Definition 4.2.

A map $f:X\to Y$ is called incompressible if the maps
$f_{*}:\pi_{1}(X)\to\pi_{1}(Y)$ induced by $f$ are injective (for any choices
of basepoints).

A subspace $X\subset Y$ is called incompressible if the inclusion $X\to Y$ is
incompressible.

Note that by the long exact sequence (6), the induced map
$f_{*}:\pi_{1}(X)\to\pi_{1}(Y)$ is injective if and only the boundary map
$\pi_{2}(f)\to\pi_{1}(X)$ vanishes.

We can now define topologically unobstructed Lagrangian immersions.

######  Definition 4.3.

A Lagrangian immersion $\iota:L\to M$ is topologically unobstructed if $\iota$
is incompressible and does not bound continuous teardrops.

######  Remark 4.4.

  1. (i)

We emphasize that in Definition 4.3 we do not assume that the immersion
$\iota$ has generic self-intersections. The reason is that it will be
convenient for us to allow cobordisms with immersed ends to belong to the
class of topologically unobstructed cobordisms.

  2. (ii)

The cases of interest for this paper are $M=\Sigma$ and
$M=\mathbb{C}\times\Sigma$, where $\Sigma$ is a closed surface of genus $g\geq
2$. In these cases we have $\pi_{2}(M)=0$, hence incompressibility of an
immersion $\iota$ is equivalent to $\pi_{2}(\iota)=0$.

In the case of a surface $\Sigma$, it follows from Lemma 2.2 of [Abo08] that
an immersion $S^{1}\to\Sigma$ is topologically unobstructed if and only if it
is _unobstructed_ in the sense of Definition 2.1 of [Abo08], that is if its
lifts to the universal cover $\widetilde{\Sigma}$ are proper embeddings. See
Lemma 4.6 for a generalization. Note also that, since $\pi_{1}(\Sigma)$ has no
torsion, an immersion $S^{1}\to\Sigma$ is incompressible if and only if it is
non-contractible.

######  Proposition 4.5.

Let $M$ be a symplectically aspherical manifold. Let
$\iota_{V}:V\to\mathbb{C}\times M$ be a topologically unobstructed Lagrangian
cobordism with embedded ends and generic self-intersections. Then $(V,J)$ is
quasi-exact for any admissible $J$.

###### Proof.

By assumption, $V$ does not bound any continuous teardrops. Moreover, since
the map $\pi_{2}(\iota_{V})\to\pi_{1}(V)$ is trivial and $M$ is symplectically
aspherical, disks with boundary on $V$ have zero symplectic area. Hence, there
are no non-constant $J$-holomorphic disks with boundary on $V$. ∎

We now give a useful reformulation of topological unobstructedness, which is a
straightforward generalization of Lemma 2.2 of [Abo08]. Let $\iota:L\to M$ be
an immersion and denote by $\pi:\widetilde{L}\to L$ and $\rho:\widetilde{M}\to
M$ the universal covers of $L$ and $M$. By a _lift_ of $\iota$ to
$\widetilde{M}$, we mean a lift of $\iota\circ\pi$ to $\widetilde{M}$.

######  Lemma 4.6.

Let $L$ be a compact manifold. An immersion $\iota:L\to M$ is topologically
unobstructed if and only if its lifts to $\widetilde{M}$ are proper
embeddings.

###### Proof.

The case of immersed curves on surfaces is Lemma 2.2 of [Abo08]. The general
case follows from the same argument. Since the proof is short, we include it
here.

Observe that $\iota$ is topologically obstructed if and only if there is a
path $\gamma$ in $L$ such that

  1. (i)

either $\gamma$ has distinct endpoints (in the case of a teardrop) or is a
non-contractible loop (in the case of a disk), and

  2. (ii)

$\iota\circ\gamma$ is a contractible loop in $M$.

By lifting to the universal covers, paths in $L$ satisfying conditions (i) and
(ii) correspond to paths $\widetilde{\gamma}$ in $\widetilde{L}$ with distinct
endpoints such that $\widetilde{\iota}\circ\widetilde{\gamma}$ is a loop. Such
paths exist if and only if $\widetilde{\iota}$ has self-intersections.

Hence, we have proved that $\iota$ is topologically unobstructed if and only
if its lifts are injective. To finish the proof, we observe that an injective
lift of a proper immersion is also a proper immersion, hence a proper
embedding. ∎

###  4.2. Topological obstruction of suspensions

We now prove a criterion for the topological unobstructedness of Lagrangian
suspensions.

######  Lemma 4.7.

Let $\iota:L\times[0,1]\to\mathbb{C}\times M$ be the suspension of an exact
Lagrangian homotopy $(\phi_{t})_{t\in[0,1]}$. If $\phi_{t}$ is topologically
unobstructed for every $t\in[0,1]$, then $\iota$ is topologically
unobstructed.

###### Proof.

This follows easily from the fact the each slice $L\times\\{t\\}$ is a strong
deformation retract of $L\times[0,1]$. Indeed, this readily implies that
$\iota$ is incompressible if and only if $\phi_{t}$ is incompressible for all
$t$. Moreover, if $\iota$ bounds a teardrop $u$, then by definition of the
suspension the endpoints of the boundary lift of $u$ must be in the same slice
$L\times\\{t\\}$. Hence, applying the deformation retraction to the boundary
lift yields a teardrop with boundary on $\phi_{t}$.

∎

###  4.3. Stability under perturbations

The incompressibility of an immersion $\iota$ is obviously invariant under
homotopies of $\iota$. On the other hand, the non-existence of teardrops with
boundary on an immersion is generally not preserved by homotopies or even
regular homotopies. Nevertheless, this property is preserved by sufficiently
small deformations, as we now show.

######  Lemma 4.8.

Let $L$ be a compact manifold, possibly with boundary. Suppose that the
immersion $\iota:L\to M$ does not bound teardrops. Then any immersion
sufficiently close to $\iota$ in the $C^{1}$ topology does not bound
teardrops.

###### Proof.

Assume, in view of a contradiction, that there is a sequence of immersions
$\theta_{n}:L\to M$ converging to $\iota$ in the $C^{1}$ topology, such that
each $\theta_{n}$ bounds a teardrop. This means that there exist sequences
$(p_{n})$ and $(q_{n})$ in $L$ and a sequence of paths $\gamma_{n}$ from
$p_{n}$ to $q_{n}$ with the following properties:

  1. (i)

$\theta_{n}(p_{n})=\theta_{n}(q_{n})$ for all $n$.

  2. (ii)

$q_{n}\neq p_{n}$ for all $n$.

  3. (iii)

The loop $\theta_{n}\circ\gamma_{n}$ is contractible in $M$.

By compactness, we can assume that $p_{n}\to p$ and $q_{n}\to q$ for some
points $p,q\in L$. Then $\iota(p)=\iota(q)$. Moreover, since
$\theta_{n}\to\iota$ in the $C^{1}$ topology, we must have $p\neq q$
(otherwise the existence of the sequences $p_{n}$ and $q_{n}$ would contradict
that $\iota$ is an immersion).

For every $n$, choose a path $\widetilde{\gamma}_{n}$ from $p$ to $q$ so that
$d_{C^{0}}(\gamma_{n},\widetilde{\gamma}_{n})\to 0$ as $n\to\infty$ (where the
$C^{0}$ distance is computed with respect to some fixed Riemannian metric).
Then we also have
$d_{C^{0}}(\theta_{n}\circ\gamma_{n},\iota\circ\widetilde{\gamma}_{n})\to 0$
as $n\to\infty$ since $\theta_{n}\to\iota$ in the $C^{0}$ topology. In
particular, for $n$ large enough the loops $\theta_{n}\circ\gamma_{n}$ and
$\iota\circ\widetilde{\gamma}_{n}$ are freely homotopic. Therefore the loop
$\iota\circ\widetilde{\gamma}_{n}$ is contractible in $M$ and we conclude that
$\iota$ bounds a teardrop. ∎

######  Corollary 4.9.

Let $\iota:L\to M$ be a topologically unobstructed immersion, where $L$ is
compact. Then any immersion sufficiently close to $\iota$ in the $C^{1}$
topology is topologically unobstructed.

Recall that a Lagrangian immersion has _generic self-intersections_ if it has
no triple points and each double point is transverse. It is well known that
immersions $L\to M$ with generic self-intersections are dense in the space of
all Lagrangian immersions. As a consequence, we obtain the following
approximation result.

######  Corollary 4.10.

Let $\iota:L\to M$ be a topologically unobstructed immersion, where $L$ is
compact. Then there is a $C^{1}$-close Lagrangian immersion which is
topologically unobstructed and has generic self-intersections.

In the case of a cobordism $V$ with immersed ends, this corollary allows us to
perturb $V$ so that its ends are generic, and so that the only non-generic
self-intersections of $V$ are the intervals of double points corresponding to
the double points of its ends. The double points over the ends of $V$ will be
handled by a more careful choice of perturbations, which are described in the
next section.

###  4.4. Concatenations of cobordisms along immersed ends

Let $V:L\rightsquigarrow(L_{1},\ldots,L_{r})$ be a cobordism and let
$V^{\prime}$ be a cobordism that has a negative end modelled over $L$. Then
$V$ and $V^{\prime}$ can be concatenated by gluing them along the ends
corresponding to $L$, producing a cobordism $V\\#V^{\prime}$. Assume that $V$
and $V^{\prime}$ are topologically unobstructed. Then $V\\#V^{\prime}$ needs
not be topologically unobstructed, since a disk or teardrop on
$V\\#V^{\prime}$ needs not correspond to a disk or teardrop on $V$ or
$V^{\prime}$. Despite this, we will prove that, after a suitable perturbation,
the concatenation $V\\#V^{\prime}$ does not bound $J$-holomorphic disks or
teardrops for appropriate choices of almost complex structures $J$. The
precise statement is as follows.

######  Proposition 4.11.

Let $V$ be a Lagrangian cobordism with embedded ends which is the
concatenation of topologically unobstructed cobordisms. Then $V$ is exact
homotopic relative to its boundary to a quasi-exact cobordism.

Before proving the proposition, we describe the class of perturbations that
will be used in the proof. We will use a construction from [BC21, Section
3.2.1] which replaces the self-intersections of an immersed end of a cobordism
by transverse double points. The outcome of this perturbation is a _cobordism
with bottlenecks_ in the sense of [MW18]. The presence of these bottlenecks is
the key feature that allows to control the behaviour of $J$-holomorphic maps.

Let $\iota_{V}:V\to\mathbb{C}\times M$ be an immersed cobordism. For
notational convenience, we replace $V$ by its extension as in Section 3.2.
Consider a positive end of $\iota_{V}$, which for definiteness we assume is
lying over the interval
$I=[1-\varepsilon,\infty)\times\\{0\\}\subset\mathbb{C}$ for some small
$\varepsilon>0$. This means that there is a Lagrangian immersion
$\iota_{L}:L\to M$ and a proper embedding $j:[0,\infty)\times L\to V$ such
that $V|_{I}=j([1-\varepsilon,\infty)\times L)$ and

$\iota_{V}\circ j(x,p)=(x,0,\iota_{L}(p))$

for $x\geq 1-\varepsilon$. From now on, we identify $[0,\infty)\times L$ with
its image in $V$. Moreover, we assume that $\iota_{L}$ has generic self-
intersections and that $\iota_{V}$ does not have double points in
$[0,1-\varepsilon)\times L$.

The perturbations we consider are of the following form. Extend $\iota_{V}$ to
a symplectic immersion $\Psi:U\to\mathbb{C}\times M$, where $U$ is a
neighborhood of the zero-section in $T^{*}V$. The perturbed immersion will be
obtained by composing $\Psi$ with a Hamiltonian isotopy of the zero-section
inside $U$.

To describe the relevant Hamiltonian isotopy, consider a smooth function
$h:[0,\infty)\to\mathbb{R}$ that has the profile shown in Figure 3. More
precisely, $h(x)$ vanishes on $[0,1-2\varepsilon]$, has a unique non-
degenerate critical point in $(1-2\varepsilon,\infty)$ located at $x=1$, and
is affine with positive slope for $x>2$.

Figure 3. Left: the behaviour of the profile function $h$. Right: the
projection to $\mathbb{C}$ of a cobordism after perturbation of a positive
end.

Fix a double point of $\iota_{L}$ and pick an ordering $(P_{-},P_{+})$ of the
two preimages. Let $D_{\pm}\subset L$ be disjoint open disks centered at
$P_{\pm}$. Fix bump functions $\chi_{\pm}$ on $D_{\pm}$ such that
$\chi_{\pm}\geq 0$, $\chi\equiv 1$ near $P_{\pm}$ and $\chi\equiv 0$ near
$\partial D_{\pm}$. We now define functions $H_{\pm}:[0,\infty)\times
D_{\pm}\to\mathbb{R}$ by setting $H_{\pm}(x,p)=\pm h(x)\,\chi_{\pm}(p)$. Doing
this for each pair of double points and extending by zero, we obtain a
function $H$ on $V$. Finally, extend $H$ to a neighborhood of $V$ in $T^{*}V$
by pulling back by the projection $T^{*}V\to V$.

Consider the family of Lagrangian immersions
$\iota_{t}=\Psi\circ\phi^{t}_{H}|_{V}$ for small $t$. By our choice of $h$,
for $t>0$ the projection of $\iota_{t}$ to $\mathbb{C}$ has the shape shown in
Figure 3. The key feature is that for $t$ small enough each pair of intervals
of double points of $\iota_{V}$ inside $[0,\infty)\times L$ has been replaced
by two pairs of transverse double points. One of these pairs projects to the
"bottleneck" at $x=1$, while the other pair is a small perturbation of the
double points $(1-\varepsilon,P_{\pm})$ of $\iota_{V}$.

Assuming that $\iota_{V}$ does not bound teardrops, then by Lemma 4.8, for $t$
small enough $\iota_{t}$ also does not bound teardrops. We will always assume
that $t$ is chosen small enough so that this is the case.

In the case of a negative end of $V$, the perturbation is defined in a similar
way, with the profile function $h$ replaced by $x\mapsto-h(-x)$.

We now proceed to the proof of Proposition 4.11.

###### Proof of Proposition 4.11..

We only provide details for the case where $V$ is the concatenation of two
cobordisms; the general case is similar.

Let $V$ be the concatenation of topologically unobstructed cobordisms $V_{0}$
and $V_{1}$ along a matching end modelled over the immersed Lagrangian $L$.
After generic perturbations, we may assume that $L$ has generic self-
intersections and that the only non-generic self-intersections of $V_{0}$ and
$V_{1}$ are along the end corresponding to $L$. By Lemma 4.8, for a small
enough perturbation the resulting cobordisms are still topologically
unobstructed.

Next, perturb $V$ by splicing together compatible perturbations of the ends of
$V_{0}$ and $V_{1}$ which were concatenated, as described in the beginning of
this section. The outcome is a cobordism (which we still call $V$) with
generic self-intersections and two bottlenecks $\zeta_{\pm}\in\mathbb{C}$ as
in Figure 4. The bottlenecks separate the cobordism in three parts which we
label $V_{\ell}$, $V_{c}$ and $V_{r}$. Note that $V_{\ell}$, $V_{r}$ and
$V_{c}$ are topologically unobstructed since they are small perturbations of
$V_{0}$, $V_{1}$ and a product cobordism $I\times L$, respectively.

Figure 4. The projection to $\mathbb{C}$ of the cobordism obtained by
perturbing the concatenation of $V_{0}$ and $V_{1}$.

Let $J$ be a compatible almost complex structure on $\mathbb{C}\times M$ with
the following properties:

  1. (i)

The projection $\pi_{\mathbb{C}}$ is $(J,i)$-holomorphic outside $K\times M$
for some compact set $K\subset\mathbb{C}$.

  2. (ii)

The projection $\pi_{\mathbb{C}}$ is $(J,i)$-holomorphic on $U_{\pm}\times M$,
where $U_{\pm}$ is a neighborhood of the bottleneck $\zeta_{\pm}$.

We claim that $(V,J)$ is quasi-exact. Indeed, if there is a $J$-holomorphic
teardrop with boundary on $V$, then by the open mapping theorem this teardrop
cannot cross the bottlenecks, i.e. it must have boundary on one of the three
parts $V_{\ell}$, $V_{c}$ or $V_{r}$. This contradicts that these immersions
are topologically unobstructed. By the same argument, $V$ does not bound non-
constant $J$-holomorphic disks. ∎

##  5\. Topological obstruction of surgery cobordisms in dimension 2

We now turn to the computation of $\Omega_{\operatorname{unob}}(\Sigma)$ for a
closed surface $\Sigma$ of genus $g\geq 2$, which will take up the rest of
this paper. As a first step, in this section we prove topological
unobstructedness results for the cobordisms associated to the surgery of
immersed curves in $\Sigma$. These results will form the basis of the proofs
that the cobordisms appearing in the computation of
$\Omega_{\operatorname{unob}}(\Sigma)$ are unobstructed. We note that the
proofs in this section make heavy use of special features of the topology of
surfaces.

Throughout this section, we fix immersed curves $\alpha$ and $\beta$ in
$\Sigma$ that are in general position and intersect at a point $s\in\Sigma$.
In Section 5.1, we give a precise description of the Lagrangian surgery of
$\alpha$ and $\beta$ at $s$ and of the associated surgery cobordism, which we
denote $S(\alpha,\beta;s)$. In Section 5.2 we consider teardrops with boundary
$S(\alpha,\beta;s)$, and in Section 5.3 we characterize the existence of non-
trivial disks with boundary on $S(\alpha,\beta;s)$.

###  5.1. Construction of the surgery cobordism

We recall the construction of the surgery cobordism $S(\alpha,\beta;s)$ in
order to fix notations for the proofs of the obstruction results of the next
sections. We closely follow the construction given by Biran and Cornea [BC13,
Section 6.1], although our presentation differs slightly since we will need
some control over the double points of the surgery cobordism in order to
investigate teardrops.

We start with a local model, which is the surgery of the Lagrangian subspaces
$\mathbb{R}$ and $i\mathbb{R}$ in $\mathbb{C}$. Fix an embedding
$c:\mathbb{R}\to\mathbb{C}$ with the following properties. Writing
$c=c_{1}+ic_{2}$, we have for some $\varepsilon>0$

  * •

$c(t)=t$ for $t\leq-\frac{\varepsilon}{2}$ and $c(t)=it$ for
$t\geq\frac{\varepsilon}{2}$,

  * •

$c_{1}^{\prime}(t)>0$ and $c_{2}^{\prime}(t)>0$ for
$t\in(-\frac{\varepsilon}{2},\frac{\varepsilon}{2})$,

  * •

$c_{1}^{\prime\prime}(t)<0$ and $c_{2}^{\prime\prime}(t)>0$ for
$t\in(-\frac{\varepsilon}{2},\frac{\varepsilon}{2})$.

The local model for the surgery is then the Lagrangian
$\mathbb{R}\\#i\mathbb{R}=c(\mathbb{R})\cup-c(\mathbb{R})$.

To define the local model for the surgery cobordism, consider the Lagrangian
embedding

$\displaystyle\psi:\mathbb{R}\times S^{1}$
$\displaystyle\longrightarrow\mathbb{C}^{2}$ $\displaystyle(t,(x,y))$
$\displaystyle\longmapsto(c(t)x,c(t)y).$

The model for the surgery cobordism is the Lagrangian $1$-handle $\psi(W)$,
where

$W=\\{(t,(x,y)):x\geq 0,|tx|\leq\varepsilon\\}.$ (7)

Note that $\psi(W)$ is a submanifold of $\mathbb{C}^{2}$ with boundary
$(\\{-\varepsilon\\}\times\mathbb{R})\cup(\\{i\varepsilon\\}\times
i\mathbb{R})\cup(\\{0\\}\times\mathbb{R}\\#i\mathbb{R})$.

Suppose now that $\alpha$ and $\beta$ are immersed curves in $\Sigma$ in
general position. Let $s$ be a positive intersection222According to the
grading convention of Section 2.3.4, this means that $s$ has degree $1$ in
$CF(\alpha,\beta)$. for the pair $(\alpha,\beta)$. Let $B(0,\varepsilon)$ be
the open disk of radius $\varepsilon$ in $\mathbb{C}$. For a small enough
$\varepsilon>0$, fix a Darboux chart $\Phi:B(0,\varepsilon)\to U\subset\Sigma$
centered at $s$ with the property that $\Phi^{-1}\circ\alpha$ parametrizes
$\mathbb{R}$ in the positive direction and $\Phi^{-1}\circ\beta$ parametrizes
$i\mathbb{R}$. The surgery $\alpha\\#_{s}\beta$ is defined by taking the union
$\alpha\cup\beta$ and replacing its intersection with $U$ by the local model
$\mathbb{R}\\#i\mathbb{R}$. Note that the resulting immersion is independent
of the above choices up to Hamiltonian isotopy. Moreover, since $s$ is
positive the surgery has a canonical orientation which is compatible with
those of $\alpha$ and $\beta$.

To construct the surgery cobordism, start with the product cobordisms
$\widehat{\alpha}=[-1,0]\times\alpha$ and $\widehat{\beta}=i[0,1]\times\beta$.
These cobordisms are oriented so that $[-1,0]$ is oriented from $-1$ to $0$,
and $i[0,1]$ is oriented from $i$ to $0$. Let
$\widehat{\iota}:S\to\mathbb{C}\times\Sigma$ be the Lagrangian immersion
obtained from $\alpha\cup\beta$ by removing its intersection with
$B(0,\varepsilon)\times U$ and gluing the handle $\psi(W)$ using the Darboux
chart $\operatorname{id}\times\Phi:B(0,\varepsilon)\times B(0,\varepsilon)\to
B(0,\varepsilon)\times U$. Then $\widehat{\iota}$ is a Lagrangian immersion of
a pair of pants whose restriction to the boundary coincides with the
immersions $\\{-1\\}\times\alpha$, $\\{i\\}\times\beta$ and
$\\{0\\}\times(\alpha\\#_{s}\beta)$. Note that the handle $\psi(W)$ is
compatible with the orientations of $\widehat{\alpha}$ and $\widehat{\beta}$,
so that $S$ is canonically oriented.

The immersion $\widehat{\iota}$ is almost the required cobordism
$\alpha\\#_{s}\beta\rightsquigarrow(\alpha,\beta)$, except that it is not
cylindrical near the boundary component that projects to $0\in\mathbb{C}$. We
perturb $\widehat{\iota}$ to make it cylindrical using the argument of [BC13,
Section 6.1].

To describe the perturbation, we first introduce some notations. Write

$P=S-\widehat{\iota}^{-1}(B(0,\varepsilon)\times U),$

so that $\widehat{\iota}$ coincides over $P$ with restrictions of the products
$\widehat{\alpha}$ and $\widehat{\beta}$. Let $C$ be the boundary component of
$S$ that projects to $0\in\mathbb{C}$. Let $\mathcal{U}$ be a collar
neighborhood of $C$. Taking $\mathcal{U}$ smaller if necessary, assume that
there is an isomorphism $\mathcal{U}\cong(-\delta,0]\times S^{1}$ that extends
the obvious identification over $P$.

Extend the immersion $\widehat{\iota}:S\to\mathbb{C}\times\Sigma$ to a
symplectic immersion $\varphi:\mathcal{N}\to\mathbb{C}\times\Sigma$, where
$\mathcal{N}$ is a convex neighborhood of the zero-section in $T^{*}S$. We
make the following assumptions on $\varphi$. First, taking $\mathcal{N}$
smaller if necessary, we may assume that $\varphi$ is an embedding on each
fiber and that $\varphi$ does not have self-intersections that project to the
subset $\widehat{\iota}^{-1}(\mathbb{C}\times\overline{U})$. Secondly, we
choose $\varphi$ so that over $P$ it is compatible with the splitting of
$\widehat{\iota}$ into a product.

Write $\gamma=\alpha\\#_{s}\beta$ and consider the immersed Lagrangian
submanifold $I\times\gamma$, where $I=\\{x+iy\in\mathbb{C}:y=-x\\}$. Taking
$\mathcal{U}$ smaller if necessary, we may assume that
$\varphi^{-1}(I\times\gamma)$ is the graph of a closed $1$-form $\eta$ on
$\mathcal{U}$ that vanishes on $C$. Since $\mathcal{U}$ deformation retracts
onto $C$, $\eta$ is exact. Write $\eta=dG$ for a function $G$ on
$\mathcal{U}$. Let $r$ be the collar coordinate on $\mathcal{U}$ and let
$\chi(r)$ be an increasing smooth function such that $\chi(r)=0$ near
$-\delta$ and $\chi(r)=1$ on a neighborhood of $0$. Extend the function
$\chi(r)G$ to $S$ by $0$. Finally, the desired immersion
$\iota:S\to\mathbb{C}\times\Sigma$ is the composition of $\varphi$ with the
graph of $d(\chi G)$, which is then isotoped to make its ends horizontal.

######  Notation.

In the remainder of this section, we will use the following notations to
describe $S(\alpha,\beta;s)$. We denote by $S$ the domain of the cobordism and
by $\iota:S\to\mathbb{C}\times\Sigma$ the immersion of $S$. We denote by $A$,
$B$ and $C$ the boundary components of $S$ which correspond, respectively, to
the curves $\alpha$, $\beta$ and $\gamma:=\alpha\\#_{s}\beta$. We call
$H=\iota^{-1}(\mathbb{C}\times U)$ the handle region, where $U$ is the Darboux
chart around $s$ used to define the surgery. We call
$K=\iota^{-1}(\mathbb{C}\times\\{s\\})$ the core of the handle. See Figure 5
for a schematic representation of $S$.

Figure 5. Schematic representation of the domain of a surgery cobordism. The
handle $H$ is shaded. On the complement of $H$, the cobordism is equivalent to
a product.

The following properties of $S(\alpha,\beta;s)$ are straightforward
consequences of the construction.

######  Lemma 5.1.

The surgery cobordism $S(\alpha,\beta;s)$ satisfies the following properties:

  1. (i)

All the double points of $\iota$ belong to $S\setminus H$.

  2. (ii)

Over $S\setminus H$, $\iota$ is equivalent to restrictions of the product
immersions $\lambda\times\alpha$ and $\lambda^{\prime}\times\beta$, where
$\lambda$ and $\lambda^{\prime}$ are embedded paths in $\mathbb{C}$.

###### Proof.

Property (i) follows from the fact that $H$ does not contain double points of
the original immersion $\widehat{\iota}$. Our assumptions about the Weinstein
neighborhood $\varphi$ imply that this is still true after perturbation.

Property (ii) follows from the fact that the $\Sigma$ component of
$\widehat{\iota}$ agrees with the $\Sigma$ component of $I\times\gamma$ over
$P$, and that the Weinstein immersion $\varphi$ is chosen to respect the
splitting of $\iota$ into a product over $P$. It follows that over $P$ the
$1$-form $\eta$ used to define the perturbation is given by $\eta=f(r)dr$,
where $r$ is the collar coordinate on $\mathcal{U}$. This implies that the
perturbed immersion has the claimed form over $S\setminus H$. ∎

###  5.2. Teardrops on surgery cobordisms

By construction, the projection of the surgery cobordism $S(\alpha,\beta;s)$
to $\Sigma$ consists of $\alpha\cup\beta$ and a small neighborhood of $s$. In
particular, the projection deformation retracts onto $\alpha\cup\beta$.
Therefore, the projection of a polygon with boundary on $S(\alpha,\beta;s)$
should correspond, via this deformation, to a polygon with boundary on
$\alpha\cup\beta$ that may have additional corners at $s$. This leads to the
following definition.

######  Definition 5.2.

Let $\alpha$ and $\beta$ be immersed curves intersecting at $s$. A polygon
with boundary on $\alpha$ and $\beta$ is called an $s$-marked teardrop if it
has one corner at a point $c$ distinct from $s$, and all remaining corners at
$s$.

See Figure 6 for some examples of $s$-marked teardrops. Note that in the
preceding definition we allow the case where there is no corner at $s$; the
polygon is then an ordinary teardrop on one of the curves.

Figure 6. Left: curves on the four-holed sphere that bound an $s$-marked
triangle. Center and right: curves on the torus that bound an $s$-marked
square, with the square drawn in the universal cover of the torus.

To relate teardrops on $S(\alpha,\beta;s)$ with $s$-marked teardrops on
$\alpha\cup\beta$, we will use the following lemma, which makes the above
deformation argument precise.

######  Lemma 5.3.

There is a strong deformation retraction $r:S\times[0,1]\to S$ of $S$ onto
$A\cup B\cup K$ that satisfies

$\pi_{\Sigma}\circ\iota\circ r_{s}(p)=\pi_{\Sigma}\circ\iota(p)$ (8)

for all $s\in[0,1]$ and for all $p\in S\setminus H$.

###### Proof.

Recall that the cobordism $\iota$ is obtained by perturbing an immersion
$\widehat{\iota}$. Moreover, over $S\setminus H$ we have
$\pi_{\Sigma}\circ\iota=\pi_{\Sigma}\circ\widehat{\iota}$. Hence, it suffices
to construct the required deformation retraction for $\widehat{\iota}$.

For the immersion $\widehat{\iota}$, the claim follows from a standard
argument, using the fact that $\widehat{\iota}$ is obtained from the product
cobordisms $\widehat{\alpha}$ and $\widehat{\beta}$ by attaching the index $1$
handle $H$. This can be made precise by the following general argument.
Observe that the function $(x+iy,p)\mapsto x-y$ on $\mathbb{C}\times\Sigma$
restricts to a Morse function $f$ on $S$ with a single critical point of index
$1$, which lies on $K$. The required deformation retraction can then be
obtained by using the negative gradient flow of $f$ away from $K$ and a
straight-line homotopy near $K$, as explained in [Mil65, Theorem 3.14]. ∎

The purpose of condition (8) in the previous lemma is to control the behaviour
of the double points of $\iota$ throughout the deformation.

######  Proposition 5.4.

Let $\alpha$ and $\beta$ be immersed curves intersecting at $s$. Then the
surgery cobordism $S(\alpha,\beta;s)$ bounds a teardrop if and only if
$\alpha$ and $\beta$ bound an $s$-marked teardrop.

###### Proof.

Suppose first that $S(\alpha,\beta;s)$ bounds a teardrop $u$. We see the
boundary lift of $u$ as a path $\delta:[0,1]\to S$ with distinct endpoints and
with $\iota(\delta(0))=\iota(\delta(1))$. Since the handle $H$ does not
contain double points of $\iota$, the endpoints of $\delta$ lie in $S\setminus
H$. By applying the retraction of Lemma 5.3, we obtain a new path
$\delta^{\prime}:=r_{1}\circ\delta$ on $A\cup B\cup K$ whose endpoints are
distinct and belong to $A\cup B$. By a further homotopy, we can assume that
this path is locally embedded. By condition (8) of Lemma 5.3, the path
$\lambda:=\pi_{\Sigma}\circ\iota\circ\delta^{\prime}$ is a loop which is
homotopic relative endpoints to the loop $\pi_{\Sigma}\circ\iota\circ\delta$.
It follows that $\lambda$ is null-homotopic, hence is the boundary of a disk
$v$ with boundary on $\alpha\cup\beta$. The disk $v$ is an $s$-marked
teardrop; indeed, it has one corner corresponding to projection of the corner
of $u$, and the remaining corners at $s$ correspond to the (finitely many)
times when $\lambda$ crosses $K$.

Conversely, suppose that $\alpha\cup\beta$ bounds an $s$-marked teardrop $v$
with corner at $c$. Identify $A$ and $B$ with the domains of $\alpha$ and
$\beta$, respectively. Then the boundary lifts of $v$ can be seen as paths in
$A\cup B$ with endpoints at the preimages of $s$ and $c$. Whenever $v$ has a
corner that maps to $s$, the boundary lifts on each side of the corner can be
glued together by concatenating them with a path going along the core $K$.
Since all the corners of $v$ except one are mapped to $s$, by doing this we
obtain a single path $\delta$ in $A\cup B\cup K$ with endpoints on the two
preimages of the other corner $c$.

If the endpoints of $\delta$ lie on the same component of $A\cup B$, then
$\iota\circ\delta$ is a loop. This loop is null-homotopic in
$\mathbb{C}\times\Sigma$ since by construction
$\pi_{\Sigma}\circ\iota\circ\delta$ is a reparametrization of
$v|_{\partial\mathbb{D}}$. Hence $\iota\circ\delta$ extends to a disk
$u:\mathbb{D}\to\mathbb{C}\times\Sigma$ with boundary on $\iota(S)$. This disk
is a teardrop since the path $\delta$ is a boundary lift of $u$.

If the endpoints of $\delta$ de not lie on the same component of $A\cup B$, we
close up $\iota\circ\delta$ into a loop in the following way. By Lemma 5.3,
for each endpoint of $\delta$, there is a path in
$(\pi_{\Sigma}\circ\iota)^{-1}(c)$ that connects that endpoint to the boundary
component $C$. Concatenating $\delta$ with these paths yields a path
$\delta^{\prime}$ with endpoints on $C$ with the property that
$\iota\circ\delta^{\prime}$ is a loop. By the same argument as in the previous
case, $\delta^{\prime}$ is the boundary lift of a teardrop on $\iota$. ∎

Next, we give a simple algebraic obstruction to representing a class in
$\pi_{2}(\Sigma,\alpha\cup\beta,s)$ by an $s$-marked teardrop. Since $\alpha$
and $\beta$ are in general position, we can see the union $\alpha\cup\beta$ as
a $4$-valent oriented graph (i.e. a $1$-dimensional CW complex) embedded in
$\Sigma$, whose vertices are the intersections and self-intersections of the
curves. For each vertex $v$ of this graph, there is an integral cellular
$1$-cocycle $\rho_{v}$ supported on the edges incident to $v$, whose values
are prescribed by Figure 7. This cocycle can be thought of as giving an
algebraic count of the number of corners that a cycle has at $v$.

Figure 7. The definition of the cocycle $\rho_{v}$ that counts the algebraic
number of corners of a cycle at $v$.

For a class $A\in\pi_{2}(\Sigma,\alpha\cup\beta,s)$, we denote by $\partial
A\in H_{1}(\alpha\cup\beta;\mathbb{Z})$ the image of its boundary by the
Hurewicz morphism.

######  Lemma 5.5.

Suppose that the class $A\in\pi_{2}(\Sigma,\alpha\cup\beta,s)$ can be
represented by an $s$-marked teardrop with corner at $c$. Then
$\rho_{c}(\partial A)=\pm 1$ and $\rho_{v}(\partial A)=0$ for every vertex $v$
distinct from $c$ and $s$.

###### Proof.

Suppose that $A$ can be represented by a polygon $u$ that has no corners at
the vertex $v$. Let $G$ be the graph obtained by pulling apart the two
branches of $\alpha\cup\beta$ meeting at $v$. To be precise, $G$ is obtained
from $\alpha\cup\beta$ by adding a vertex $v^{\prime}$ and has the same set of
edges, except that the two $\beta$ edges incident to $v$ in $\alpha\cup\beta$
are incident to $v^{\prime}$ in $G$. That $u$ has no corner at $v$ means that
$\partial u$ factors through the quotient map $G\to\alpha\cup\beta$ that
identifies $v^{\prime}$ and $v$. Since the pullback of $\rho_{v}$ to $G$ is
exact, we conclude that $\rho_{v}(\partial A)=0$.

Suppose now that $A$ can be represented by a polygon $u$ that has one corner
at the vertex $c$. Let $E$ and $E^{\prime}$ be the edges of $\alpha\cup\beta$
visited by $\partial u$ before and after passing through the corner $c$, and
suppose that their endpoints are $\\{c,p\\}$ and $\\{c,q\\}$, respectively.
Since $\partial u$ has a corner at $c$, one of these edges has $\rho_{c}=\pm
1$, and the other one has $\rho_{c}=0$. The loop $\partial u$ is homotopic to
the concatenation of a simple path going from $p$ to $c$ along $E$, a simple
path going from $c$ to $q$ along $E^{\prime}$, and a path going from $q$ back
to $p$ that has no corners at $c$. Hence $\rho_{c}([\partial
u])=\rho_{c}(E)\pm\rho_{c}(E^{\prime})=\pm 1$. ∎

###  5.3. Disks on surgery cobordisms

As a consequence of Lemma 5.3, we also obtain the following criterion for the
existence of non-trivial disks on surgery cobordisms.

######  Proposition 5.6.

Let $\alpha$ and $\beta$ be immersed curves intersecting at $s$. Then the
surgery cobordism $S(\alpha,\beta;s)$ is incompressible if and only if
$\alpha$ and $\beta$ span a free group of rank $2$ in $\pi_{1}(\Sigma,s)$.

###### Proof.

We equip $S$ with the basepoint $x$ which is the intersection of $A$ and $K$.
By retracting $S$ onto $A\cup B\cup K$ as in Lemma 5.3 and then collapsing $K$
to a point, we obtain a homotopy equivalence $S\to S^{1}\vee S^{1}$ that makes
the following diagram commute up to homotopy:

${(S,x)}$${(\mathbb{C}\times\Sigma,\iota(x))}$${(S^{1}\vee
S^{1},x_{0})}$${(\Sigma,s).}$$\scriptstyle{\simeq}$$\scriptstyle{\iota}$$\scriptstyle{\pi_{\Sigma}}$$\scriptstyle{\alpha\vee\beta}$
(9)

Here, $\alpha$ and $\beta$ are seen as loops based at $s$, the wedge sum
identifies the two preimages of $s$, and $x_{0}$ is the wedge point.

Since the vertical maps in Diagram (9) are homotopy equivalences, we deduce
that $\iota$ is incompressible if and only if the map $\mu:\pi_{1}(S^{1}\vee
S^{1},x_{0})\to\pi_{1}(\Sigma,s)$ induced by $\alpha\vee\beta$ is injective.

The image of $\mu$ is the subgroup $\langle[\alpha],[\beta]\rangle$ of
$\pi_{1}(\Sigma,s)$ generated by $\alpha$ and $\beta$. Therefore, if $\mu$ is
injective then $\langle[\alpha],[\beta]\rangle\cong\pi_{1}(S^{1}\vee
S^{1},x_{0})$, which is a free group of rank $2$. Conversely, if
$\langle[\alpha],[\beta]\rangle$ is a free group of rank $2$, then $\mu$ maps
a free group of rank $2$ onto a free group of rank $2$. Since free groups of
finite rank are Hopfian333Recall that a group $G$ is _Hopfian_ if any
epimorphism $G\to G$ is an isomorphism., $\mu$ is an isomorphism on its image.
Hence, we conclude that $\mu$ is injective if and only if
$\langle[\alpha],[\beta]\rangle$ is a free group of rank $2$. ∎

For a surface of genus $g\geq 2$, the subgroup of $\pi_{1}$ generated by two
elements is always free. This is a special case of the following result of
Jaco.

######  Theorem 5.7 ([Jac70, Corollary 2]).

Let $\Sigma$ be a surface with $\chi(\Sigma)\leq 0$. Then any subgroup of
$\pi_{1}(\Sigma)$ generated by $k$ elements, where $k<2-\chi(\Sigma)$, is a
free group.

In the setting of Proposition 5.6, this implies that the subgroup of
$\pi_{1}(\Sigma,s)$ generated by $\alpha$ and $\beta$ is a free group of rank
$2$ or less. The rank is $0$ or $1$ if and only if there is some non-trivial
relation $[\alpha]^{p}=[\beta]^{q}$ in $\pi_{1}(\Sigma,s)$. Hence we can
reformulate Proposition 5.6 as follows.

######  Proposition 5.8.

Let $\alpha$ and $\beta$ be immersed curves intersecting at $s$. Then the
surgery cobordism $S(\alpha,\beta;s)$ is incompressible if and only if there
are no integers $p$ and $q$ with $(p,q)\neq(0,0)$ such that
$[\alpha]^{p}=[\beta]^{q}$ in $\pi_{1}(\Sigma,s)$.

We will often make use of the following special case of the preceding results.

######  Corollary 5.9.

Let $\alpha$ and $\beta$ be immersed curves intersecting at $s$. If
$\alpha\cdot\beta\neq 0$, then the surgery cobordism $S(\alpha,\beta;s)$ is
incompressible.

###### Proof.

The hypothesis implies that the intersection pairing restricted to
$\langle[\alpha],[\beta]\rangle$ is non-trivial. Hence,
$\langle[\alpha],[\beta]\rangle$ cannot be a free group of rank $0$ or $1$. By
Theorem 5.7, it must be a free group of rank $2$, and the conclusion follows
from Proposition 5.6. ∎

##  6\. Lagrangian cobordism invariants of curves

In this section, we define the Lagrangian cobordism invariants of curves on
symplectic surfaces that give rise to the morphism

$\Omega_{\operatorname{unob}}(\Sigma)\to
H_{1}(S\Sigma;\mathbb{Z})\oplus\mathbb{R}$

of Theorem B. These invariants are the same as those considered by Abouzaid
[Abo08], who showed that they induce well-defined morphisms on
$K_{0}\operatorname{DFuk}(\Sigma)$. Our aim in this section is to show that
these invariants also descend to morphisms on
$\Omega_{\operatorname{imm}}(\Sigma)$ (and hence on
$\Omega_{\operatorname{unob}}(\Sigma)$). Moreover, in Section 6.3, we compute
these invariants in some cases which are relevant to the proof of Theorem B.

###  6.1. Discrete invariants

For a symplectic manifold $M^{2n}$, we write
$\mathcal{L}^{\operatorname{or}}(M)$ for the Grassmannian of oriented
Lagrangian subspaces of $TM$. This is a fiber bundle with fiber $\Lambda_{n}$,
the oriented Lagrangian Grassmannian of $\mathbb{C}^{n}$. Recall that
$\Lambda_{n}\cong U(n)/SO(n)$. In the case of a surface $\Sigma$, we identify
$\mathcal{L}^{\operatorname{or}}(\Sigma)$ with the unit tangent bundle of
$\Sigma$, which we denote $S\Sigma$. We associate to an immersed curve
$\gamma:S^{1}\to\Sigma$ the class $[\widetilde{\gamma}]\in
H_{1}(S\Sigma;\mathbb{Z})$, where $\widetilde{\gamma}:S^{1}\to S\Sigma$ is the
Gauss map of $\gamma$. We will show that this gives rise to a morphism
$\Omega_{\operatorname{imm}}(\Sigma)\to H_{1}(S\Sigma;\mathbb{Z})$.

To see this, consider the stabilization map
$i:\mathcal{L}^{\operatorname{or}}(\Sigma)\to\mathcal{L}^{\operatorname{or}}(\mathbb{C}\times\Sigma)$
that sends an oriented Lagrangian subspace $U\subset T_{p}\Sigma$ to the
Lagrangian subspace $\mathbb{R}\oplus U$ of
$T_{(z,p)}(\mathbb{C}\times\Sigma)$, equipped with the product orientation.
Here, $z\in\mathbb{C}$ is fixed, but the map does not depend on this choice up
to homotopy.

######  Lemma 6.1.

The stabilization map $i$ induces an isomorphism on $H_{1}(-;\mathbb{Z})$.

###### Proof.

It suffices to prove that stabilization induces an isomorphism on $\pi_{1}$.
To see this, recall that the determinant map $\Lambda_{n}\to S^{1}$ coming
from the identification $\Lambda_{n}\cong U(n)/SO(n)$ induces an isomorphism
on $\pi_{1}$. The stabilization map $\Lambda_{n}\to\Lambda_{n+1}$ commutes
with the determinant, hence it also induces an isomorphism on $\pi_{1}$. The
result now follows from comparison of the long exact sequences of homotopy
groups associated to the fiber bundles
$\mathcal{L}^{\operatorname{or}}(\Sigma)$ and
$\mathcal{L}^{\operatorname{or}}(\mathbb{C}\times\Sigma)$. ∎

It follows from Lemma 6.1 that the class $[\widetilde{\gamma}]$ is invariant
under Lagrangian cobordisms in the following sense.

######  Proposition 6.2.

Let $V:(\gamma_{1},\ldots,\gamma_{r})\rightsquigarrow\emptyset$ be an oriented
Lagrangian cobordism between immersed curves in $\Sigma$. Let
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{r}$ be the Gauss maps of
the ends of $V$. Then

$\sum_{k=1}^{r}\;[\widetilde{\gamma}_{k}]=0\qquad\text{in
}H_{1}(S\Sigma;\mathbb{Z}).$

###### Proof.

The cobordism has a Gauss map
$V\to\mathcal{L}^{\operatorname{or}}(\mathbb{C}\times\Sigma)$ which extends
the stabilizations of the Gauss maps of its ends. This yields the relation

$\sum_{k=1}^{r}\;i_{*}[\widetilde{\gamma}_{k}]=0\qquad\text{in
}H_{1}(\mathcal{L}^{\operatorname{or}}(\mathbb{C}\times\Sigma);\mathbb{Z}).$

The conclusion now follows from Lemma 6.1. ∎

It follows from Proposition 6.2 that the homology class of the Gauss map
descends to a morphism $\Omega_{\operatorname{imm}}(\Sigma)\to
H_{1}(S\Sigma;\mathbb{Z})$.

Since $S\Sigma$ is an oriented circle bundle, its homology can be computed
from the Gysin sequence

${0}$${\mathbb{Z}_{\chi(\Sigma)}}$${H_{1}(S\Sigma;\mathbb{Z})}$${H_{1}(\Sigma;\mathbb{Z})}$${0,}$$\scriptstyle{p_{*}}$
(10)

where the first map sends $1$ to the class of a fiber and $p:S\Sigma\to\Sigma$
is the projection. This sequence splits since $H_{1}(\Sigma;\mathbb{Z})$ is a
free abelian group. It will be useful for us to fix a choice of splitting of
(10). A splitting map
$\mu:H_{1}(S\Sigma;\mathbb{Z})\to\mathbb{Z}_{\chi(\Sigma)}$ will be called a
winding number morphism.

######  Remark 6.3.

The set of splittings of (10) is a torsor over

$\operatorname{Hom}(H_{1}(\Sigma;\mathbb{Z}),\mathbb{Z}_{\chi(\Sigma)})\cong
H^{1}(\Sigma;\mathbb{Z}_{\chi(\Sigma)}).$

In the terminology of Seidel [Sei00], they correspond to the
$\mathbb{Z}_{2\chi(\Sigma)}$-gradings of $\Sigma$ that factor through the
oriented Lagrangian Grassmannian. Winding numbers of curves on closed surfaces
first appeared in the works of Reinhart [Rei60] and Chillingworth [Chi72].

###  6.2. Holonomy

We recall the definition of the holonomy of an immersed curve in a surface of
genus $g\neq 0$, following [Abo08]. First, fix a primitive $\lambda$ of
$p^{*}\omega$, where as before $p:S\Sigma\to\Sigma$ denotes the projection of
the unit tangent bundle. Such a primitive exists whenever $\chi(\Sigma)\neq 0$
(see Corollary A.3 for a proof of a more general claim). The holonomy of an
immersed curve $\gamma$ is then defined as

$\operatorname{Hol}_{\lambda}(\gamma):=\int_{S^{1}}\widetilde{\gamma}^{*}\lambda,$

where $\widetilde{\gamma}$ is the lift of $\gamma$ to $S\Sigma$.

The holonomy of curves admits a straightforward generalization to a class of
real-valued cobordism invariants of Lagrangian immersions in _monotone_
symplectic manifolds. For the benefit of the reader, this theory is developed
in Appendix A. As in the case of surfaces, the basic observation that leads to
these invariants is that for a monotone manifold $M$ the form $p^{*}\omega$ is
exact, where $p:\mathcal{L}M\to M$ is the Lagrangian Grassmannian bundle (see
Corollary A.3). As a special case of the general theory developed in Appendix
A, we prove that holonomy is invariant under oriented Lagrangian cobordisms.
The precise statement is as follows.

######  Proposition 6.4.

Suppose that there is an oriented Lagrangian cobordism
$(\gamma_{1},\ldots,\gamma_{r})\rightsquigarrow\emptyset$. Then

$\sum_{k=1}^{r}\;\operatorname{Hol}_{\lambda}(\gamma_{k})=0.$

###### Proof.

This is a special case of Corollary A.10. ∎

By Proposition 6.4, holonomy descends to a morphism
$\operatorname{Hol}_{\lambda}:\Omega_{\operatorname{imm}}(\Sigma)\to\mathbb{R}$.
From now on, we fix a choice of primitive $\lambda$ and suppress it from the
notation.

We will often make use of the following relationship between holonomy and
regular homotopies. Recall that a regular homotopy $\phi:S^{1}\times
I\to\Sigma$ is a Lagrangian homotopy and therefore has a well-defined flux
class $\operatorname{flux}(\phi)\in H^{1}(S^{1};\mathbb{R})$, which by
definition is the class that satisfies

$\operatorname{flux}(\phi)\cdot[S^{1}]=\int_{S^{1}\times I}\phi^{*}\omega.$

######  Lemma 6.5.

Let $\phi:S^{1}\times I\to\Sigma$ be a regular homotopy from $\alpha$ to
$\beta$. Then

$\operatorname{Hol}(\beta)-\operatorname{Hol}(\alpha)=\int_{S^{1}\times
I}\phi^{*}\omega=\operatorname{flux}(\phi)\cdot[S^{1}].$

###### Proof.

The regular homotopy $\phi$ lifts to a homotopy from $\widetilde{\alpha}$ to
$\widetilde{\beta}$ in $S\Sigma$. The result then follows from the Stokes
Theorem. ∎

###  6.3. Winding and holonomy of bounding curves

In this section, we compute the winding number and holonomy of curves that
bound surfaces in $\Sigma$. We start with the following lemma.

######  Lemma 6.6.

Suppose that $\sigma$ is a positively oriented fiber of $S\Sigma$. Then

$\int_{\sigma}\lambda=-\frac{\operatorname{area}(\Sigma)}{\chi(\Sigma)}.$

###### Proof.

Suppose that $\sigma$ is the fiber over $x$. Let $v$ be a vector field on
$\Sigma$ that has a unique zero at $x$. By the Poincaré-Hopf Theorem, the
index of this zero is $\chi(\Sigma)$.

Let $D$ be a smoothly embedded closed disk containing $x$. Consider the
section $\widetilde{v}:\Sigma\setminus\operatorname{int}D\to S\Sigma$ given by
$\widetilde{v}=v/|v|$. By choosing a trivialization of $S\Sigma$ over $D$, we
can find a homotopy in $p^{-1}(D)$ from the loop $\widetilde{v}|_{\partial D}$
to the loop $\sigma$ iterated $-\chi(\Sigma)$ times. Moreover, the area of
this homotopy with respect to $p^{*}\omega$ is $\operatorname{area}(D)$.
Hence, by the Stokes Theorem, we have

$-\chi(\Sigma)\,\int_{\sigma}\lambda=\int_{\widetilde{v}|_{\partial
D}}\lambda+\operatorname{area}(D)=\operatorname{area}(\Sigma\setminus
D)+\operatorname{area}(D)=\operatorname{area}(\Sigma).$

∎

######  Lemma 6.7.

Suppose that the curves $\gamma_{k}$, $k=1,\ldots,r$, form the oriented
boundary of a compact surface $P\subset\Sigma$. Then,

$\displaystyle\sum_{k=1}^{r}\operatorname{Hol}(\gamma_{k})=\operatorname{area}(P)-\frac{\chi(P)}{\chi(\Sigma)}\,\operatorname{area}(\Sigma),$
$\displaystyle\sum_{k=1}^{r}\mu(\gamma_{k})=\chi(P)\mod\chi(\Sigma).$

###### Proof.

Let $v$ be a vector field on $P$ that agrees with the derivatives
$\gamma_{k}^{\prime}$ over $\partial P$ and has a unique zero at a point
$x\in\operatorname{int}P$. By the Poincaré-Hopf Theorem, the index of $v$ at
$x$ is $\chi(P)$.

Let $\sigma$ be the fiber of $S\Sigma$ over $x$, with the positive
orientation. Arguing as in the proof of Lemma 6.6, the chain
$\sum_{k}\widetilde{\gamma}_{k}-\chi(P)\sigma$ is the boundary of a surface
whose area with respect to $p^{*}\omega$ equals $\operatorname{area}(P)$. As a
consequence, using Lemma 6.6 we obtain

$\sum_{k=1}^{r}\operatorname{Hol}(\gamma_{k})=\operatorname{area}(P)+\chi(P)\,\int_{\sigma}\lambda=\operatorname{area}(P)-\frac{\chi(P)}{\chi(\Sigma)}\operatorname{area}(\Sigma).$

Likewise, since $\mu(\sigma)=1$ by definition, we have

$\sum_{k=1}^{r}\mu(\gamma_{k})=\chi(P)\,\mu(\sigma)=\chi(P).$

∎

As a consequence, we obtain the following

######  Corollary 6.8.

The morphism $\Omega_{\operatorname{unob}}(\Sigma)\to
H_{1}(S\Sigma;\mathbb{Z})$ is surjective.

##  7\. Relations between isotopic curves

In this section, we describe how isotopic curves are related in
$\Omega_{\operatorname{unob}}(\Sigma)$. We let $\mathcal{I}$ be the subgroup
of $\Omega_{\operatorname{unob}}(\Sigma)$ generated by all the elements of the
form $[\widetilde{\alpha}]-[\alpha]$, where $\widetilde{\alpha}$ and $\alpha$
are isotopic curves. The main result of this section is the following
computation of $\mathcal{I}$.

######  Proposition 7.1.

The holonomy morphism restricts to an isomorphism
$\mathcal{I}\cong\mathbb{R}$.

As an immediate consequence, we deduce that

$\Omega_{\operatorname{unob}}(\Sigma)\cong\mathbb{R}\oplus\Omega_{\operatorname{unob}}(\Sigma)/\mathcal{I}.$

Therefore, the computation of $\Omega_{\operatorname{unob}}(\Sigma)$ reduces
to the computation of the quotient
$\Omega_{\operatorname{unob}}(\Sigma)/\mathcal{I}$.

######  Definition 7.2.

The quotient

$\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)=\Omega_{\operatorname{unob}}(\Sigma)/\mathcal{I}$

is called the reduced unobstructed cobordism group of $\Sigma$.

We emphasize that, by definition, isotopic curves become equal in
$\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$. This will play a crucial
role in the proofs of Section 8 and Section 9 by allowing us to put curves in
minimal position by isotopies.

###  7.1. Holonomy and Hamiltonian isotopies

As a first step towards the proof of Proposition 7.1, we show that holonomy
completely determines whether two isotopic curves are Hamiltonian isotopic.

######  Lemma 7.3.

Let $\alpha$ and $\beta$ be isotopic simple curves with
$\operatorname{Hol}(\alpha)=\operatorname{Hol}(\beta)$. Then $\alpha$ and
$\beta$ are Hamiltonian isotopic.

###### Proof.

Suppose first that the curves are non-separating. By Lemma 6.5, the holonomy
condition implies that any isotopy from $\alpha$ to $\beta$ has zero flux.
Moreover, since the curves are non-separating, they induce surjective maps
$H^{1}(\Sigma;\mathbb{R})\to H^{1}(S^{1};\mathbb{R})$. By a standard argument,
this condition implies that an isotopy from $\alpha$ to $\beta$ extends to an
ambient symplectic isotopy with zero flux; see Lemma 6.6 of [Sol13] for a
detailed proof. The result now follows from the well-known fact that a
symplectic isotopy with zero flux is homotopic relative to its endpoints to a
Hamiltonian isotopy; see Theorem 10.2.5 of [MS17].

Suppose now that the curves are separating. Then, by Lemma 6.7, the surfaces
bounded by the curves have the same area. The conclusion then follows from a
standard Moser isotopy argument. In the case of contractible curves, a
detailed proof is given in [AS01, Appendix A]. The same proof applies to the
case of separating curves; for completeness, we sketch the argument from
[AS01].

Choose a diffeomorphism $\phi$ isotopic to the identity such that
$\phi(\alpha)=\beta$. Let $S$ and $S^{\prime}$ be the surfaces bounded by
$\alpha$ and $\beta$, respectively. Then $\phi(S)=S^{\prime}$, hence

$\int_{S}\phi^{*}\omega=\int_{S^{\prime}}\omega=\int_{S}\omega$

since $S$ and $S^{\prime}$ have the same area. By a Moser isotopy argument
(see [AS01, Lemma A.3]), we can find an isotopy $\psi_{t}$ such that
$\psi_{1}^{*}(\phi^{*}\omega)=\omega$ and $\psi_{t}(S)=S$. Then the
composition $\theta=\phi\circ\psi_{1}$ is symplectic, isotopic to the identity
and satisfies $\theta(S)=S^{\prime}$. It follows again from Moser isotopy that
the inclusion
$\operatorname{Symp}(\Sigma)\hookrightarrow\operatorname{Diff}^{+}(\Sigma)$ is
an isomorphism on $\pi_{0}$, hence there is a path $(\theta_{t})$ in
$\operatorname{Symp}(\Sigma)$ with $\theta_{0}=\operatorname{id}$ and
$\theta_{1}=\theta$. The restriction of $(\theta_{t})$ to $\alpha$ is an exact
Lagrangian path, hence $\alpha$ and $\beta$ are Hamiltonian isotopic. ∎

For future use, we also record a version of Lemma 7.3 that applies to
topologically unobstructed curves.

######  Lemma 7.4.

Let $\alpha$ and $\beta$ be homotopic topologically unobstructed curves with
$\operatorname{Hol}(\alpha)=\operatorname{Hol}(\beta)$. Then there is an exact
homotopy $(\gamma_{t})_{t\in[0,1]}$ such that $\gamma_{0}=\alpha$,
$\gamma_{1}=\beta$ and each curve $\gamma_{t}$ is topologically unobstructed.

###### Proof.

Consider the covering map $\rho:\widehat{\Sigma}\to\Sigma$ associated to the
subgroup of $\pi_{1}(\Sigma,\alpha(0))$ generated by $[\alpha]$. Equip
$\widehat{\Sigma}$ with the symplectic form $\rho^{*}\omega$.

By definition, $\alpha$ lifts to a closed curve $\widehat{\alpha}$ in
$\widehat{\Sigma}$. Moreover, a homotopy from $\alpha$ to $\beta$ lifts to a
homotopy from $\widehat{\alpha}$ to a closed curve $\widehat{\beta}$ which is
a lift of $\beta$. Since $\alpha$ and $\beta$ are unobstructed, by Lemma 4.6
their lifts to the universal cover of $\Sigma$ are embedded. This implies that
$\widehat{\alpha}$ and $\widehat{\beta}$ are simple curves.

Since $\widehat{\alpha}$ and $\widehat{\beta}$ are homotopic non-contractible
simple curves, they are isotopic by Theorem 2.1 of [Eps66]. Any isotopy
between them has zero flux since its projection to $\Sigma$ is a regular
homotopy from $\alpha$ to $\beta$, and this homotopy has zero flux by the
holonomy assumption and Lemma 6.5. Moreover, $\widehat{\alpha}$ represents a
generator of $\pi_{1}(\widehat{\Sigma})\cong\mathbb{Z}$, hence the map
$H^{1}(\widehat{\Sigma};\mathbb{R})\to H^{1}(S^{1};\mathbb{R})$ induced by
$\widehat{\alpha}$ is an isomorphism. As in the first case of the proof of
Lemma 7.3, it follows that $\widehat{\alpha}$ and $\widehat{\beta}$ are
Hamiltonian isotopic. The projection of a Hamiltonian isotopy from
$\widehat{\alpha}$ to $\widehat{\beta}$ yields an exact homotopy from $\alpha$
to $\beta$ through unobstructed curves. ∎

###  7.2. Proof of Proposition 7.1

In this section, we prove Proposition 7.1. First, we show that
$\operatorname{Hol}|_{\mathcal{I}}$ is surjective, which is a consequence of
the following lemma.

######  Lemma 7.5.

Let $\alpha$ be a non-separating simple curve. Then for any $x\in\mathbb{R}$,
there exists a simple curve $\widetilde{\alpha}$ isotopic to $\alpha$ such
that $\operatorname{Hol}(\widetilde{\alpha})-\operatorname{Hol}(\alpha)=x.$

###### Proof.

There is a curve $\beta$ that intersects $\alpha$ transversely in a single
point. Let $A\cong S^{1}\times[0,1]$ be a small tubular neighborhood of
$\beta$ such that $\alpha\cap A$ is a ray of the annulus. Then, the required
curve $\widetilde{\alpha}$ can be obtained by wrapping $\alpha$ around the
annulus by the flow of a vector field of the form
$f(r)\frac{\partial}{\partial\theta}$, where $f(r)\geq 0$ is a non-constant
function that vanishes near $r=0$ and $r=1$. ∎

It remains to prove that $\operatorname{Hol}|_{\mathcal{I}}$ is injective,
which is the hardest part of the proof. We will deduce it from the following
proposition, which is a generalization of Lemma 7.3.

######  Proposition 7.6.

Suppose that $(\alpha,\widetilde{\alpha})$ and $(\beta,\widetilde{\beta})$ are
two pairs of isotopic simple curves with
$\operatorname{Hol}(\widetilde{\alpha})-\operatorname{Hol}(\alpha)=\operatorname{Hol}(\widetilde{\beta})-\operatorname{Hol}(\beta)$.
Then, in $\Omega_{\operatorname{unob}}(\Sigma)$ we have

$[\widetilde{\alpha}]-[\alpha]=[\widetilde{\beta}]-[\beta].$

This result appears as Proposition 5.4 and Lemma 5.15 in [Per19]. As explained
in Remark 3.5, the extension of Perrier’s result to the present case requires
additional verifications to show unobstructedness of the relevant cobordisms.
These verifications are straightforward applications of the obstruction
criteria proved in Section 4 and Section 5. For the benefit of the reader, we
include in Appendix B a proof of Proposition 7.6 that closely follows the
proof in [Per19] and incorporates these verifications.

Using Proposition 7.6, we can now finish the proof of Proposition 7.1.

###### Proof of Proposition 7.1..

By Lemma 7.5, $\operatorname{Hol}|_{\mathcal{I}}$ is surjective. To prove
injectivity, we claim that any element of $\mathcal{I}$ can be written as
$[\widetilde{\alpha}]-[\alpha]$ for some isotopic curves $\widetilde{\alpha}$
and $\alpha$ (in other words, the subset of elements of the form
$[\widetilde{\alpha}]-[\alpha]$ is already a subgroup). Indeed, a general
element of $\mathcal{I}$ can be written as a sum
$\sum_{k=1}^{N}[\widetilde{\beta}_{k}]-[\beta_{k}]$, where
$\widetilde{\beta}_{k}$ and $\beta_{k}$ are isotopic. Fix a non-separating
curve $\alpha_{0}$. By Lemma 7.5, we can find curves $\alpha_{k}$ for
$k=1,\ldots,N$, such that $\alpha_{k}$ is obtained from $\alpha_{k-1}$ by an
isotopy of area
$\operatorname{Hol}(\widetilde{\beta}_{k})-\operatorname{Hol}(\beta_{k})$.
Then, by Proposition 7.6, we have

$\sum_{k=1}^{N}[\widetilde{\beta}_{k}]-[\beta_{k}]=\sum_{k=1}^{N}[\alpha_{k}]-[\alpha_{k-1}]=[\alpha_{N}]-[\alpha_{0}].$

This proves the claim. The injectivity of $\operatorname{Hol}|_{\mathcal{I}}$
now follows directly from Proposition 7.6. ∎

##  8\. Action of the mapping class group

In Section 7, we showed how to reduce the computation of
$\Omega_{\operatorname{unob}}(\Sigma)$ to the computation of the reduced
cobordism group $\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$ (see
Definition 7.2). The action of the group of symplectic diffeomorphisms
$\operatorname{Symp}(\Sigma)$ on $\Omega_{\operatorname{unob}}(\Sigma)$
descends to an action on $\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$.
By definition, the component of the identity
$\operatorname{Symp}_{0}(\Sigma)\subset\operatorname{Symp}(\Sigma)$ acts
trivially on $\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$. Therefore,
the action of $\operatorname{Symp}(\Sigma)$ descends to an action of the
_symplectic mapping class group_
$\operatorname{Mod}_{\omega}(\Sigma)=\operatorname{Symp}(\Sigma)/\operatorname{Symp}_{0}(\Sigma)$
on $\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$. The goal of this
section is to describe this action. This will be a key ingredient in the
computation of $\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$, as it will
allow us to determine a simple set of generators of this group (see
Proposition 9.2).

The main result of this section is the computation of the action of
_symplectic Dehn twists_ , which are generators of
$\operatorname{Mod}_{\omega}(\Sigma)$. For a simple curve $\alpha$, we denote
by $T_{\alpha}\in\operatorname{Mod}_{\omega}(\Sigma)$ the Dehn twist around
$\alpha$ (the definition of $T_{\alpha}$ is recalled in Section 8.1).

######  Theorem 8.1.

Let $\alpha$ and $\beta$ be two simple curves in $\Sigma$. Then in
$\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$ there is the relation

$T_{\alpha}[\beta]=[\beta]+(\alpha\cdot\beta)[\alpha].$ (11)

Theorem 8.1 is proved in two steps. The first step is to prove the theorem in
the special case where the curves intersect twice with zero algebraic
intersection. This is carried out in Section 8.2. Then, in Section 8.3, we
show how to deduce the general case of Theorem 8.1 from the special case by an
inductive argument.

######  Remark 8.2.

On the side of $K_{0}\operatorname{DFuk}$, the existence of the Dehn twist
relation (11) is a consequence of the well-known _Seidel exact triangle_ in
$\operatorname{DFuk}$

${HF^{*}(S,L)\otimes S}$${L}$${T_{S}L}$$\scriptstyle{\left[1\right]}$ (12)

where $T_{S}$ is the Dehn twist around a framed Lagrangian sphere $S$; see
[Sei08, Corollary 17.18]. See also [Abo08] for the existence of the triangle
(12) in the case of curves on surfaces that intersect minimally. We remark
that the methods used in the present work do not recover the exact triangle
(12) on the level of $\operatorname{DFuk}$, but only the induced relation in
$K$-theory.

###  8.1. Dehn twists

We recall the definition of the Dehn twist $T_{\alpha}$ around a simple curve
$\alpha$. Choose a symplectic embedding $\psi:[-\delta,\delta]\times
S^{1}\to\Sigma$ such that $\psi|_{\\{0\\}\times S^{1}}=\alpha$, where the
annulus is equipped with the standard symplectic form $dt\wedge d\theta$.
Consider the symplectic diffeomorphism of the annulus given by

$(t,\theta)\longmapsto(t,\theta-f(t)),$

where $f:[-\delta,\delta]\to\mathbb{R}$ is a smooth increasing function which
is $0$ near $-\delta$ and $1$ near $\delta$. Using the embedding $\psi$, this
map extends by the identity to a symplectomorphism $T_{\alpha}$ of $\Sigma$,
called a (right-handed) _symplectic Dehn twist_ around $\alpha$. The class of
this map in $\operatorname{Mod}_{\omega}(\Sigma)$ does not depend on the
choices of $\psi$ and $f$ above. We will also denote this class by the symbol
$T_{\alpha}$, whenever no confusion arises.

######  Remark 8.3.

We warn the reader that our convention for the direction of the Dehn twist is
opposite to that of [Abo08] and [Per19].

By Moser’s Theorem, the inclusion of $\operatorname{Symp}(\Sigma)$ inside the
group of orientation-preserving diffeomorphisms
$\operatorname{Diff}^{+}(\Sigma)$ induces an isomorphism between
$\operatorname{Mod}_{\omega}(\Sigma)$ and the classical mapping class group
$\operatorname{Mod}(\Sigma)=\operatorname{Diff}^{+}(\Sigma)/\operatorname{Diff}_{0}(\Sigma)$,
where $\operatorname{Diff}_{0}(\Sigma)$ is the subgroup of diffeomorphisms
isotopic to the identity. In what follows, we will identify
$\operatorname{Mod}_{\omega}(\Sigma)$ and $\operatorname{Mod}(\Sigma)$ using
this isomorphism.

It is a well-known theorem of Dehn and Lickorish that
$\operatorname{Mod}(\Sigma)$ is generated by the classes of finitely many Dehn
twists. It follows that $\operatorname{Mod}_{\omega}(\Sigma)$ is generated by
symplectic Dehn twists.

###  8.2. Proof of Theorem 8.1: curves intersecting twice

In this section, we prove the following special case of Theorem 8.1.

######  Proposition 8.4.

Let $\alpha$ and $\beta$ be simple curves which are in minimal position and
intersect in two points of opposite signs. Then, in
$\widetilde{\Omega}_{\operatorname{unob}}(\Sigma)$ we have
$T_{\alpha}[\beta]=[\beta]+(\alpha\cdot\beta)[\alpha].$

Proposition 8.4 is proved by realizing the Dehn twist as an iterated surgery.
This is a well-known procedure; see for example [FM12, Section 3.1.1]. We will
follow the description given by Perrier in [Per19, Proposition 2.15], which we
recall in order to fix some notation.

Let $\alpha$ and $\beta$ be curves satisfying the hypothesis of Proposition
8.4. Let $N$ be a closed regular neighborhood of $\alpha\cup\beta$.444To be
more precise, we mean that $N=A\cup B$, where $A$ and $B$ are closed tubular
neighborhoods of $\alpha$ and $\beta$, respectively. Moreover, $A$ and $B$ are
chosen small enough so that $A\cap B$ is a disjoint union of disks, and inside
each such disk the curves are in standard position (i.e. look like the
coordinate axes of $\mathbb{R}^{2}$). Note that $N$ is a four-holed sphere. We
perform the following surgeries, which are illustrated in Figure 8. Let
$s_{1}$ be the positive intersection of $(\alpha,\beta)$, and $q_{1}$ the
negative intersection. The first step is to form the surgery
$\sigma:=\alpha\\#_{s_{1}}\beta$. Next, let $\widetilde{\alpha}$ be a
$C^{1}$-small Hamiltonian perturbation of $\alpha^{-1}$ which intersects
$\sigma$ in two points. This perturbation is chosen so that one of these
intersection points, denoted $s_{2}$, lies on the arc of $\beta$ near $q_{1}$,
and the other, denoted $q_{2}$, lies on the arc of $\alpha$ near $q_{1}$.
Finally, we let $\tau:=\widetilde{\alpha}\\#_{s_{2}}\sigma$. The size of the
surgery handle is chosen big enough so that $\tau$ is embedded.

Figure 8. The surgery procedure used in the proof of Proposition 8.4.

It is easy to check that $\tau$ is isotopic to $T_{\alpha}\beta$. We denote
the traces of the surgeries by $S_{1}=S(\alpha,\beta;s_{1})$ and
$S_{2}=S(\widetilde{\alpha},\sigma;s_{2})$. Concatenating $S_{1}$ and $S_{2}$
produces an immersed cobordism
$\tau\rightsquigarrow(\widetilde{\alpha},\alpha,\beta)$. To prove Proposition
8.4, we will show that there is an unobstructed cobordism with the same ends.
By Proposition 4.11, it suffices to check that the surgery cobordisms $S_{1}$
and $S_{2}$ constructed above are topologically unobstructed. Hence, to finish
the proof of Proposition 8.4, it suffices to show the following lemma.

######  Lemma 8.5.

Assume that $\alpha$ and $\beta$ are in minimal position. Then the surgery
cobordisms $S_{1}$ and $S_{2}$ constructed above are topologically
unobstructed.

Before giving the proof of the lemma, we make some preliminary observations.
It will be convenient to perform the necessary computations inside the regular
neighborhood $N$. To relate computations in $N$ to those in $\Sigma$, we will
use the fact that $N$ is incompressible in $\Sigma$ (recall from Definition
4.2 that this means that the maps $\pi_{1}(N)\to\pi_{1}(\Sigma)$ induced by
the inclusion are injective).

######  Lemma 8.6.

Assume that $\alpha$ and $\beta$ are in minimal position. Then the regular
neighborhood $N$ of $\alpha\cup\beta$ is incompressible in $\Sigma$.

###### Proof.

Since the curves $\alpha$ and $\beta$ are in minimal position, it follows from
the Bigon Criterion [FM12, Proposition 1.7] that they do not bound bigons. In
particular, this implies that the boundary circles of $N$ are non-contractible
in $\Sigma$. The incompressibility of $N$ is then a consequence of the
following well-known fact.

######  Lemma 8.7.

Let $M$ be a smooth compact manifold and $X\subset M$ a smooth codimension $0$
compact submanifold with boundary. If $\partial X$ is incompressible in $M$,
then $X$ is incompressible in $M$.

We refer to the appendix of [GT23] for an elegant proof of this lemma. In the
case of surfaces, $\partial X$ is the union of embedded circles. Since the
fundamental group of an orientable surface has no torsion, the condition that
$\partial X$ be incompressible is equivalent to the non-contractibility of
each component of $\partial X$. Therefore, we conclude from Lemma 8.7 that the
regular neighborhood $N$ above is incompressible. ∎

The incompressibility of $N$ implies that for a subset $A\subset N$, the
inclusions induce an isomorphism $\pi_{2}(N,A,*)\to\pi_{2}(\Sigma,A,*)$. This
follows from comparison of the homotopy exact sequences of the pairs $(N,A)$
and $(\Sigma,A)$.

We now proceed to the proof that the cobordisms $S_{i}$ are topologically
unobstructed.

###### Proof of Lemma 8.5.

$S_{1}$ is topologically unobstructed.

The group $\pi_{1}(N,s_{1})$ is free of rank $3$. The classes $[\alpha]$ and
$[\beta]$ belong to a basis of $\pi_{1}(N,s_{1})$, hence span a free group of
rank $2$. Since $N$ is incompressible, they also span a free group of rank $2$
in $\pi_{1}(\Sigma,s_{1})$. Hence, by Proposition 5.6, $S_{1}$ is
incompressible.

Moreover, we have $\pi_{2}(N,\alpha\cup\beta,s_{1})=0$, hence also
$\pi_{2}(\Sigma,\alpha\cup\beta,s_{1})=0$. By Lemma 5.5, $\alpha$ and $\beta$
do not bound $s_{1}$-marked teardrops. We conclude from Proposition 5.4 that
$S_{1}$ does not bound teardrops.

$S_{2}$ is topologically unobstructed.

Again, it can be checked directly that $[\widetilde{\alpha}]$ and $[\sigma]$
belong to a basis of $\pi_{1}(N,s_{2})$. By the same argument, it follows that
$S_{2}$ is incompressible.

It remains to show that $\widetilde{\alpha}$ and $\sigma$ do not bound
$s_{2}$-marked teardrops. The group
$\pi_{2}(\Sigma,\widetilde{\alpha}\cup\sigma,s_{2})\cong\pi_{2}(N,\widetilde{\alpha}\cup\sigma,s_{2})$
is isomorphic to $\mathbb{Z}$ and is spanned by the class of the obvious
triangle with boundary on $\widetilde{\alpha}\cup\sigma$. Since this triangle
has three distinct corners, Lemma 5.5 ensures that no classes in
$\pi_{2}(\Sigma,\widetilde{\alpha}\cup\sigma,s_{2})$ can be represented by
$s_{2}$-marked teardrops.

∎

###  8.3. Proof of Theorem 8.1: general case

We now show how to deduce the general case of Theorem 8.1 from the special
case of Proposition 8.4. The proof uses an inductive argument based on the
following lemma, due to Lickorish.

######  Lemma 8.8 (Lickorish [Lic64]).

Let $\alpha$ and $\beta$ be simple curves in general position with $N\geq 2$
intersection points. In the case $N=2$, assume that the intersections have the
same sign. Then there exists a simple curve $\gamma$ with the following
properties:

  1. (i)

$\gamma$ intersects $\beta$ transversely in one point or two points of
opposite signs.

  2. (ii)

$\gamma$ intersects $\alpha$ transversely in at most $N-1$ points.

  3. (iii)