# Lagrangian cobordisms and $K$-theory of symplectic bielliptic surfaces

Álvaro Muñiz-Brea

###### Abstract

Abstract: We consider a family of closed symplectic manifolds $4$-manifolds
which we call symplectic bielliptic surfaces and study its Lagrangian
cobordism group of weakly-exact Lagrangian $G$-branes (that is, Lagrangians
equipped with a grading, a Pin structure and a $G$-local system); relations
come from Lagrangian cobordisms satisfying a tautologically unobstructedness-
type condition, also equipped with $G$-brane structures. Our first theorem
computes its subgroup generated by tropical Lagrangians. When $G$ is the
unitary group of the Novikov field, we use homological mirror symmetry to
compute the Grothendieck group of the Fukaya category and show it agrees with
our computation for the cobordism group. This leads us to conjecture that
tropical Lagrangians generate the whole cobordism group.

##  1 Introduction

###  1.1 Lagrangian cobordisms and the Grothendieck group

One of the main goals of symplectic topology is to understand and classify the
Lagrangian submanifolds of a given symplectic manifold. In its most classical
form, this problem asks for a classification of Lagrangians up to Hamiltonian
isotopy; although easy to state, this problem is remarkably hard—barely any
answers are known outside real dimension two. In an attempt to define a
coarser relation between Lagrangians that would be amenable to classification,
Arnold introduced in a series of two papers [Arn80, Arn80a] the notion of a
_Lagrangian cobordism_. A Lagrangian cobordism between Lagrangians
$L_{-},L_{+}\subset X$ is a properly embedded Lagrangian submanifold
$V\subset\C\times X$ such that, outside some compact set $K\subset\C$, one has

$V\setminus\pi^{-1}_{\C}(K)=L_{-}\times\R_{<-a_{-}}\bigsqcup
L_{+}\times\R_{>a_{+}}$

for some values $a_{\pm}\in\R_{>0}$. (It is helpful to think of $V$ via its
projection to $\C$, which is depicted in Figure 1.) When such a cobordism
exists, one says that the Lagrangians $L_{-}$ and $L_{+}$ are Lagrangian
cobordant. This definition extends straightforwardly to Lagrangian cobordisms
between Lagrangian tuples, where the cobordism must now fiber over several
horizontal rays outside some compact set.

$L_{-}\times\mathbb{R}_{<-a_{-}}$$L_{+}\times\mathbb{R}_{>a_{+}}$$\pi_{\mathbb{C}}(V)$$K$
Figure 1: The shaded region (including the two horizontal lines) depicts a
typical projection of a two-ended Lagrangian cobordism $V\subset X\times\C$ to
$\C$. We have included (in dashed lines) an example of a compact region
$K\subset\C$ outside which $V$ is product type: in $\C\setminus K$ the
projection looks like two straight lines, and living over them in $X\times\C$
we have the cylidrical Lagrangians $L_{-}\times\R_{<-a_{-}}$ and
$L_{+}\times\R_{>a_{+}}$.

There is a different but related notion of Lagrangian cobordism between
Legendrian submanifolds, where one instead tries to classify Legendrian
submanifolds in a contact manifold up to Lagrangian cobordism in the
symplectisation (see e.g. [Cha10]). While both definitions attempt to classify
objects within a given manifold by looking at Lagrangian cobordisms in a
bigger space, the invariants obtained are very different.

There are several algebraic structures that can be obtained from the notion of
Lagrangian cobordism, and many of them have been connected to the Fukaya
category. For instance, Biran-Cornea work in the compact, monotone setting and
use Lagrangian cobordisms to construct a ‘Lagrangian cobordism category’
[BC13, Section 7]. In [BC14] they show that Lagrangian cobordisms provide a
_functorial_ way to decompose one end of a cobordism as an iterated mapping
cone of the other ends, and using this they show there is a natural functor
from their Lagrangian cobordism category to a category which encodes triangle
decompositions in the derived Fukaya category. With a different set-up,
Nadler-Tanaka [NT20] work in the non-compact, exact case and construct a
stable $\infty$-category whose higher morphisms are ‘cobordisms between
cobordisms’, and they conjecture it to be equivalent to the (partially)
wrapped Fukaya category. Further work of Tanaka [Tan16] shows that there is
indeed a functor from this stable $\infty$-category to (modules over) the
wrapped Fukaya category, and that this functor is exact [Tan16a] (in
particular, exactness of the functor recovers the above-mentioned result of
Biran-Cornea).

In this paper, we focus on a linear algebra invariant going back to Arnold,
the Lagrangian cobordism group. In its most naive form, it is an abelian group
freely generated by Lagrangians and whose relations come from Lagrangian
cobordisms:

######  Definition 1.

Let $\mathcal{L}$ be a collection of Lagrangians in $X$ and
$\mathcal{L}_{cob}$ a collection of cobordisms in $X\times\C$ whose ends are
in $\mathcal{L}$. We define the Lagrangian cobordism group as the quotient

$\operatorname{\mathrm{Cob}}(X)\equiv\operatorname{\mathrm{Cob}}(X;\mathcal{L},\mathcal{L}_{cob}):=\Z^{\mathcal{L}}/\sim,$

where $\sim$ is the equivalence relation on $\Z^{\mathcal{L}}$ generated by
expressions of the form

$L_{1}^{+}+\dots+L^{+}_{k}\sim L_{1}^{-}+\dots+L_{s}^{-}$

whenever there exists a cobordism in $\mathcal{L}_{cob}$ between tuples
$(L_{1}^{+},\dots,L_{k}^{+})$ and $(L_{1}^{-},\dots,L_{s}^{-})$.

Often the Lagrangians $\mathcal{L}$ under consideration are not plain
Lagrangians but come equipped with some extra structure (for instance, one can
take $\mathcal{L}$ to be the collection of orientable Lagrangians _plus_ a
choice of orientation, or impose conditions such as exactness or
monotonicity). In this case, it makes sense to ask the cobordisms in
$\mathcal{L}_{cob}$ to carry the same extra structure, and this data should be
compatible with restriction (whenever restriction of data makes sense). A
Lagrangian cobordism $V$ equipped with some extra data $\mathcal{D}$ will then
induce a relation

$(L_{1}^{+},\mathcal{D}|_{L_{1}^{+}})+\dots+(L_{k}^{+},\mathcal{D}|_{L_{k}^{+}})\sim(L_{1}^{-},\mathcal{D}|_{L_{1}^{-}})+\dots+(L_{s}^{-},\mathcal{D}|_{L_{s}^{-}})$

in the cobordism group. Arnold himself performed the first computations of
these groups, showing that in $X=\R^{2}$ oriented immersed Lagrangians are
classified up to oriented cobordism by the Maslov index and the area they
enclose. He also computed the cobordism group of $T^{*}S^{1}$, where
Lagrangians are classified essentially by the same data (their homology class
and the area they enclose). Soon after Eliashberg [Eli84] argued that
immersed, exact Lagrangian cobordism groups are isomorphic to homotopy groups
of certain Thom spaces, and Audin [Aud85] used this to show that these
Lagrangian cobordism groups are topological.

Nonetheless, in recent years it has been discovered that this flexibility for
Lagrangian cobordism groups of immersed exact Lagrangians disappears when one
equips the Lagrangians and/or cobordism with various decorations. In a series
of two papers [BC13, BC14] Biran-Cornea brought Lagrangian cobordisms back to
the attention of the symplectic topology community by exhibiting a relation
between Lagrangian cobordisms and cone decompositions in the Fukaya category.
Whereas Eliashberg and Audin had shown that the most naive notion of a
Lagrangian cobordism reduces to algebraic topology, Biran-Cornea showed that
if the Lagrangians in $\mathcal{L}$ and $\mathcal{L}_{cob}$ are required to be
_monotone_ (with a fixed monotonicity constant) and satisfy a Gromov-Witten
type condition, then Lagrangian cobordism is no longer a flexible notion. One
of their main results is the following: if $V$ is a monotone Lagrangian
cobordism between a tuple $(L_{1},\dots,L_{k})$ and $L_{0}$, then $L_{0}$ is
generated by $L_{1},\dots,L_{k}$ in the derived Fukaya category
$D^{b}\mathcal{F}uk(X)$ (cf. [BC14, Theorem A]). It follows that the
Lagrangian cobordism group admits a map

(1) $\operatorname{\mathrm{Cob}}(X)\to K_{0}(D^{b}\mathcal{F}uk(X))$

to the Grothendieck group of the triangulated envelope of the Fukaya category
(cf. [BC14, Corollary 1.2.1]).111Recall the Grothendieck group of a
triangulated category is the free abelian group on its objects modulo
relations $Z_{1}-Z_{2}+Z_{3}=0$ whenever there is a distinguished triangle
$Z_{1}\to Z_{2}\to Z_{3}\to Z_{1}[1]$. In further work, Biran-Cornea-Shelukhin
[BCS21] take $\mathcal{L}$ to be _weakly-exact_ Lagrangians (meaning
$\omega(\pi_{2}(X,L))=0$) and $\mathcal{L}_{cob}$ to be _quasi-exact_
cobordisms (meaning there exists a compatible almost complex structure such
that they bound no holomorphic disks). Under these assumptions, they extend
the results in [BC14] and construct a map $\operatorname{\mathrm{Cob}}(X)\to
K_{0}(D^{b}\mathcal{F}uk(X))$. In this paper, we analyze a particular instance
of the following question:

_“is the map $\operatorname{\mathrm{Cob}}(X)\to K_{0}(D^{b}\mathcal{F}uk(X))$
an isomorphism?”_

By construction, the map (1) is always surjective, hence the question is
whether it is injective. The main result in [BC14] shows that cobordisms
induce cone decompositions in the Fukaya category. Injectivity would mean that
every triangle relation in the Fukaya category can be obtained from Lagrangian
cobordisms.

To date, answers to the above question (in the compact case) have only
appeared for $\dim X=2$: [Hau15] computes the Lagrangian cobordism group of
$T^{2}$, and shows the map (1) is an isomorphism in this case; and [Per19,
RF23] extend the result to higher genus surfaces. In this paper, we study the
question for a closed symplectic $4$-manifold. We work with a similar set-up
to that in [BCS21]: the collection of Lagrangians $\mathcal{L}$ will consist
of weakly-exact Lagrangians equipped a grading, a Pin structure and a local
system, and the collection $\mathcal{L}_{cob}$ will consist of Lagrangian
cobordisms carrying the same decorations and satisfying a quasi-exactness type
condition (see 26 for the precise definition of the cobordism group, and
Section 3.3 for an exposition of the extra data that we put on our
Lagrangians). A minor adaptation of the results in [BCS21] (together with
[Hau15], who works out the inclusion of the above decorations) ensures there
is a well-defined map $\operatorname{\mathrm{Cob}}(X)\to
K_{0}(D^{b}\mathcal{F}uk(X))$. We will take $X=\mathcal{K}$ to be a symplectic
bielliptic surface as in Section 4.1 and we will consider the subgroup
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\subset\operatorname{\mathrm{Cob}}(X)$
generated by Lagrangian sections and fibers. We present an explicit
computation of $\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ in 34, and
show that the composition
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\hookrightarrow\operatorname{\mathrm{Cob}}(\mathcal{K})\to
K_{0}(D^{b}\mathcal{F}uk(X))$ is an isomorphism in 35. With a slightly
different set-up, Lagrangian cobordism between non-compact Lagrangians in
exact symplectic manifolds—where the definition of a cobordism is suitably
modified—have also been considered in the literature [NT20, Bos21]. As far as
cobordism groups in this setting, [Bos21] defines a corresponding version of
the Lagrangian cobordism group and constructs an analog of the map (1) (where
the Fukaya category is replaced by the _wrapped_ Fukaya category). He also
shows that such a map is an isomorphism for punctured Riemann surfaces. In
later work, he extends his results to arbitrary Weinstein sectors, showing the
Lagrangian cobordism group is isomorphic to the middle relative homology of
the Weinstein sector [Bos23]. A consequence of his computation is that the map
(1) is _not_ an isomorphism for flexible Weinstein manifolds. We remark
however that his definition of a Lagrangian cobordism is different from ours
(due to the non-compact nature of Lagrangians in wrapped Fukaya categories).

###  1.2 Main results

Consider the symplectic manifold $T^{*}\R^{2}$ with coordinates
$(x_{1},y_{1},x_{2},y_{2})$ and the standard symplectic structure
$dx_{1}\wedge dy_{1}+dx_{2}\wedge dy_{2}$. Let $\mathcal{K}$ be the quotient
of $T^{*}\R^{2}$ by the action of $\Z^{4}$ as well as the relation

$(x_{1},y_{1},x_{2},y_{2})\sim(x_{1}+1/2,y_{1},-x_{2},-y_{2}).$

Note that all actions commute with the symplectic form, hence $\mathcal{K}$
inherits a symplectic structure from $T^{*}\R^{2}$. We call $\mathcal{K}$ a
_symplectic bielliptic surface_ (cf. 18).

The projection to the $(x_{1},x_{2})$ coordinates gives a Lagrangian torus
fibration $\pi:\mathcal{K}\to K$ over a tropical Klein bottle $K$. Let
$\det_{\Z}K=\wedge^{2}T_{\Z}K$ be the integral orientation bundle of $K$, and
denote by $\xi:=\pi^{*}\det_{\Z}K$ its pullback to $\mathcal{K}$. Given an
abelian group $G$, we consider a cobordism group
$\operatorname{\mathrm{Cob}}(\mathcal{K})$ whose generators are weakly-exact
Lagrangian $G$-branes and whose relations come from tautologically
unobstructed Lagrangian cobordisms equipped a $G$-brane structure (see Section
3.3 and 26 for precise definitions). Our first result is the following:

######  Theorem A ($=$34).

Let $\mathcal{K}$ be a symplectic bielliptic surface and $\pi:\mathcal{K}\to
K$ its Lagrangian torus fibration over a tropical Klein bottle. Let
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\subset\operatorname{\mathrm{Cob}}(\mathcal{K})$
be the subgroup generated by Lagrangian fibers and Lagrangian sections. Then
there is an isomorphism

$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\cong
H_{2}(\mathcal{K};\xi)\oplus G_{(2)}\oplus(S^{1}\oplus G)\oplus(S^{1}\oplus
G)$

for $G_{(2)}\subset G$ the subgroup of $2$-torsion elements.

Theorem A shows that $\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ is
finite-dimensional in the sense of 50—in fact, one can see directly from the
definition that it is a $2$-dimensional cobordism group (roughly speaking,
this means that $2$ is the minimum $d$ such that
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ can be parametrized by a
$d$-dimensional family of Lagrangians). We compare this with the work of
Sheridan-Smith, who show that a symplectic 4-manifold admitting a Lagrangian
of genus at least 1 has an infinite-dimensional cobordism group [SS20, Lemma
7.9]. In fact, we are able to extend the result of Sheridan-Smith to arbitrary
dimension and prove the following (see Appendix D):

######  Proposition 2.

Let $X$ be a closed symplectic manifold of dimension at least 4 with trivial
canonical bundle. If $X$ contains a Lagrangian torus of Maslov index zero,
then the cobordism group of graded Lagrangians
$\operatorname{\mathrm{Cob}}(X)$ is infinite-dimensional.

The crucial difference in Theorem A is that both Sheridan-Smith and 2 work in
the Calabi-Yau case (i.e. the canonical bundle is trivial), whereas symplectic
bielliptic surfaces have torsion but non-trivial canonical bundle. Given a
nowhere vanishing quadratic volume form $\eta^{2}\in
H^{0}((\Omega^{n})^{\otimes 2})$, a graded Lagragian with respect to
$\eta^{2}$ is naturally oriented with respect to the local system where
$\pm\eta$ takes real values (see [Sei08, Remark 11.18]). When the canonical
bundle is trivial the quadratic form $\eta^{2}$ admits a square root, hence
the local system is trivial and graded Lagrangians are naturally oriented in
the usual sense—this is the setting of [SS20] and 2. However, in this paper
the canonical bundle is non-trivial, hence Lagrangians are only oriented with
respect to some non-trivial local system (this is the local system $\xi$ of
Theorem A). Obviously, the result of Sheridan-Smith and 2 still holds for the
oriented version of the cobordism group of a bielliptic surface. The
remarkable fact is that introducing these new graded Lagrangian
cobordisms—despite eliminating some too, as not every oriented Lagrangian is
graded—creates sufficiently many new relations to get a finite-dimensional
cobordism group.

As remarked above, the result in Theorem A is, to the best of the author’s
knowledge, the first computation of a Lagrangian cobordism group in dimension
greater than two. Although Theorem A only gives a computation of a subgroup of
$\operatorname{\mathrm{Cob}}(\mathcal{K})$ (the subgroup
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ generated by tropical
Lagrangians), our second main result shows that this subgroup is indeed an
interesting subgroup:

######  Theorem B ($=$35).

Fix $G=U_{\Lambda}$ to be the unitary group of the Novikov field. Then the
composition

$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\hookrightarrow\operatorname{\mathrm{Cob}}(\mathcal{K})\to
K_{0}(D^{b}\mathcal{F}uk(X))$

is an isomorphism.

To prove Theorem B we use Abouzaid’s homological mirror symmetry result
[Abo21] to show that symplectic bielliptic surfaces are mirror to algebraic
bielliptic surfaces as defined in Section 4.1. We then show that the
composition

$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\to
K_{0}(D^{b}\mathcal{F}uk(\mathcal{K}))\hookrightarrow
K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))\xrightarrow{\sim}K_{0}(D^{b}Coh(Y))$

is an isomorphism, which yields the result. Here, we denoted by
$D^{\pi}\mathcal{F}uk(\mathcal{K})$ the split-closure of the Fukaya category.
In general, this is larger than $D^{b}\mathcal{F}uk(\mathcal{K})$, but it
follows from our argument that the inclusion
$K_{0}(D^{b}\mathcal{F}uk(\mathcal{K}))\hookrightarrow
K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))$ is an isomorphism. It is a theorem
of Thomason [Tho97] that this isomorphism of Grothendieck groups is enough to
conclude $D^{b}\mathcal{F}uk(\mathcal{K})\simeq
D^{\pi}\mathcal{F}uk(\mathcal{K})$. In particular, for bielliptic surfaces
there is no need to take the split-closure of the Fukaya category to obtain a
homological mirror symmetry equivalence.

Theorem B on its own—together with mirror symmetry considerations and the
analogous statement for Chow groups—lead us to the following conjecture:

######  Conjecture 3.

The inclusion
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\hookrightarrow\operatorname{\mathrm{Cob}}(\mathcal{K})$
is an isomorphism.

The main difficulty to address this question is our lack of understanding of
what are all the weakly-exact Lagrangians in $\mathcal{K}$.

###  1.3 Organization of the paper

The paper is organized as follows. In Section 2 we review the necessary
background on tropical geometry, including the main definitions (Section 2.1),
the construction of the Albanese variety and the Albanese morphism (Section
2.2) and the relationship between tropical and symplectic geometry (Section
2.3). The tropical manifold we study is a tropical Klein bottle, and this is
introduced in Section 2.4. Section 3 deals with the main object of this paper,
Lagrangian cobordisms. We present the main constructions of cobordisms in
Section 3.1 and use them to compute the cobordism group of $T^{2}$ Section
3.2—this will be useful for the main computations in Section 4. In the last
part of this section, Section 3.3, we explain the extra data we consider on
Lagrangians and cobordisms. The main results of the paper are in Section 4. In
Sections 4.1 and 4.2 we introduce symplectic and algebraic bielliptic surfaces
and explain how they fit into Abouzaid’s homological mirror symmetry result
[Abo21] to conclude they are homologically mirror. We then build up the
necessary results to prove the main theorems in Sections 4.3, 4.4 and 4.6. The
main theorems are in the next two sections: we compute the subgroup of the
Lagrangian cobordism group generated by tropical Lagrangians in Section 4.7
(see 34) and show it is isomorphic to the Grothendieck group of the Fukaya
category in Section 4.8 (see 35). In the last part, Section 4.9, we prove some
compatibility results involving the brane structure we put on our Lagrangians
and cobordisms—it is essential for our computations but can safely be skipped
by the reader on a first read. We also include four appendices at the end. The
first three prove (technical) results claimed in several parts of the paper,
whose proofs we decided to separate from the main text to improve the
readability of the paper and whose content might be of independent interest to
the reader. The last appendix contains the proof of 2: the content of this
Proposition ties directly to the main results of the paper, but its proof is a
simple linear algebra result and does not provide any additional insights.

#### Acknowledgments

I would like to thank my advisor Nick Sheridan for his guidance on this
project. He has been extremely generous with his time, very patient and has
provided invaluable insights. The idea to use multivalued perturbations as in
Appendix C is entirely due to him. I would also like to thank Jeff Hicks for
many helpful discussions. I owe to him much of my understanding of Lagrangian
surgery. I also thank Arend Bayer for a useful conversation that motivated the
results in Appendix B. This work was supported by an ERC Starting Grant (award
number 850713-HMS).

##  2 Tropical geometry

In this section we give a review of the tropical geometry background that will
be needed to understand some of the constructions in this paper. The
symplectic manifold we consider in this paper comes with a Lagrangian torus
fibration over a tropical affine manifold. Moreover, all the Lagrangians we
consider are _tropical Lagrangians_ : Lagrangians living over tropical
subvarieties of the base. Understanding the tropical geometry of the base is
hence essential for our computations.

###  2.1 Tropical affine manifolds

Let
$\operatorname{Aff}(\R^{n}):=\R^{n}\rtimes\operatorname*{\mathrm{GL}}(n,\Z)$
denote the group of affine transformations of $\R^{n}$ whose linear part is
integral. A tropical affine manifold of dimension $n$ is a smooth manifold
together with an atlas of charts whose transition functions belong to
$\operatorname{Aff}(\R^{n})$.

######  Example 2.1.

A choice of basis on a real vector space gives it the structure of a tropical
affine manifold. In particular, $\R^{n}$ has a natural tropical affine
structure.

Let $B_{1},B_{2}$ be tropical affine manifolds of dimension $n_{1},n_{2}$. A
smooth map $f:B_{1}\to B_{2}$ is said to be integral affine if, locally in
some charts, it is an affine map $\R^{n_{1}}\to\R^{n_{2}}$ with integer slope.
We say it is a tropical map if, locally in some charts, it is a piecewise
affine map with integer slope.

######  Example 2.2.

Let $B$ be a tropical affine manifold and $G$ be a group acting freely and
properly on $B$ by integral affine transformations. Then $B/G$ has a natural
tropical affine structure making the projection a tropical submersion.

Let $B$ be a tropical affine manifold. We say a smooth function $f:B\to\R$ is
tropical affine if it is an integral affine map $B\to\R$, where we view $\R$
as a tropical affine manifold by equipping it with its vector space tropical
affine structure as in Example 2.1. Similarly, a continuous function
$f:B\to\R$ is said to be tropical if it is piecewise integral affine. We
denote by $\operatorname{Aff}_{B}\subset C^{\infty}_{B}$ the subsheaf of
affine functions. It naturally gives a lattice bundle of integer covectors

$T^{*}_{\Z}B:=d(\operatorname{Aff}_{B})\subset T^{*}B,$

where $d:\Omega^{0}(B)\to\Omega^{1}(B)$ is the usual differential. Dually, we
get a lattice of vectors $T_{\Z}B\subset TB$ defined as

$T_{\Z}B:=\left\\{{v\in TB\,|\,\alpha(v)\in\Z\text{ for all }\alpha\in
T_{\Z}^{*}B}\right\\}.$

Both $T_{\Z}B$ and $T^{*}_{\Z}B$ form local systems of lattices over $B$.
Global sections $\alpha\in H^{0}(T^{*}_{\Z}B)$ are called _tropical
$1$-forms_.

A tropical affine manifold determines a $\Z^{n}$-lattice $T_{\Z}B\subset TB$.
However, not every $\Z^{n}$-lattice $\Lambda\subset TB$ equips $B$ with a
tropical affine structure—the lattice must be integrable. Sometimes we will
refer to a tropical affine manifold as the data $(B,\Lambda)$ of a smooth
manifold together with a $\Z^{n}$-lattice $\Lambda\subset TB$: it will always
be implicit that $\Lambda$ is integrable.

Given a tropical affine manifold $B$, a tropical affine submanifold is a
submanifold $V\subset B$ such that $TV=L\otimes\R$ for some sublattice
$L\subset T_{\Z}B|_{V}$. We remark that $L$ is not part of the data.

######  Example 2.3.

Tropical affine submanifolds of $(\R^{n},\Z^{n})$ are precisely linear
subspaces whose defining equations have integral coefficients.

###  2.2 The Albanese variety

Associated to any tropical affine manifold $B$ there is a tropical Albanese
variety $\operatorname{\mathrm{Alb}}(B)$ constructed as follows. Given an
integral $1$-form $\alpha\in H^{0}(T^{*}_{\Z}B)$ and a singular $1$-chain
$\gamma\in C_{1}(B)$, we can regard $\alpha$ as a usual differential form and
integrate $\alpha$ over $\gamma$. Since integral forms are closed (they are
locally exact), the integration pairing $\int:H^{0}(T^{*}_{\Z}B)\otimes
C_{1}(B)\to\R$ descends to a map

$\int:H^{0}(T_{\Z}^{*}B)\otimes H_{1}(B;\Z)\to\R$

over classes in the first homology of $B$. In particular, there is a map
$H_{1}(B;\Z)\to\Hom(H^{0}(T_{\Z}^{*}B),\R)$, and we define the tropical
Albanese variety of $B$ to be the quotient

(2)
$\operatorname{\mathrm{Alb}}(B):=\frac{\Hom(H^{0}(T_{\Z}^{*}B),\R)}{H_{1}(B;\Z)}.$

Associated to any smooth algebraic variety there is an (algebraic) Albanese
variety, which is an abelian variety satisfying a universal property. Whereas
this algebraic object is always smooth, the tropical Albanese variety
associated to a tropical affine manifold need not even be a manifold. However,
in all the examples we will consider in this paper it will not just be a
manifold but moreover it will come equipped with a natural structure of a
tropical affine torus.

Denote by $Z_{0}(B)$ the homologically trivial (i.e. degree zero) $0$-cycles
on $B$. There is a natural map

(3)
$\displaystyle\begin{split}\operatorname{\mathrm{alb}}:Z_{0}(B)&\to\operatorname{\mathrm{Alb}}(B)\\\
\operatorname{\mathrm{alb}}(b^{+}-b^{-})&=\int_{\gamma}-:H^{0}(T_{\Z}^{*}B)\to\R,\end{split}$

where $\gamma\in C_{1}(B)$ is any $1$-chain with $\partial\gamma=b^{+}-b^{-}$
(the quotient in Equation 2 makes the map independent of the choice of
$\gamma$). In particular, after choosing a base-point $b_{0}\in B$,
composition of the above map with the inclusion $b\in B\mapsto b-b_{0}\in
Z_{0}(B)$ gives an albanese map (still denoted by
$\operatorname{\mathrm{alb}}$)

$\operatorname{\mathrm{alb}}:B\to\operatorname{\mathrm{Alb}}(B)$

It is an isomorphism whenever $B$ is a tropical affine torus (see 4).

###  2.3 From tropical to symplectic geometry

Recall that there is a map from smooth manifolds to symplectic manifolds that
assigns to each smooth manifold $B$ its cotangent bundle $T^{*}B$. Moreover,
the canonical projection $T^{*}B\to B$ is a Lagrangian fibration (i.e. the
fibers are Lagrangian). Note that $T^{*}B$ is never compact.

When $B$ has the structure of a compact tropical affine manifold, we can
refine this construction to give a compact symplectic manifold: the quotient

$X(B):=T^{*}B/T^{*}_{\Z}B$

inherits a symplectic structure from $T^{*}B$. The projection $X(B)\to B$ is
now a Lagrangian torus fibration. There is a rich duality between the tropical
geometry of $B$ and the symplectic geometry of $X(B)$, which we now briefly
recall.

First of all, $X(B)$ possesses a broader source of Lagrangian sections. In the
cotangent bundle $T^{*}B$ one can obtain Lagrangians as graphs of closed
$1$-forms—that is, sections of the sheaf $\Omega^{1}_{c}$. In $X(B)$, the
sheaf $\Omega^{1}_{c}\cong C^{\infty}/\R$ is replaced by
$C^{\infty}/\operatorname{Aff}$, as affine functions have vanishing
differential in the quotient $X(B)=T^{*}B/T_{\Z}^{*}B$.

######  Example 2.4.

The function $f:\R\to\R,f(x)=\frac{1}{2}x^{2}$ does not descend to a section
of $C^{\infty}_{\R/\Z}/\R$, but it does give a section of
$C^{\infty}_{\R/\Z}/\operatorname{Aff}$. The corresponding Lagrangian in
$X(\R/\Z)=T^{2}$ is a geodesic of homology class $(1,1)$.

Second, in the cotangent bundle one can produce Lagrangians from submanifolds
$V\subset B$ by taking the conormal bundle $N^{*}V\subset T^{*}B$. On the
other hand, some form of rigidity appears in $X(B)$ due to the compactness of
the fibers. Without further assumptions, the projection to $X(B)$ of the
conormal lift of an arbitrary submanifold need not be a (compact) embedded
Lagrangian. Instead, under the assumption that $V\subset B$ is a _tropical_
submanifold (see Section 2.1), the quotient

$L_{V}:=N^{*}V/N^{*}_{\Z}V$

is a compact Lagrangian submanifold of $X(B)$. We call $L_{V}$ a tropical
Lagrangian and say it is a _tropical lift of $V$_. The setting in which one
can obtain a Lagrangian from a subset (not even a submanifold) of the base is
an active topic of research. Lagrangian lifts of subsets with much greater
generality than what is presented here—so-called _tropical subvarieties_ —have
been constructed; see e.g. [Mik19, Hic20, MR20, Mat21, SS21].

###  2.4 Tropical Klein bottles

Let us now introduce the tropical affine manifold that we will study in this
paper. Consider $(\R^{2},\Z^{2})$ with the standard tropical affine structure,
and let $\Lambda\subset\R^{2}$ be a lattice. Recall that the quotient of a
tropical affine manifold by a group of tropical automorphisms inherits a
tropical affine structure (see Example 2.2).

######  Definition 4.

We call the quotient

(4) $T^{2}_{\Lambda}:=\R^{2}/\Lambda$

a tropical affine torus.

######  Definition 5.

Let $T^{2}_{\Lambda}$ be a tropical affine torus and
$\psi\in\operatorname{Aff}(T^{2}_{\Lambda})$ an orientation reversing, affine
involution without fixed points. We call the quotient

(5) $K_{\Lambda,\psi}:=T^{2}_{\Lambda}/\langle\psi\rangle$

a tropical Klein bottle.

There exist tropical affine manifolds with the topology of a torus which do
not arise as a quotient of the form in 4 (see [Mis96, Section 4, Theorem A]).
Our definition only covers those tropical affine tori whose associated
symplectic manifold
$X(T^{2}_{\Lambda})=T^{*}T^{2}_{\Lambda}/T_{\Z}^{*}T^{2}_{\Lambda}$, as
defined in Section 2.3, is a symplectic 4-torus. It follows from this that
there exist tropical affine manifolds with the topology of a Klein bottle that
are not of the form (5). Again, our definition covers those tropical affine
Klein bottles whose associated symplectic manifold is a quotient of a 4-torus

Tropical Klein bottles have been classified by Sepe [Sep10]. In fact, they
classify tropical affine structures on all types of Klein bottles, not just
the restricted types we consider in this paper (cf. Section 2.4). Let us
recall the part of their result that relates to those Klein bottles in 5,
which we phrase in an equivalent way that is more convenient for our purposes:

######  Lemma 6 ([Sep10]).

Let $L\subset\R^{2}$ be a lattice and denote by $(\R^{2},L)$ the smooth
manifold $\R^{2}$ together with the tropical affine structure defined by $L$.
Every tropical Klein bottle is isomorphic to a unique ‘standard’ Klein bottle,
i.e. one of the form $K_{L}:=((\R^{2},L)/\Z^{2})/\Z_{2}$, where $\Z_{2}$ acts
as above with

(6) $A=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\quad c=(1/2,0).$

Only those integral affine structures $L$ that are preserved by this standard
$A$ give well-defined tropical Klein bottles (cf. Example 2.2). In other
words, the map

$\left\\{\begin{array}[]{cc}\text{isomorphism classes of}\\\ \text{tropical
Klein bottles}\end{array}\right\\}\to\\{L\subset\R^{2}\text{ lattice}\\}$

is an injection but not a surjection, and its image is precisely those
lattices that are symmetric with respect to the involution
$(x,y)\mapsto(x,-y)$. These lattices are easily classified: choosing
$l_{1},l_{2}\in L$ primitive lattice points lying in both axis, one can easily
check using the $A$-invariance of $L$ that it has to be of the form
$L_{1}=\Z\langle l_{1},l_{2}\rangle$ or $L_{2}=\Z\langle
l_{1},1/2(l_{1}+l_{2})\rangle$. The corresponding Klein bottles will be
denoted by $K_{1}$ and $K_{2}$, and correspond to the families in equations
(2) and (4) in [Sep10, Theorem 3.1].

The two types of Klein bottles obtained in the above remark give rise to
complex surfaces $TK_{i}/T_{\Z}K_{i}$ corresponding to bielliptic surfaces of
type I and II in the classification of complex bielliptic surfaces (see for
instance [Bea96, List VI.20] for such classification, originally due to
Bagnera-de Franchis). Although such classification contains a total of seven
types, only the first two admit maximal degenerations and thus are relevant
from the point of view of mirror symmetry.

It follows from 6 and Section 2.4 that tori $T^{2}_{\Lambda}$ admitting a free
$\Z_{2}$ orientation reversing integral affine action are ‘rational’, in the
sense that there is a basis of $\Lambda$ with rational slope (although not
necessarily rational coordinates).

##  3 Lagrangian cobordisms

Recall from Section 1.1 that a (planar) Lagrangian cobordism from
$(L_{1}^{+},\dots,L_{k}^{+})$ to $(L_{1}^{-},\dots,L_{s}^{-})$ is a properly
embedded Lagrangian submanifold $V\subset X\times\C$ such that

$V\setminus\pi^{-1}_{\C}(K)=\left(\bigsqcup_{i=1}^{k}L^{+}_{i}\times\gamma_{i}^{+}\right)\bigsqcup\left(\bigsqcup_{j=1}^{s}L^{-}_{j}\times\gamma_{j}^{-}\right),$

for some compact region $K\subset\C$ and horizontal curves
$\gamma^{\pm}_{j}(t)=(\pm t,j),\pm t>a_{j}^{\pm}$. Given collections
$\mathcal{L}$ and $\mathcal{L}_{cob}$ of Lagrangians in $X$ and Lagrangian
cobordisms in $X\times\C$ (all of whose ends are in $\mathcal{L}$), one
defines the Lagrangian cobordism group as the free abelian group on
$\mathcal{L}$ modulo the relations generated by $\mathcal{L}_{cob}$ (see 1).
More generally, the collection of Lagrangians and cobordisms can be equipped
with various extra decorations (e.g. orientations) or required to satisfy some
property (e.g. monotonicity), and in this case the cobordism group is defined
as the free abelian group on the decorated Lagrangians with the desired
property modulo the relations generated by the decorated cobordisms, where the
decorations must restrict appropriately (cf. Section 1.1).

In this section we review the main constructions of Lagrangian cobordisms and
detail the decorations we will put on our Lagrangians and cobordisms. In
Section 3.1 we present the Lagrangian cobordism associated to a Hamiltonian
isotopy (so-called suspension cobordism) and several types of surgeries. We
use these to compute the Lagrangian cobordism group of $T^{2}$ in Section 3.2,
whose cobordism relations will be used in Section 4. Section 3.3 recalls the
definition of a Pin structure, a grading and a local system—these form the
decorations that we will put on our Lagrangians in Section 4.

###  3.1 Cobordism constructions

There are essentially two main sources of Lagrangian cobordisms: Hamiltonian
isotopies and Lagrangian surgery.222A third construction due to Sheridan-Smith
constructs cobordisms from tropical curves [SS21]. However, this construction
produces _cylindrical_ cobordisms— Lagrangian submanifolds in $X\times\C^{*}$
that fiber over rays near $0$ and $\infty$. Every cobordism in $X\times\C$
gives a cylindrical cobordism by quotienting $\C$ by a large enough imaginary
translation, but not every cylindrical cobordism gives a cobordism in
$X\times\C$. We briefly recall these.

######  Lemma 7 ([ALP94]).

If $L_{0}$ and $L_{1}$ are Hamiltonian isotopic, then there is a Lagrangian
cobordism between $L_{0}$ and $L_{1}$.

The second main cobordism source is that of Lagrangian surgery. Lagrangian
surgery produces, from two (embedded) Lagrangians $L_{0},L_{1}$ intersecting
transversely at some point $p$, a third (embedded) Lagrangian $L_{0}\\#L_{1}$
which agrees with $L_{0}$ and $L_{1}$ outside an arbitrarily small
neighborhood of $p$. The Lagrangians $L_{0},L_{1}$ and $L_{0}\\#L_{1}$ are
furthermore related by a Lagrangian cobordism. We now recall the construction
of $L_{0}\\#L_{1}$, which is classical and goes back to Lalonde-Sikorav [LS91]
in the $2$-dimensional case and to Polterovich [Pol91] in arbitrary dimension.

Choose a Darboux chart $\varphi:U\to\C^{n}$ around $p\in U$ such that
$\varphi(L_{0})=\R^{n}$ and $\varphi(L_{1})=(i\R)^{n}$. Consider a curve
$\gamma:\R\to\C$ such that

  * •

$\gamma(t)=t$ for $t\in(-\infty,-1]$;

  * •

$\gamma(t)=it$ for $t\in[1,\infty)$;

  * •

$\gamma^{\prime}(t)\in\R_{+}\times i\R_{+}\subset\C$ for $t\in(-1,1)$.

We define the surgery neck $N$ to be

$N:=\gamma\cdot
S^{n-1}=\\{(\gamma(t)x_{1},\dots,\gamma(t)x_{n})\in\C^{n}\,|\,\sum_{i=1}^{n}x_{i}^{2}=1\\}$

and the surgery of $L_{0}$ and $L_{1}$ to be $L_{1}\\#L_{2}:=[(L_{1}\cup
L_{2})\setminus U]\cup\varphi^{-1}(N)$. Biran-Cornea show that the Lagrangians
$L_{i}$ and their surgery are related by a Lagrangian cobordism:

######  Proposition 8.

Let $L_{0},L_{1}\subset X$ be Lagrangians intersecting transversely at a
single point $p$. There is a cobordism with ends $L_{0},L_{1}$ and
$L_{0}\\#L_{1}$.

######  Proof 3.1.

See [BC13, Lemma 6.1.1].

Since the surgery construction is local, one immediately gets:

######  Corollary 9.

Let $L_{0}$ and $L_{1}$ be Lagrangians intersecting transversely at a finite
collection of points. Let $L_{0}\\#L_{1}$ be the Lagrangian obtained by
performing the above surgery at each intersection point.333When $L_{0}$ and
$L_{1}$ intersect at a unique point, the Lagrangian $L_{0}\\#L_{1}$ is
topologically a connect sum. This is no longer true when the intersection
consists of more than one point. Then there is a Lagrangian cobordism with
ends $L_{0},L_{1}$ and $L_{0}\\#L_{1}$.

One can show that when performing surgery at a single point, $L_{0}\\#L_{1}$
is independent, up to Hamiltonian isotopy, of the specific curve $\gamma$ used
to construct the surgery neck. This is no longer the case when there are
multiple intersection points. In particular, byconcatenating two cobordisms
with different surgery necks we obtain cobordant (embedded) Lagrangians that
need not be Hamiltonian isotopic. This shows that Lagrangian cobordism is a
strictly coarser equivalence relation than Hamiltonian isotopy.

It follows from 9 that (embedded) Lagrangian cobordisms, without extra
decorations, are topological. Namely, there is an $h$-principle for Lagrangian
immersions, showing they are governed by algebraic topology. One can then
perturb the Lagrangian immersion by a Hamiltonian isotopy to make all the
intersections transverse, and applying the surgery cobordism yields an
embedded Lagrangian cobordism. However, once we equip Lagrangians and
cobordisms with extra data (see Section 1.1) some rigidity appears. This was
first noticed by Chekanov [Che97], who showed that monotone Lagrangian
cobordisms are no longer flexible. Monotone Lagrangians are precisely the
Lagrangians considered in the original work of Biran-Cornea [BC13, BC14],
where they show that monotonicity imposes enough rigidity on Lagrangian
cobordisms to preserve Floer theory (meaning that monotone Lagrangian
cobordant Lagrangians are isomorphic objects in the Fukaya category).

A generalization of the above surgery cobordism to non-transverse
intersections has been constructed by Hicks [Hic19, Hic20]. He shows that if
$L_{0}$ and $L_{1}$ are Lagrangians whose intersection can be modeled on the
intersection of the zero section and the graph of a convex function, then
there exists a Lagrangian $L_{0}\\#L_{1}$ agreeing with $L_{0}$ and $L_{1}$
outside a neighborhood of the intersection region; furthermore, there is a
cobordism between $L_{0},L_{1}$ and $L_{0}\\#L_{1}$ [Hic20, Corollary 3.5].
Now assume $X(B)\to B$ comes equipped with a Lagrangian torus fibration. Let
$L_{0}=\Gamma_{0}$ be the zero section and
$L_{1}=\Gamma(d\phi):=\Gamma(d\tilde{\phi})$ be the graph of (a smoothing of)
a _tropical polynomial_ —a piecewise linear function $\phi:B\to\R$ with
integer slope which is furthermore convex ($\tilde{\phi}$ denotes a suitable
smoothing, see [Hic20, Section 3.2]). Then Hicks uses his surgery construction
to show that the projection of the surgery Lagrangian
$\Gamma_{0}\\#\Gamma(d\phi)$ to $B$ lives arbitrarily close to the _tropical
hypersurface_ $V(\phi)$ (the locus of points in $B$ where $\phi$ is not
smooth) [Hic20, Theorem 3.17].

We will use the following instance of Hicks’ construction in our computations
in the next sections. The tropical hypersurface $V(\phi)$ will be a collection
of disjoint, codimension-$1$ tropical submanifolds $V_{1},\dots,V_{k}\subset
B$. Each $V_{i}$ comes with a ‘weight’ $w_{i}\in\Z$ that records the ‘amount
of bending’ of $\phi$ at $V_{i}$. Let $L_{i}=N^{*}V_{i}/N_{\Z}^{*}V_{i}$ be
the Lagrangian lift of $V_{i}$ and write $L_{\phi}:=\sum_{i}w_{i}L_{i}$. A
surgery cobordism shows that $\Gamma_{0}\\#\Gamma(d\phi)\sim L_{\phi}$ in the
cobordism group.444An interesting point is that a cobordism from
$\Gamma_{0}\\#\Gamma(d\phi)$ to $L_{\phi}$ does _not_ exist; rather, there is
a cobordism between $(\Gamma_{0}\\#\Gamma(d\phi),\Gamma_{0})$ and
$(L_{\phi},\Gamma_{0})$. Let us state the precise result we will need for
further reference:

######  Proposition 10.

Consider the Lagrangian torus fibration $X(B)\to B$, and let $\Gamma_{0}$ be
the zero section. Let $\phi$ be a convex piecewise linear function such that
$V(\phi)=V_{1}\sqcup\dots\sqcup V_{k}$ is a collection of disjoint tropical
submanifolds and let $L_{\phi}$ be defined as above. Then the relation

$\Gamma_{0}+L_{\phi}\sim\Gamma(d\phi)$

holds in $\operatorname{\mathrm{Cob}}(X(B))$.

In fact, in our computations the sections $\Gamma(d\phi)$ will be product-
type, meaning the function $\phi$ will only depend on one of the coordinates
$(x,y)\in K$. One could then pull back such a section to $T^{2}\times
T^{2}\to\mathcal{K}$ and surger it with the $0$-section by considering the
product $V\times\Gamma_{0}\subset(T^{2}\times\C)\times T^{2}$ (or
$\Gamma_{0}\times V\subset T^{2}\times(T^{2}\times\C)$) of a standard surgery
cobordism $V\subset T^{2}\times\C$ in one of the $T^{2}$-factors with the
$0$-section $\Gamma_{0}$ in the other factor. This cobordism is invariant
under the covering group of the cover $T^{2}\times
T^{2}\times\C\to\mathcal{K}\times C$, thus descends to $\mathcal{K}\times\C$
and recovers the cobordism relation of 10.

###  3.2 The Lagrangian cobordism group of $T^{2}$

In this section we recall the computation of the Lagrangian cobordism group of
$T^{2}$ whose generators are oriented embedded non-contractible Lagrangians
and whose relations come from oriented embedded cobordisms. This will not just
exhibit the use of Hamiltonian isotopies and Lagrangian surgeries to compute
cobordism groups, but in fact will be essential in our computation of the
cobordism group of a bielliptic surface in Section 4. The computation of
$\operatorname{\mathrm{Cob}}(T^{2})$ was first done in [Hau15] using
homological mirror symmetry. We will present a computation that only requires
standard symplectic topology arguments.

Denote by $\operatorname{\mathrm{Cob}}(T^{2})_{hom}$ the subgroup of
$\operatorname{\mathrm{Cob}}(T^{2})$ generated by homologically trivial formal
sums of Lagrangians (this is the kernel of the cycle-class map
$\operatorname{\mathrm{Cob}}(T^{2})\to H_{1}(T^{2};\Z)$, which exists because
all Lagrangians and cobordisms are oriented—cf. Section 4.4). The following
Lemma is an application of Stokes’ theorem.

######  Lemma 11.

There is a well-defined map
$\Phi:\operatorname{\mathrm{Cob}}(T^{2})_{hom}\to\R/(\Z\,area(T^{2}))\cong
S^{1}$ sending a homologically trivial collection of Lagrangians
$L_{1}+\dots+L_{k}$ to the area $\omega(u)$ of a topological 2-chain with
$\partial u=L_{1}+\dots+L_{k}$.

We now state the main cobordism relations in $T^{2}$, which will be used to
compute $\operatorname{\mathrm{Cob}}(T^{2})$ and will also be useful later in
Section 4.

######  Lemma 12.

The following relations hold in $\operatorname{\mathrm{Cob}}(T^{2})$:

  * •

For any $a,b,\theta\in\R/\Z$, we have

(7) $S^{1}\times\\{a\\}-S^{1}\times\\{a+\theta\\}\sim\\{b\\}\times
S^{1}-\\{b+\theta\\}\times S^{1}.$

  * •

For any $a,\theta\in S^{1}$, we have

(8) $S^{1}\times\\{0\\}-S^{1}\times\\{\theta\\}\sim
S^{1}\times\\{a\\}-S^{1}\times\\{a+\theta\\}.$

######  Proof 3.2.

The first relation is equivalent to $S^{1}\times\\{a\\}+\\{b+\theta\\}\times
S^{1}\sim S^{1}\times\\{a+\theta\\}+\\{b\\}\times S^{1}$, or using surgery to
$(S^{1}\times\\{a\\})\\#(\\{b+\theta\\}\times
S^{1})\sim(S^{1}\times\\{a+\theta\\})\\#(\\{b\\}\times S^{1})$. These two
Lagrangians can be easily seen to be Hamiltonian isotopic by flux
considerations. The second relation is an immediate corollary of the first:
both sides are cobordant to $\\{b\\}\times S^{1}-\\{b+\theta\\}\times S^{1}$
for any $b\in S^{1}$.

With this one can then show the following:

######  Proposition 13.

The map $\Phi$ of 11 is an isomorphism.

######  Proof 3.3.

Surjectivity is obvious: for any $\theta\in S^{1}$, we have
$S^{1}\times\\{0\\}-S^{1}\times\\{\theta\\}=\partial(S^{1}\times[0,\theta])$
and $\text{area }(S^{1}\times[0,\theta])=\theta$. To prove injectivity we
proceed as follows. Let
$L_{1}+\dots+L_{k}\in\ker\Phi\subset\operatorname{\mathrm{Cob}}(T^{2})_{hom}$;
we must prove that $L_{1}+\dots+L_{k}=0$ in
$\operatorname{\mathrm{Cob}}(T^{2})$. We first write each $L_{i}$ as an
iterated surgery of $m_{i}$ horizontal lines and $n_{i}$ vertical lines, where
$(m_{i},n_{i})=[L_{i}]\in H_{1}(T^{2};\Z)$ (such surgery yields a Lagrangian
in the same homology class as $L_{i}$, and by sliding one of the horizontal or
vertical Lagrangians we can arrange the surgery to be Hamiltonian isotopic to
$L_{i}$). Hence we may assume that all the $L_{i}$ are either horizontal or
vertical straight lines. If $L^{\pm}_{i_{1}},\dots,L^{\pm}_{i_{l}}$ are
vertical Lagrangians with homology class $[L^{\pm}_{i_{j}}]=(0,\pm 1)$, using
Equation 7 we may turn each pair $L^{\pm}_{i_{j}}$ into a cobordant pair of
horizontal Lagrangians. We have now reduced to the case where we have a
collection of horizontal Lagrangians $L_{1}^{\pm},\dots,L_{n}^{\pm}$ with
homology class $[L_{i}^{\pm}]=(\pm 1,0)$. By Equation 8, for each
$i=2,\dots,n$ we may slide the pair $L_{i}^{\pm}$ so that $L_{i}^{+}$ lies
over $L_{i-1}^{-}$, thus cancelling. Doing this for all $i$ leads the two
Lagrangians $L_{1}^{+}$ and $L_{n}^{-}$, and since they bound area
$k\,area(T^{2})$ for some $k\in\Z$ (they are in the kernel of $\Phi$) they
must lie on top of eachother. Thus all the Lagrangians cancel and we are left
with the zero-sum on $\operatorname{\mathrm{Cob}}(T^{2})$, as wanted.

######  Corollary 14.

The Lagrangian cobordism group of closed, embedded, oriented and non-
contractible Lagrangians of $T^{2}$ is

$\operatorname{\mathrm{Cob}}(T^{2})\cong\Z^{2}\oplus S^{1}.$

######  Proof 3.4.

Consider the short exact sequence

$0\to\operatorname{\mathrm{Cob}}(T^{2})_{hom}\to\operatorname{\mathrm{Cob}}(T^{2})\xrightarrow{cyc}H_{1}(T^{2};\Z)\to
0,$

which splits after a choice of base point $p\in T^{2}$. Using 13 the result
follows.

For Lagrangians equipped with $G$-local systems, there is an extension of
$\Phi$ to a map $\operatorname{\mathrm{Cob}}^{loc}(T^{2})_{hom}\to S^{1}\oplus
G$ that records the product of the monodromies of each of the local systems. A
similar argument to that of 13 shows this map is an isomorphism, thus one
obtains:

######  Corollary 15.

For any abelian group $G$, the Lagrangian cobordism group of closed, embedded,
oriented and non-contractible Lagrangians of $T^{2}$ equipped with $G$-local
systems is

$\operatorname{\mathrm{Cob}}(T^{2})\cong\Z^{2}\oplus(S^{1}\oplus G).$

###  3.3 Extra data

Recall from Section 1.1 that one often equips Lagrangians and cobordisms with
various decorations (this is essential to pass from a flexible to a rigid
notion). In this section we discuss the extra structure that we will consider
on all our Lagrangians: gradings, Pin structures and local systems. The first
two are necessary to define a $\Z$-graded Fukaya category with signs (i.e.
away of fields of characteristic $2$) [Sei08, Fuk+09]. The third decoration is
a $G$-local system on each Lagrangian, where $G$ is some abelian group. When
$G=U_{\Lambda}$ is the unitary group of the Novikov field, it is expected that
this data is needed to have a homological mirror symmetry equivalence between
the Fukaya category (without taking the split closure) and the derived
category of coherent sheaves of its mirror. We will refer to Lagrangians with
this extra structure—that is, a grading, a Pin structure and a $G$-local
system—as Lagrangian $G$-branes (or simply Lagrangian branes). We work with an
arbitrary $G$ because our computation of the cobordism group works in that
generality (see 34). However, to compute the Grothendieck group and show that
it agrees with the cobordism group (35), we need to apply homological mirror
symmetry and thus we specialize to the case $G=U_{\Lambda}$.

We start with gradings. This is some extra data on the Lagrangians that allows
us to define a $\Z$-graded Fukaya category (meaning the morphism spaces are
$\Z$-graded vector spaces). For a complete exposition on gradings see [Sei08,
Section (11j)] or Seidel’s original paper [Sei00].

Let $X^{2n}$ be a symplectic manifold such that $2c_{1}(X)=0$. This means the
complex line bundle $(\Omega^{n})^{\otimes 2}$ of holomorphic quadratic volume
forms is trivial. Let $\eta^{2}:X\to(\Omega^{n})^{\otimes 2}$ be a nowhere
zero section (note this is the same data as a section of $\Omega^{n}$ defined
up to a sign $\pm 1$, which we denote by $\pm\eta$). Given a Lagrangian
submanifold $L\subset X$, there is an associated (squared) phase map
$\alpha_{L}:L\to S^{1}$ given by

$\alpha_{L}(p)=\frac{\eta_{p}^{2}(vol_{L,p}^{\otimes
2})}{|\eta_{p}^{2}(vol_{L,p}^{\otimes 2})|}$

where $vol_{L,p}$ is a generator of $\wedge^{n}T_{p}L$. The cohomology class
$\mu_{L}\in H^{1}(L;\Z)$ determined by $\alpha_{L}$ is called the Maslov class
of $L$. By definition, $\mu_{L}$ is the obstruction to the existence of a lift
$\tilde{\alpha}_{L}:L\to\R$ to the universal cover $\R\to S^{1},t\mapsto
e^{i2\pi t}$.

######  Definition 16.

Let $L\subset X$ be a Lagrangian with $\mu_{L}=0$. A pair
$(L,\tilde{\alpha}_{L})$ is called a graded Lagrangian.

The phase map $\alpha_{L}$ depends on the homotopy class of the trivialization
$\eta^{2}$. Whether a Lagrangian admits a grading or not will therefore also
depend on such choice.

As explained in [Sei08, Remark 11.18], a graded Lagrangian is naturally
oriented with respect to the $\Z$-local system $\xi\subset\wedge^{n}_{\C}TX$
where $\pm\eta$ takes integer values. Hence, every graded Lagrangian defines a
class in the homology group $H_{n}(X;\xi)$.

Next we turn to (twisted) Pin structures, again following [Sei08]. These are
needed to orient the moduli-spaces that define the $A_{\infty}$ operations, so
that one can define a Fukaya category over a field of characteristic different
from two. First recall that there is a group $Pin_{n}$, constructed explicitly
as a subgroup of the Clifford algebra of $\R^{n}$, that admits a $2$-to-$1$
map $Pin_{n}\to O_{n}$. A _Pin structure_ on an $n$-dimensional manifold $M$
is a principal $Pin_{n}$-bundle $P\to M$ together with an isomorphism
$P\times_{Pin_{n}}\R^{n}\cong TM$, where $Pin_{n}$ acts on $\R^{n}$ via the
composition $Pin_{n}\to
O_{n}\hookrightarrow\operatorname*{\mathrm{GL}}(\R^{n})$. The obstruction to
the existence of a Pin structure is the second Stiefel-Whitney class
$w_{2}(M)\in H^{2}(M;\Z_{2})$. More generally, given some class $w\in
H^{2}(M;\Z_{2})$ and an oriented vector bundle $E\to M$ with $w_{2}(E)=w$, one
can define a _twisted Pin structure_ as a Pin structure on $TM\oplus E$. The
obstruction to the existence of a Pin structure is now $w_{2}(E)-w_{2}(M)$.

If $M=X$ is a symplectic manifold and $E\to X$ is an oriented vector bundle,
one can define a Fukaya category whose objects are Lagrangians equipped with a
Pin structure on the bundle $TL\oplus E|_{L}$. This structure is enough to
orient the moduli-spaces defining the $A_{\infty}$-operations, and hence to
define a Fukaya category away of characteristic $2$ [Sei08, Fuk+09].

Lastly, let us recall that, for an (abelian) group $G$, a $G$-local system on
a manifold $M$ is the data of a group homomorphism

$\xi:\pi_{1}(M)\to G.$

Geometrically, one can think of a local system as a principal $G$-bundle with
a _flat_ connection. Namely, the fibers of a $G$-principal bundle give an
assignment of principal homogenous $G$-space $\xi_{p}$ to each $p\in M$. The
data of a flat connection gives, via parallel transport, isomorphisms
$\xi_{\gamma(0)}\cong\xi_{\gamma(1)}$ for every path $\gamma:[0,1]\to M$.
Furthermore, these isomorphisms depend only on the homotopy class of $\gamma$
relative to its endpoints. Considering isomorphisms associated to loops
returns the map $\xi:\pi_{1}(M)\to G$.

Suppose that $\pi:X\to B$ comes with a Lagrangian torus fibration. In this
case, there are natural choices of $\eta^{2}:X\to(\Omega^{n})^{\otimes 2}$ and
$w\in H^{2}(X;\Z_{2})$ to define gradings and twisted Pin structures. Namely,
we choose $\eta^{2}$ to be the complexification of a section of
$(\wedge^{n}T^{*}B)^{\otimes 2}$, and $w=\pi^{*}(w_{2}(B))$ to be the pullback
of the second Stiefel-Whitney class of $B$. In full generality one can
consider $E:=\pi^{*}(TB\oplus(\det B)^{\oplus 3})$ (this choice ensures that
$w_{2}(E)=\pi^{*}w_{2}(B)$ while making $E$ orientable), but when $w=0$ we
will simply take $E=0$ to be the trivial vector bundle. The obstruction to a
twisted Pin structure (a Pin structure on $TL\oplus E|_{L}$) is then
$w_{2}(L)+(\pi^{*}w_{2}(B))|_{L}$. Note these choices ensure that fibers and
sections admit a brane structure. Furthermore, with this choice of $\eta^{2}$
the $\Z$-local system $\xi$ of Section 3.3 is the pullback
$\pi^{*}(\wedge^{n}T^{*}_{\Z}B)$ of the integral orientation bundle of the
base. Hence every graded Lagrangian is oriented with respect to
$\pi^{*}(\wedge^{n}T^{*}_{\Z}B)$.

Summarizing, the Lagrangians we consider will be _Lagrangian $G$-branes_—that
is, they will carry the data of a grading, a twisted Pin structure and a
$G$-local system. A cobordism between such Lagrangians is then a Lagrangian
cobordism $V\subset X\times\C$ carrying also a $G$-brane structure;555The
almost complex structure on $X\times\C$ used to define gradings is chosen to
be of the form $J_{X\times\C}=J_{X}\times j_{\C}$ at infinity, where $j_{\C}$
is the standard complex structure on $\C$. it induces a relation between the
ends equipped with the restricted brane structure. The fact that we can
restrict Pin structures and gradings follows from the product-type structure
of the cobordism and complex structure at infinity, together with the fact
that the ends are all horizontal (thus, the composition of the phase map of
the cobordism with the inclusion of one of its ends coincides with the phase
map of the end).

##  4 The Lagrangian cobordism group of a bielliptic surface

In this section we prove the main results of this paper. The first, see 34, is
a computation of the subgroup
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\subset\operatorname{\mathrm{Cob}}(\mathcal{K})$
generated by Lagrangian sections and fibers of the projection
$\pi:\mathcal{K}\to K$, where $\mathcal{K}$ is a symplectic bielliptic surface
as defined in Section 4.1. Using this computation together with Abouzaid’s
homological mirror symmetry result [Abo21], we show in 35 that
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ is naturally isomorphic to
the Grothendieck group of the Fukaya category.

###  4.1 Algebraic and symplectic bielliptic surfaces

We start by defining the algebraic varieties and symplectic manifolds that we
call bielliptic surfaces. On the algebraic side, we always work over an
algebraically closed field of characteristic zero.

In the Kodaira-Enriques classification of surfaces, a bielliptic surface (also
known as a hyperelliptic surface) is a minimal surface $Y$ characterised by

$\kappa(Y)=0,\qquad q(Y)=1,\qquad p_{g}(Y)=0$

where $\kappa$ denotes the Kodaira dimension, $q(Y)=h^{1}(Y,\mathcal{O}_{Y})$
is the irregularity and $p_{g}(Y)=h^{2}(Y,\mathcal{O}_{Y})$ is the geometric
genus (this condition $p_{g}(Y)=0$ can be omitted in characteristic zero).
Since $q(Y)=1$ equals the dimension of the Albanese variety of $Y$,
$\operatorname{\mathrm{Alb}}(Y)$ is an elliptic curve; moreover, the Albanese
morphism $\operatorname{\mathrm{alb}}:Y\to\operatorname{\mathrm{Alb}}(Y)$ is
an elliptic fibration and all fibers are smooth [BM77]. One can show that
there exists another elliptic fibration $Y\to\mathbb{P}^{1}$ that is
transverse to $\operatorname{\mathrm{alb}}$. This justifies the name
_bielliptic_ , as these surfaces admit two (transverse) elliptic fibrations.
These two fibrations can be seen explicitly, as follows from the following
structure theorem for bielliptic surfaces (see [BM77, Theorem 4, §3]):

######  Theorem/Definition 17.

Let $Y$ be a bielliptic surface. There exist elliptic curves $E_{1},E_{2}$ and
a finite group $G\subset E_{1}$ of translations of $E_{1}$ acting also on
$E_{2}$ such that $Y=(E_{1}\times E_{2})/G$. The quotient $E_{1}/G$ is again
an elliptic curve and $E_{2}/G\cong\mathbb{P}^{1}$.

It follows from Serre duality that $h^{0}(Y,\mathcal{K}_{Y})=0$, so in
particular the canonical bundle $\mathcal{K}_{Y}$ is not trivial; nonetheless,
it can be proved using the canonical bundle formula for elliptic fibrations
and the structure theorem above that it is either two-, three-, four- or six-
torsion.

We remark that the quotient $E_{1}/G$ is again an elliptic curve, and the
projection $Y\to E_{1}/G$ is a fiber bundle (all fibers being smooth elliptic
curves isomorphic to $E_{2}$). In fact, one can identify $E_{1}/G$ (together
with the projection $\pi_{1}:Y\to E_{1}/G$) with the Albanese variety of
$Y$.666This property actually provides a third (equivalent) definition of
bielliptic surfaces: a bielliptic surface is an algebraic surface such that
the Albanese morphism $Y\to\operatorname{\mathrm{Alb}}(Y)$ is an elliptic
fibration. On the other hand, the projection $\pi_{2}:Y\to E_{2}/G$ is an
elliptic fibration over $E_{2}/G\cong\mathbb{P}^{1}$. This fibration is
however not smooth: the generic fiber is isomorphic to $E_{1}$, but over the
branching points of the projection $E_{2}\to E_{2}/G$ we have non-reduced
fibers, their multiplicity being equal to that of the associated branching
point. For more details on the algebraic geometry of bielliptic surfaces we
refer the reader to [BM77].

We now introduce symplectic bielliptic surfaces. In Section 4.2 we explain how
symplectic and algebraic bielliptic surfaces are homologically mirror as an
instance of Abouzaid’s homological mirror symmetry result [Abo14, Abo17,
Abo21]. Recall Abouzaid’s result is proved using family Floer theory, whose
input is a symplectic manifold together with a Lagrangian torus fibration over
some tropical affine manifold. In our case, we take the tropical affine
manifold to be a Klein bottle (see 5) and construct a symplectic manifold
$X(K)$ as explained in Section 2.3:

######  Definition 18.

Let $K$ be a tropical Klein bottle. The total space $\mathcal{K}:=X(K)$ of the
Lagrangian torus fibration $X(K)\to K$ is called a symplectic bielliptic
surface.

The following Lemma shows this definition agrees with the symplectic of 17,
hence the name symplectic bielliptic surface. Namely, we have the following:

######  Lemma 19.

Every symplectic bielliptic surface is the quotient of the product of two
symplectic $2$-tori by the action of a finite group of symplectomorphisms.

######  Proof 4.1.

Recall from 5 that every tropical Klein bottle is the quotient of a tropical
affine torus $T^{2}_{\Lambda}$ by a finite group. The discussion of Section
2.4 shows that there is a tropical morphism $T\to T^{2}_{\Lambda}$ from a
product tropical torus $T\simeq S^{1}\times S^{1}$ onto $T^{2}_{\Lambda}$
which is a diffeomorphism of the underlying smooth manifolds (but might not be
a tropical isomorphism). The composition $T\to T^{2}_{\Lambda}\to K$ is a
local diffeomorphism and also a tropical map, thus it induces a map

$X(T)\simeq X(S^{1})\times X(S^{1})\to\mathcal{K}$

which can be seen to be a projection $X(T)\to X(T)/G$. Hence every symplectic
bielliptic surface can be expressed as a quotient

$\mathcal{K}\simeq((T^{2},\omega_{1})\times(T^{2},\omega_{2}))/G$

of two symplectic $2$-tori by a finite group of symplectomorphisms.

To be more precise, if $(q_{i},p_{i})$ are Arnold-Liouville coordinates on the
Lagrangian torus fibration $(T^{2},\omega_{i})\to S^{1}_{q_{i}}$, then the
group $G$ can either be:

  1. 1.

$G=\Z_{2}$, acting by

$(q_{1},p_{1},q_{2},p_{2})\mapsto(q_{1}+\theta,p_{1},-q_{2},-p_{2})$

and $2\theta=0\neq\theta$; or

  2. 2.

$G=\Z_{2}\oplus\Z_{2}$, where the first generator acts as before and the
second as

$(q_{1},p_{1},q_{2},p_{2})\mapsto(q_{1},p_{1}+\phi_{1},q_{2},p_{2}+\phi_{2}),$

and again $2\phi_{i}=0\neq\phi_{1}\phi_{2}$.

Note that while the first action modifies the topology of the base of the
Lagrangian torus fibration $T^{2}\times T^{2}\to S^{1}\times S^{1}$, the
second only modifies its tropical affine structure.

As mentioned in Section 2.4, these two symplectic bielliptic surfaces
correspond to algebraic bielliptic surfaces of type I and II in the
classification of algebraic bielliptic surfaces. The classification of complex
bielliptic surfaces (see e.g. [Bea96, List VI.20]) includes seven families,
with more possible groups than $G=\Z_{2}$ or $G=\Z_{2}\oplus\Z_{2}$. However,
it is only in two of these families that the complex structure can be deformed
in a maximally degenerating way. Hence, these are the only two that make sense
for mirror symmetry, where to a symplectic manifold one associates a maximal
degeneration of an algebraic-geometric object.

###  4.2 Homological mirror symmetry for bielliptic surfaces

Let $\mathcal{K}$ be a symplectic bielliptic surface. Recall that, by
definition, $\mathcal{K}$ comes equipped with a Lagrangian torus fibration
$\pi:\mathcal{K}\to K$. We observe that the fibration is smooth (i.e. there
are no singular fibers) and that $\pi_{2}(K)=0$. This puts us in the setting
to apply the results in [Abo21], where a (rigid analytic) mirror is
constructed and homological mirror symmetry is proved for such a pair.

Denote by $\check{\mathcal{K}}$ the rigid analytic mirror of $\mathcal{K}$.
Set-wise, $\check{\mathcal{K}}$ is the disjoint union

(9) $\check{\mathcal{K}}:=\bigsqcup_{p\in K}H^{1}(F_{p};U_{\Lambda}),$

where $F_{p}:=\pi^{-1}(p)\subset\mathcal{K}$ is the Lagrangian torus fiber
over $p\in\mathcal{K}$ and $U_{\Lambda}\subset\Lambda^{*}$ is the unitary
subgroup of the Novikov field $\Lambda$. The space $\check{\mathcal{K}}$ can
be endowed with the structure of a rigid analytic space [Abo14]. Examining
Equation 9 and using that a tropical Klein bottle is the quotient of a
tropical $2$-torus (5), one sees that the rigid analytic space
$\check{\mathcal{K}}$ is the quotient of a product
$\mathcal{E}_{1}\times\mathcal{E}_{2}$ of two Tate curves by the action of a
finite group $G$, and each of these Tate curves is the analytification of an
algebraic elliptic curve $E_{i}$ over the Novikov field. Hence we have that

$\check{\mathcal{K}}=(\mathcal{E}_{1}\times\mathcal{E}_{2})/G=(E_{1}^{an}\times
E_{2}^{an})/G=(E_{1}\times E_{2}/G)^{an}$

is the analytification of an algebraic variety $Y:=(E_{1}\times E_{2})/G$.
Note that $Y$ is an algebraic bielliptic surface (see 17). One can then apply
the rigid analytic GAGA principle to obtain an equivalence of categories

$D^{b}Coh_{an}(\check{\mathcal{K}})\cong D^{b}Coh(Y),$

so that combining this with the homological mirror symmetry statement of
[Abo21] we obtain:

######  Proposition 20.

Let $\mathcal{K}$ be a symplectic bielliptic surface and
$D^{\pi}\mathcal{F}uk(\mathcal{K})$ the split-closed derived Fukaya category
of $\mathcal{K}$. Then there is an algebraic bielliptic surface $Y$ over the
Novikov field $\Lambda$ and an equivalence of triangulated categories

$D^{\pi}\mathcal{F}uk(\mathcal{K})\cong D^{b}Coh(Y).$

Strictly speaking, [Abo21] proves there is a fully faithful embedding
$D^{\pi}\mathcal{F}uk(\mathcal{K})\hookrightarrow
D^{b}Coh_{an}(\check{\mathcal{K}})$ of the Fukaya category into the category
of (twisted) coherent sheaves, with no discussion of essential surjectivity.
However, for a smooth quasi-projective variety $Y$ we have that
$D^{b}Coh(Y)\simeq Perf(Y)$ is idempotent-complete. In this case, essential
surjectivity is equivalent to showing that the image of the family Floer
functor split-generates $D^{b}Coh(Y)$. It is a result of Orlov [Orl08] that
sufficiently many powers of any ample line bundle are enough to split-generate
$D^{b}Coh(Y)$, so it suffices to show such an ample line bundle exists in the
image of the family Floer embedding. One can check that the construction in
[Abo14] of the complex of coherent sheaves mirror to a Lagrangian section is
indeed a line bundle, and that when the function defining the section is
strictly convex such line bundle is ample. As $K$ admits a strictly convex
Lagrangian section by 21, we indeed have an equivalence
$D^{\pi}\mathcal{F}uk(\mathcal{K})\simeq D^{b}Coh_{an}(\check{\mathcal{K}})$.
In fact, for symplectic bielliptic surfaces we have that
$D\mathcal{F}uk(\mathcal{K})\cong D^{\pi}\mathcal{F}uk(\mathcal{K})$—see
Section 1.2. Hence we obtain a homological mirror symmetry equivalence
$D\mathcal{F}uk(\mathcal{K})\cong D^{b}Coh(Y)$ with no need of taking split-
closure on the A-side.

###  4.3 Lagrangian sections in bielliptic surfaces

Let $\mathcal{K}$ be a symplectic bielliptic surface (recall this is the total
space of a Lagrangian torus fibration over a Klein bottle). From now on we fix
$K=K_{1}$ to be a tropical Klein bottle of type I (see Section 2.4). Hence
$\mathcal{K}=T^{*}K_{1}/T^{*}_{\Z}K_{1}$ will always be a symplectic
bielliptic surface mirror to an algebraic bielliptic surface of type I (see
Section 2.4). In this section we classify Lagrangian sections and compute
their homology class.

Recall from Section 2.3 that Lagrangian sections in
$\mathcal{K}=T^{*}K/T^{*}_{\Z}K$ are in one-to-one correspondence with
elements in $H^{0}(C^{\infty}_{K}/\operatorname{Aff}_{K})$. Among these,
global sections of $H^{0}(C^{\infty}_{K})$ give Lagrangian sections that are
Hamiltonian isotopic to the zero section. Thus, the group

(10) $H^{1}(\operatorname{Aff})\cong
H^{0}(C^{\infty}/\operatorname{Aff})/H^{0}(C^{\infty})$

parametrizes Lagrangian sections up to Hamiltonian isotopy. We now determine
this group:

######  Lemma 21.

Let $K$ be a tropical Klein bottle. Then
$H^{1}(\operatorname{Aff})\cong\Z^{2}\oplus\Z_{2}\oplus S^{1}$, with an
element $(m,n,[l],e^{2\pi i\theta})$ represented by the section
$f_{m\theta}^{nl}\in H^{0}(C^{\infty}_{K}/\operatorname{Aff}_{K})$ defined by

(11)
$f_{m\theta}^{nl}(x,y):=\frac{m}{2}x^{2}+\frac{n}{2}y^{2}+\frac{l}{2}y+\theta
x.$

######  Proof 4.2.

Take an element in $H^{0}(C^{\infty}_{K}/\operatorname{Aff}_{K})$. Pulling it
back to $H^{0}(C^{\infty}_{\R^{2}}/\operatorname{Aff}_{\R^{2}})$ and lifting
it to a smooth function, we are looking at functions $f\in
H^{0}(C^{\infty}_{\R^{2}})$ satisfying the periodicity conditions

(12) $\displaystyle f(x,y)-f(x+1,-y)$ $\displaystyle=g_{1}(x,y)$ (13)
$\displaystyle f(x,y)-f(x,y+1)$ $\displaystyle=g_{2}(x,y)$

for some affine functions $g_{1},g_{2}\in\operatorname{Aff}(\R^{2})$. Now note
that

$g_{1}(x,y)+g_{2}(x+1,-y)=f(x,y)-f(x+1,-y+1)=g_{1}(x+1,y-1)-g_{2}(x,y-1),$

and this compatibility guarantees the existence of a quadratic function
$f_{q}$ with the same periodicity as $f$. In particular, the difference
$f-f_{q}$ descends to a well-defined smooth function on $K$, hence $f$ and
$f_{q}$ define the same class in $H^{1}(\operatorname{Aff})$. Thus, we may
assume that the element in $H^{0}(C^{\infty}_{K}/\operatorname{Aff}_{K})$ is
given by a quadratic polynomial $f$, and similarly that it vanishes at a
preferred point $0\in K$. Then we want to find quadratic polynomials

$f(x,y)=\sum_{\begin{subarray}{c}i,j=0\\\ 0<i+j\leq
2\end{subarray}}^{2}a_{ij}x^{i}y^{j}$

that satisfy the periodicity conditions (12) and (13). These imply
$2a_{20},2a_{02},2a_{01}\in\Z$ and $a_{11}=0$. Since adding an affine function
does not change the sections in
$H^{0}(C^{\infty}_{K}/\operatorname{Aff}_{K})$, we mod out by
$(a_{01},a_{20})\in\Z^{2}$. This proves that
$H^{1}(\operatorname{Aff})\cong\Z^{2}\oplus\Z_{2}\oplus S^{1}$, with an
element $(m,n,[l],e^{2\pi i\theta})$ represented by the function
$f_{m\theta}^{nl}$ of Equation 11.

We will now show that $f_{m\theta}^{n[l]}$ can be approximated by (a smoothing
of) a piecewise-linear function with integer slope, in the sense that they
define the same class in $H^{1}(\operatorname{Aff})$. We first construct the
approximation and then prove in 22 that it defines the same class as
$f_{m\theta}^{n[l]}$.

Let $\mathfrak{f}$ and $\mathfrak{f}_{\phi}$ be the unique (up to a constant)
piecewise linear functions with integer slope $\R\to\R$ such that:

  * •

if $x$ is a smooth point of $\mathfrak{f}$, then
$\mathfrak{f}^{\prime}(x+1)=\mathfrak{f}^{\prime}(x)+1$, and
$\mathfrak{f}^{\prime}(x)=0$ for $x\in(-\frac{1}{2},\frac{1}{2})$;

  * •

$\mathfrak{f}_{\phi}(x+1)=\mathfrak{f}_{\phi}(x)+\phi$,
$\mathfrak{f}_{\phi}^{\prime}(x)=0$ for $x\in(0,\phi)$ and
$\mathfrak{f}_{\phi}^{\prime}(x)=1$ for $x\in(1-\phi,1)$.

From these we define a new piecewise linear function
$\mathfrak{f}^{nl}_{m\theta}:\R^{2}\to\R$ by

(14)
$\mathfrak{f}^{nl}_{m\theta}(x,y):=(m\mathfrak{f}+\mathfrak{f}_{\theta})(x)+(n\mathfrak{f}+\mathfrak{f}_{l})(y).$

It follows from the definition of $\mathfrak{f}$ and $\mathfrak{f}_{\phi}$
that $\mathfrak{f}^{nl}_{m\theta}$ satisfies periodicity equations as in
(12)-(13). Hence—after a suitable smoothing as in [Hic20, Section 3.2]—it
gives a well-defined element in
$H^{0}(C^{\infty}_{K}/\operatorname{Aff}_{K})$.

######  Lemma 22.

The functions $f^{nl}_{m\theta}$ and $\mathfrak{f}^{nl}_{m\theta}$ have the
same class in $H^{1}(\operatorname{Aff})$.

######  Proof 4.3.

Recall that $H^{1}(\operatorname{Aff})\cong
H^{0}(C^{\infty}/\operatorname{Aff})/H^{0}(C^{\infty})$, so we must show that
the difference $f^{nl}_{m\theta}-\mathfrak{f}^{nl}_{m\theta}$ gives a well-
defined smooth function on $K$. This is equivalent to showing that the
periodicity functions $g_{i}$ of $f^{nl}_{m\theta}$ are the same as those of
$\mathfrak{f}^{nl}_{m\theta}$. Note that
$f^{nl}_{m\theta}(x,y)=h_{m\theta}(x)+g^{nl}(y)$ decomposes as a sum of one-
variable functions, and that the periodicity conditions (12)-(13) split for
each summand. Therefore, it is sufficient to prove that for
$\mathfrak{f}_{m\phi}:=m\mathfrak{f}+\mathfrak{f}_{\phi}$ and
$h_{m\phi}(x):=\frac{m}{2}x^{2}+\phi x$ one has

(15) $\displaystyle\mathfrak{f}_{m\phi}(x)-\mathfrak{f}_{m\phi}(x+1)$
$\displaystyle=h_{m\phi}(x)-h_{m\phi}(x+1)$ (16)
$\displaystyle\mathfrak{f}_{m\phi}(x)-\mathfrak{f}_{m\phi}(-x)$
$\displaystyle=h_{m\phi}(x)-h_{m\phi}(-x)\quad\text{(if }\phi=1/2).$

Since the function $\mathfrak{f}$ (resp. $\mathfrak{f}_{\phi}$) satisfies
Equations 15 and 16 whenever $m=1,l=0$ (resp. $m=0,l=1$), the result follows
by linearity.

######  Corollary 23.

The Lagrangian sections $\Gamma(df_{m\theta}^{nl})$ and
$\Gamma(d\mathfrak{f}^{nl}_{m\theta})$ are Hamiltonian isotopic.

Next we compute the homology class of an arbitrary section (recall from
Section 3.3 and Section 3.3 that we take homology with coefficients in the
$\Z$-local system $\xi=\pi^{*}\det_{\Z}K$, where
$\det_{\Z}K=\wedge^{2}T_{\Z}K$ is the integral orientation bundle). For this
we first prove an important cobordism relation, which will also be useful
later on. In the remainder of this section we work with undecorated
cobordisms, as these are enough to give homology relations. Nonetheless, we
show in Section 4.9 that the cobordisms we consider admit brane structures and
work out the restriction of the brane structures to the ends. This will be
essential for the results in 30.

Write $f_{m\theta}:=\mathfrak{f}_{m\theta}^{00}$ and
$f^{nl}:=\mathfrak{f}_{00}^{nl}$ for (smoothings of) the functions defined in
Equation 14. Writing $\Gamma_{0}$ for the zero-section and $L_{f_{m\theta}}$
for the Lagrangian lift of the non-smooth locus of $f_{m\theta}$, we can use
10 to obtain a cobordism relation777In 10 the functions defining the
Lagrangian sections are assumed to be convex. The functions $f_{m\theta}$ we
consider are only convex when $m\geq 0$. However, it is easy to see that the
claimed cobordism relation still holds if we revert the orientation of
$L_{f_{m\theta}}$ when $m<0$.

(17) $\Gamma(df_{m\theta})\sim L_{f_{m\theta}}+\Gamma_{0}$

and similarly for $\Gamma^{n[l]}$. Now consider
$\Gamma_{m\theta}^{n[l]}=\Gamma(df_{m\theta}^{nl})$. We write $\otimes$ for
the fiber-wise addition operation defined in [Sub10] (see Section 4.9 for more
details on this operation). Note that the operation $\otimes$ has the property

$\Gamma(d\varphi_{1})\otimes\Gamma(d\varphi_{2})=\Gamma(d(\varphi_{1}+\varphi_{2}))$

for $\varphi_{i}\in H^{0}(C^{\infty}/\operatorname{Aff})$. In particular,
writing $\Gamma_{m\theta}:=\Gamma_{m\theta}^{0[0]}$ and
$\Gamma^{n[l]}:=\Gamma_{00}^{n[l]}$, we can decompose
$\Gamma_{m\theta}^{n[l]}$ as a product

(18) $\Gamma_{m\theta}^{n[l]}=\Gamma_{m\theta}\otimes\Gamma^{n[l]}.$

######  Lemma 24.

There is a cobordism relation $\Gamma_{m\theta}^{n[l]}\sim
L_{f_{m\theta}}\otimes L_{f^{nl}}+L_{f_{m\theta}}+L_{f^{nl}}+\Gamma_{0}$.

######  Proof 4.4.

Recalling that $\Gamma_{m\theta}$ (resp. $\Gamma^{n[l]}$) is Hamiltonian
isotopic to $\Gamma(df_{m\theta})$ (resp. $\Gamma(df^{nl})$), we can combine
Equation 18 with Equation 17 to get cobordism relations

$\displaystyle\begin{split}\Gamma_{m\theta}^{n[l]}&=\Gamma_{m\theta}\otimes\Gamma^{n[l]}\\\
&\sim(L_{f_{m\theta}}+\Gamma_{0})\otimes(L_{f^{nl}}+\Gamma_{0})\\\
&=L_{f_{m\theta}}\otimes
L_{f^{nl}}+L_{f_{m\theta}}+L_{f^{nl}}+\Gamma_{0}.\end{split}$

We are using implicitly that Lagrangian correspondences induce homomorphisms
between Lagrangian cobordism groups [HH22, Appendix C]. In full generality,
this is only true if either all the Lagrangians are undecorated (so that one
can apply surgeries to resolve possible self-intersections arising from
geometric composition, see Section 3.1) or if one is content with working with
immersed cobordisms. However, homomorphisms of cobordism groups obtained by
fiberwise summing with a Lagrangian section are actually induced by global
symplectomorphisms, thus giving well-defined group homomorphisms even at the
embedded level. A detailed study of the transformation of the brane structures
under fiberwise addition can be found in Section 4.9.

We can now compute the homology class of the sections
$\Gamma^{n[l]}_{m\theta}$. In Appendix A we show that
$H_{2}(\mathcal{K};\xi)\cong\Z^{4}\oplus\Z_{2}$ and give explicit generators.
Let us recall what these are. Consider the tropical submanifolds

(19) $\\{0\\},K,C_{1}=\\{x=0\\},C_{2}=\\{y=1/2\\},C_{3}=\\{y=0\\}$

inside $K$. As in Section 2.3, for a tropical submanifold $V\subset K$ we
denote by $L_{V}=N^{*}V/N^{*}_{\Z}V$ the Lagrangian in $\mathcal{K}$ living
over $V$. Then the $\Z$-factors in $H_{2}(\mathcal{K};\xi)$ are generated by
$L_{\\{0\\}}$ (a fiber), $L_{K}$ (the zero-section) and the Lagrangians
$L_{C_{1}}$ and $L_{C_{2}}$, whereas the $\Z_{2}$-factor is generated by the
difference $L_{C_{2}}-L_{C_{3}}$. Using this, we prove the following:

######  Corollary 25.

The homology class of $\Gamma_{m\theta}^{n[l]}$ is

$[\Gamma_{m\theta}^{n[l]}]=(mn,1,m,n,[l])\in\Z^{4}\oplus\Z_{2}\cong
H_{2}(\mathcal{K};\xi).$

######  Proof 4.5.

Using 24 it is enough to compute the homology class of the terms appearing on
the right-hand side. By construction of $f_{m\theta}$, the tropical
submanifold $V(f_{m\theta})$—the locus where $f_{m\theta}$ is not smooth, each
component equipped with a multiplicity as explained before 10—is homologous to
$m$ copies of $C_{1}$, whereas $V(f^{nl})$ is a combination of $n$ copies of
$C_{2}$ and $l$ copies of the difference $C_{2}-C_{3}$. In particular, the
term $L_{f_{m\theta}}\otimes L_{f^{nl}}$ in 24—which lives over the
intersection $V(f_{m\theta})\cap V(f^{nl})$—is a collection of $m\cdot n$
fibers. It then follows that

$\displaystyle[\Gamma_{m\theta}^{n[l]}]$
$\displaystyle=(mn,0,0,0,[0])+(0,0,m,0,[0])+(0,0,0,n,[l])+(0,1,0,0,[0])$
$\displaystyle=(mn,1,m,n,[l]),$

which proves the Corollary.

###  4.4 The cycle class map

We start by defining the precise cobordism group that we will consider
(namely, the classes $\mathcal{L}$ and $\mathcal{L}_{cob}$ in 1). Given a
symplectic manifold $X$, we say a Lagrangian $L\subset X$ is weakly exact if
$\omega(\pi_{2}(\mathcal{K},L))=0$, and we say it is tautologically
unobstructed or quasi-exact if there exists some almost complex structure $J$
on $X$ such that $L$ bounds no non-constant $J$-holomorphic disks. We denote
by $(L,\alpha,P,\eta)$ the data of a Lagrangian $G$-brane, where
$L\subset\mathcal{K}$ is a Lagrangian, $\alpha:L\to\R$ its grading, $P\to L$ a
principal $Pin_{2}$-bundle giving the Pin structure and $\eta:\pi_{1}(L)\to G$
a local system. Here, gradings and twisted Pin structures are defined with
respect to the natural choices explained in Section 3.3; that is, we grade
Lagrangians with respect to a quadratic volume form given by complexifying a
section of $(\wedge^{2}T^{*}K)^{\otimes 2}$, and a twisted Pin structure is in
this case a regular Pin structure (as $w_{2}(K)=0$). The same notation is used
for cobordisms.

######  Definition 26.

Let $\mathcal{L}=\\{(L,\alpha,P,\eta)\\}$ be the collection of weakly-exact
Lagrangian $G$-brane, and $\mathcal{L}_{cob}=\\{(V,\alpha,P,\eta)\\}$ be the
collection of Lagrangian cobordisms $V\subset\mathcal{K}\times\C$ carrying a
$G$-brane structure, all of whose ends are weakly-exact, and such that their
pullback to the cover $T^{4}\times\C\to\mathcal{K}\times\C$ is tautologically
unobstructed. The Lagrangian cobordism group of $\mathcal{K}$ is the group

$\operatorname{\mathrm{Cob}}(\mathcal{K}):=\operatorname{\mathrm{Cob}}(\mathcal{K};\mathcal{L},\mathcal{L}_{cob})/\sim,$

where $\sim$ is the equivalence relation generated by expressions of the form

$(L,\alpha,P,\eta)\sim(L,\alpha,P\otimes\beta,\eta\otimes\beta)$

for any $\Z_{2}$-local system $\beta$ on $L$.

The weaker condition that the pullback to $T^{4}\times\C$ (instead of the
cobordism itself) is tautologically unobstructed is explained further in
Section 4.8. Let us briefly justify the additional relation imposed in 26.
Given any Lagrangian brane $(L,\alpha,P,\eta)$, one can tensor both the
$Pin_{2}$-bundle $P$ and the local system with a $\Z_{2}$-local system $\beta$
to obtain a new Lagrangian brane $(L,\alpha,P\otimes\beta,\eta\otimes\beta)$.
These Lagrangian branes are naturally isomorphic in the Fukaya category,
whereas there is no reason to expect they should be related by a Lagrangian
cobordism. Hence, to hope for an isomorphism between the cobordism group and
the Grothendieck group of the Fukaya category, we impose this additional
relation in the cobordism group.

We now introduce the first invariant of cobordism classes. In Section 3.3 and
Section 3.3 we argued that every graded Lagrangian in $\mathcal{K}$ is
oriented with respect to the local system $\pi^{*}(\det_{\Z}K)$. Since
cobordisms are also graded and compatible with restriction of gradings, there
is a natural _cycle class map_

$cyc:\operatorname{\mathrm{Cob}}(\mathcal{K})\to H_{2}(\mathcal{K};\xi)$

that assigns to each Lagrangian its twisted fundamental class. This map is
compatible with the _open-closed_ map
$\mathcal{OC}:HH_{0}(\mathcal{F}uk(\mathcal{K}))\to
H_{2}(\mathcal{K};\xi)$,888This version of the open-closed map (mapping to
twisted homology) has not appeared in the literature before. It is not needed
for our proofs and we just include it for context. in the sense that there is
a commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
19.6667pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathrm{Cob}}(\mathcal{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
43.6667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
70.59143pt\raise-25.18748pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{cyc}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
139.70995pt\raise-34.75381pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
43.6667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K_{0}(\mathcal{F}uk(\mathcal{K}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
111.51428pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{\mathcal{T}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 126.78166pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
126.78166pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{HH_{0}(\mathcal{F}uk(\mathcal{K}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
160.85994pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{\mathcal{OC}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 160.85994pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
70.22418pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
139.70995pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H_{2}(\mathcal{K};\xi)}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Here $\mathcal{T}$ denotes the Dennis trace map, which exists for any
triangulated category and sends an object to its identity endomorphism, and
the map $\operatorname{\mathrm{Cob}}(\mathcal{K})\to
K_{0}(\mathcal{F}uk(\mathcal{K}))$ is a Biran-Cornea type map as in Equation 1
which is further discussed in Section 4.8.

###  4.5 Tropical cobordism group and the refined cycle class map

The computation in Appendix A shows that
$H_{2}(\mathcal{K};\xi)\cong\Z^{4}\oplus\Z_{2}$ and reflects an interesting
phenomenon. We showed in Section 4.2 that $\mathcal{K}$ is mirror to an
algebraic bielliptic surface over the Novikov field. The (co)homology of
bielliptic surfaces has been studied (see for instance [Ser90] for the complex
case), and one finds it does not match our computation of
$H_{2}(\mathcal{K};\xi)$ (it does up to torsion). Namely, for an algebraic
bielliptic surface of type I one finds
$H^{\text{even}}(Y)\cong\Z^{4}\oplus\Z_{2}^{2}$—an extra $\Z_{2}$ factor
appearing with respect to $H_{2}(\mathcal{K};\xi)$. Moreover, in [Ber+19] the
authors exhibit explicit generators for the torsion part of
$H^{\text{even}}(Y)$, and these happen to be (linear combinations of) lifts of
the tropical divisors $\left\\{{y=0}\right\\}$ and $\left\\{{y=1/2}\right\\}$.
By mirror symmetry, these torsion divisors correspond to Lagrangian lifts
equipped with torsion local systems. These Lagrangians have the same
underlying Lagrangian submanifold, hence are not distinguished by the naive
map $\operatorname{\mathrm{Cob}}(\mathcal{K})\to H_{2}(\mathcal{K};\xi)$. We
will therefore use a refined cycle class map that distinguishes these two
Lagrangians, just as the mirror cycle class map distinguishes the
corresponding divisors.

Although it is possible to define a refinement of $cyc$ on the whole cobordism
group, for convenience we will consider a subgroup
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\subset\operatorname{\mathrm{Cob}}(\mathcal{K})$
and define the refinement only on this subgroup. Note that, after choosing a
Pin structure on $K$, Lagrangian fibers and sections of the projection
$\mathcal{K}\to K$ come equipped with a preferred Pin structure: for a section
$\Gamma$ there is a natural diffeomorphism $\Gamma\cong K$ given by the
projection, whereas the tangent space to the fibers is naturally isomorphic to
the (co)tangent space of $K$. Furthermore, fibers and sections are weakly
exact, since their pullback to the universal cover $\R^{2}\to\mathcal{K}$ are
exact Lagrangians.999In fact, for $L$ a fiber or a section we get an even
stronger property, sometimes referred to as _topological unobstructedness_ :
the relative homotopy groups $\pi_{2}(\mathcal{K},L)$ vanish. To summarize,
fibers and sections are part of $\mathcal{L}$, and they come with a preferred
Pin structure.

######  Definition 27.

The _tropical cobordism group_ is the subgroup
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\subset\operatorname{\mathrm{Cob}}(\mathcal{K})$
generated by Lagrangian fibers and Lagrangian sections of $\pi:\mathcal{K}\to
K$ equipped with the preferred Pin structure.

We now introduce the following construction to refine the cycle class map. Let
$\iota:L\hookrightarrow\mathcal{K}$ be a Lagrangian brane (in particular, $L$
is oriented with respect to $\xi=\pi^{*}(\wedge^{2}T_{\Z}T^{*}K)$ by Section
3.3). Recall that the twisted version of Poincaré duality gives an isomorphism
$H_{i}(M;\zeta)\cong H^{n-i}(M;\zeta^{\vee}\otimes or_{M})$ for any local
system $\zeta$ on a manifold $M$. We consider the composition

$H_{3}(\mathcal{K};\xi)\cong
H^{1}(\mathcal{K};\xi^{\vee})\xrightarrow{i^{*}}H^{1}(L;\xi^{\vee})\cong
H^{1}(L;or_{L}),$

where in the first isomorphism we have used that $or_{\mathcal{K}}\cong\Z$ (as
for any symplectic manifold), the second isomorphism follows from $L$ being
$\xi$-oriented and the third isomorphism is twisted Poincaré duality for $L$.
The above composition gives an assignment

$\displaystyle H_{3}(\mathcal{K};\xi)$ $\displaystyle\to H_{1}(L;\Z)$
$\displaystyle C$ $\displaystyle\mapsto C_{L}.$

Now suppose $L$ carries a $G$-local system, i.e. a homomorphism
$\eta:\pi_{1}(L)\to G$.

######  Lemma 28.

There is a well-defined pairing

(20)
$\displaystyle\begin{split}\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\otimes
H_{3}(\mathcal{K};\xi)&\to G\\\ (L,\eta)\otimes
C&\mapsto\eta(C_{L})\end{split}$

where $\eta(C_{L})$ is the product of the monodromies of the local system
$\eta$ around the connected components of the $1$-cycle $C_{L}\subset L$.

######  Proof 4.6.

We show that the map does not depend on the cobordism class. First note that
the extra relation imposed in 26 is trivial on the subgroup
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$, so that compatibility with
this relation is immediate. Let $(L_{i},\eta_{i}),i=1,\dots,k$ be the ends of
a cobordism $(V,\eta_{V})$ and let $C\in H_{3}(\mathcal{K};\xi)$. Consider the
transverse intersection $(C\times\C)\pitchfork V\subset\mathcal{K}\times\C$.
This is a $2$-cycle with boundary $C_{L_{1}}\cup\dots\cup C_{L_{k}}$, and we
equip it with a local system given by restricting $\eta_{V}$. The product of
the monodromies along the boundary of this cycle must vanish, and since
$\eta_{V}|_{L_{i}}=\eta_{i}$ we get

$1=\prod_{i}\eta_{V}(C_{L_{i}})=\prod_{i}\eta_{i}(C_{L_{i}}).$

This proves that the map (20) does not depend on a representative of the
cobordism class, hence it is well-defined.

Denote by $H_{3}(\mathcal{K};\xi)_{(2)}\subset H_{3}(\mathcal{K};\xi)$ the
subgroup of $2$-torsion elements. A similar computation to that of Appendix A
shows that $H_{3}(\mathcal{K};\xi)_{(2)}\cong\Z_{2}$. Restricting the map (20)
to $H_{3}(\mathcal{K},\xi)_{(2)}$, we obtain a map

(21)
$\Psi:\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\to\Hom(H_{3}(\mathcal{K};\xi)_{(2)},G)\cong
G_{(2)}.$

######  Definition 29.

We call the map

$\widetilde{cyc}=cyc\oplus\Psi:\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\to
H_{2}(\mathcal{K};\xi)\oplus G_{(2)}$

the refined cycle class map.

Intersecting Lagrangians with other classes in $H_{3}(\mathcal{K};\xi)$
records monodromies of the local systems around different $1$-cycles. These
will be detected by an Albanese-type map defined in Section 4.6, hence we
forget them here. Denote by
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\subset\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$
the kernel of $\widetilde{cyc}$. It consists of the following Lagrangians:

######  Proposition 30.

The subgroup
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\subset\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$
is generated by Lagrangian branes supported on the differences
$F_{b^{+}}-F_{b^{-}}$ and $L_{f_{m^{+}\theta^{+}}}-L_{f_{m^{-}\theta^{-}}}$.
Here, all Lagrangians are equipped with the preferred Pin structure.101010We
already argued that fibers and sections have a preferred Pin structure. Given
a cobordism between $\Gamma_{0},\Gamma_{m\theta}$ and $L_{f_{m\theta}}$
together with a Pin structure extending the preferred one on $\Gamma_{0}$ and
$\Gamma_{m\theta}$, one can simply define the preferred Pin structure on
$L_{f_{m\theta}}$ to be that induced by the cobordism. The fact that the
surgery cobordism admits a unique Pin structure restricting to the preferred
one on $\Gamma_{0}$ and $\Gamma_{m\theta}$ is proved in 40.

######  Proof 4.7.

We first analyze the kernel of
$cyc:\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\to
H_{2}(\mathcal{K};\xi)$. By 25, a general element of $\ker(cyc)$ will be a
finite sum of the form

(22)
$\sum_{i=1}^{N^{+}}F_{b^{+}_{i}}-\sum_{i=1}^{N^{-}}F_{b^{-}_{i}}+\sum_{(m_{i}^{\pm},n_{i}^{\pm},l_{i}^{\pm}\theta_{i}^{\pm})}(\Gamma_{m_{i}^{+}\theta_{i}^{+}}^{n_{i}^{+}[l_{i}^{+}]}-\Gamma_{m_{i}^{-}\theta_{i}^{-}}^{n_{i}^{-}[l_{i}^{-}]})$

for some $b_{i}^{\pm}\in K$ and
$(m_{i}^{\pm},n_{i}^{\pm},l_{i}^{\pm},\theta_{i}^{\pm})\in\Z^{2}\oplus\Z_{2}\oplus
S^{1}$ satisfying:

(23) $\displaystyle\sum(m_{i}^{+}n_{i}^{+}-m_{i}^{-}n_{i}^{-})$
$\displaystyle=N^{+}-N^{-}.$ (24) $\displaystyle\sum(m_{i}^{+}-m_{i}^{-})$
$\displaystyle=0$ (25) $\displaystyle\sum(n_{i}^{+}-n_{i}^{-})$
$\displaystyle=0$ (26) $\displaystyle\sum(l_{i}^{+}-l_{i}^{-})$
$\displaystyle=0\quad(\text{mod }2)$

We want to apply cobordism relations as those in 24. However, recall that this
Lemma was proved for undecorated cobordisms (as we were only interested in the
homology classes of the underlying Lagrangians), whereas relations in
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ involve cobordisms carrying
$G$-brane structures and whose pullback to $T^{4}\times\C$ is tautologically
unobstructed (cf. 26). Once we equip our Lagrangians with this extra
structure—and fix it on the end $\Gamma^{n[l]}_{m\theta}$—there might be some
obstruction to the existence of a Lagrangian cobordism (recall the Lagrangian
cobordism itself must carry such extra structure, and that the restriction to
the ends of the cobordism must agree with the extra structure specified on the
Lagrangians). The key to prove 30 will then be to show that, given a grading,
the preferred Pin structure and a local system on $\Gamma^{n[l]}_{m\theta}$,
the cobordism relations of 24 hold in
$\operatorname{\mathrm{Cob}}(\mathcal{K})$.

In Section 4.9 we show that 24 can be enhanced to take into account this extra
structure. Namely, Lemmas 37-38 show that, given any grading on
$\Gamma^{n[l]}_{m\theta}$, there are gradings on $\Gamma_{m\theta}$ and
$\Gamma^{n[l]}$ such that Equation 18 holds when we put the preferred Pin
structure on all three sections, whereas 43 shows that the surgery cobordism
of Equation 17 is tautologically unobstructed when pulled back to
$T^{4}\times\C$ and can be equipped with a $G$-brane structure restricting
appropriately to the ends. The upshot is that, given a $G$-brane structure on
$\Gamma^{n[l]}_{m\theta}$, 24 holds when we equip all the Lagrangians in the
right-hand side with their natural Pin structures. Furthermore, one can choose
the local system on $L_{f^{nl}}$ to be trivial (see 41).

With this in mind, one can conclude as follows. Apply the cobordism relation
of 24 to all the sections in Equation 22, and note that, since the local
system and Pin structure on the Lagrangians $L_{f^{n[l]}}$ are all the same,
Equation 26 translates into an algebraic cancellation of these Lagrangians.
Hence any null-homologous combination as in Equation 22 consists of fibers,
tropical Lagrangians $L_{f_{m\theta}}$ plus a cobordism class of the form

$\mathbb{L}=\sum_{i}((\Gamma_{0},\eta^{+}_{i\Z_{2}})-(\Gamma_{0},\eta^{-}_{i\Z_{2}})).$

Now note that $\ker(\widetilde{cyc})=\ker(cyc)\cap\ker(\Psi)$, where $\Psi$ is
the map defined in Equation 21. We have that $\Psi(\mathbb{L})=0$ if and only
if $\prod_{i}\eta^{+}_{i\Z_{2}}=\prod_{i}\eta^{-}_{i\Z_{2}}$. Under this
condition, one can iteratively surger copies of $\Gamma_{0}$ with $L_{f^{10}}$
to obtain a Lagrangian $L:=\Gamma_{0}\\#\dots\\#\Gamma_{0}\\#L_{f^{10}}$ and a
relation

$\displaystyle\sum_{i}(\Gamma_{0},\eta^{+}_{i\Z_{2}})+(L_{f^{10}},\mathbf{1})$
$\displaystyle\sim(L,\prod_{i}\eta^{+}_{i\Z_{2}})$
$\displaystyle=(L,\prod_{i}\eta^{-}_{i\Z_{2}})$
$\displaystyle\sim\sum_{i}(\Gamma_{0},\eta^{-}_{i\Z_{2}})+(L_{f^{10}},\mathbf{1}).$

This is equivalent to $\mathbb{L}\sim 0$, which concludes the proof.

###  4.6 Symplectic Albanese map

We will need one more tool to prove Theorem A, namely a modification of an
Abel-Jacobi type map introduced by Sheridan-Smith in [SS21, Lemma 2.10]. They
define, for any tropical affine torus $B$, a map

(27)
$\displaystyle\begin{split}\operatorname{\mathrm{alb}}_{fib}:\operatorname{\mathrm{Cob}}_{fib}(X(B))_{\hom}&\to\operatorname{\mathrm{Alb}}(B)\\\
F_{b^{+}}-F_{b^{-}}&\mapsto\operatorname{\mathrm{alb}}(b^{+}-b^{-})\end{split}$

by looking at the flux swept by a homologically trivial collection of fibers.
Here, $\operatorname{\mathrm{alb}}$ is the map of Equation 3;
$\operatorname{\mathrm{Cob}}_{fib}(X(B))$ is a cobordism group generated by
fibers of the projection $X(B)\to B$ and whose relations come from cobordism
all of whose ends are fibers; and
$\operatorname{\mathrm{Cob}}_{fib}(X)_{\hom}$ denotes the kernel of the degree
map $\operatorname{\mathrm{Cob}}_{fib}(X)\to\Z$. The following Proposition
provides two generalizations: it shows their map is also well-defined for a
symplectic bielliptic surface (which fibers over a _quotient_ of a tropical
affine torus), and it extends their map to the whole kernel of the cycle class
map (not just the part generated by fibers). To state it, let us recall from
30 that $\operatorname{\mathrm{Cob}}(\mathcal{K})_{0}$ is generated by
Lagrangians of the form $F_{b^{+}}-F_{b^{-}}$ and
$L_{f_{m^{+}\theta^{+}}}-L_{f_{m^{-}\theta^{-}}}$.

######  Proposition 31.

There is a well-defined map

(28)
$\displaystyle\begin{split}\operatorname{\mathrm{alb}}:\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}&\to\operatorname{\mathrm{Alb}}(K)\\\
\sum_{i}(F_{b_{i}^{+}}-F_{b_{i}^{-}})&\mapsto\operatorname{\mathrm{alb}}(\sum(b_{i}^{+}-b_{i}^{-}))\\\
\sum_{i}(L_{f_{m_{i}^{+}\theta_{i}^{+}}}-L_{f_{m_{i}^{-}\theta_{i}^{-}}})&\mapsto
0.\end{split}$

######  Proof 4.8.

We first recall the basic idea of the proof of [SS21, Lemma 2.10] that proves
the existence of the map (27). For such a map to be well-defined, one must
show that if $\left\\{{F_{b_{i}}-F_{b_{i}^{\prime}}}\right\\}$ are the ends of
a Lagrangian cobordism $V$, then the homomorphism

$\operatorname{\mathrm{alb}}\left(\sum(b_{i}^{+}-b_{i}^{-})\right):H^{0}(T^{*}_{\Z}K)\to\R$

coincides with integration over some class $\beta\in H_{1}(B;\Z)$. Sheridan-
Smith construct such class by ‘closing up’ the Lagrangian cobordism to an
$(n+1)$-cycle in $X(B)\times\C$ (attaching cylinders fibering over paths from
$b_{i}$ to $b_{i}^{\prime}$), and then projecting the homology class of this
cycle to $H_{1}(B)$ via the Künneth map $H_{n+1}(X(B))\to H_{1}(B)\otimes
H_{n}(F)\cong H_{1}(B)$. An application of Stokes’ theorem and the fact that
$V$ is Lagrangian then yield the result.

For the map (28) to be well-defined, we must show that if
$V\subset\mathcal{K}\times\C$ is a cobordism with ends
$\mathbb{F}=\\{F_{b_{i}^{+}}-F_{b_{i}^{-}}\\}$ and
$\mathbb{L}=\\{L_{f_{m^{+}\theta^{+}}}-L_{f_{m^{-}\theta^{-}}}\\}$, then
$\operatorname{\mathrm{alb}}(\sum(b_{i}^{+}-b_{i}^{-}))=0$. In other words,
there exists a class $\beta\in H_{1}(K;\Z)$ such that
$\operatorname{\mathrm{alb}}(\sum(b_{i}^{+}-b_{i}^{-}))=\int_{\beta}$. We
first close up the cobordism to obtain a ($\xi$-oriented) $3$-cycle. Choose
paths in $K$ from $b_{i}^{+}$ to $b_{i}^{-}$; taking their preimage we get a
$3$-chain $W_{1}\subset\mathcal{K}$ whose boundary is $\mathbb{F}$. On the
other hand, the collection $\mathbb{L}$ is a linear combination of Lagrangian
lifts of tropical circles $\left\\{{x=\theta}\right\\}$, and since
$\mathbb{L}$ is homologically trivial there are as many positive as negative
ones. We group them in pairs $\theta_{i}^{\pm}$ and choose isotopies
$\nu_{i}^{s}=\left\\{{x=\theta_{i}^{s}}\right\\}$ between the curves
$\left\\{{x=\theta_{i}^{+}}\right\\}$ and
$\left\\{{x=\theta_{i}^{-}}\right\\}$. Writing
$W_{2}^{i}=\cup_{s}L_{\eta^{i}_{s}}$ for the union of the Lagrangian lifts of
the curves $\eta^{s}_{i}$, we obtain a $3$-chain $W_{2}=\cup_{i}W_{2}^{i}$
whose boundary is $\mathbb{L}$. The union $V+W_{1}+W_{2}$ is a $\xi$-oriented
$3$-cycle. We will use this $3$-cycle to show that the extension by zero of
Equation 28 gives a well-defined map
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\to\operatorname{\mathrm{Alb}}(K)$.

Let $\alpha\in H^{0}(T^{*}_{\Z}K)$ be a globally defined tropical $1$-form,
which corresponds to some global section of the local system given by the
homology of the fibers. Under this correspondence, the value of
$\operatorname{\mathrm{alb}}(\sum_{i}(b_{i}^{+}-b_{i}^{-}))(\alpha)$ can be
computed as the symplectic area of the $2$-chain
$W_{1}\pitchfork([K]\times\alpha\times\C)$ (note this is now a chain with
$\Z$-coefficients). Here, we use the notation $[K]\times\alpha\in
H_{3}(\mathcal{K};\xi)$ to denote the submanifold of $\mathcal{K}$ obtained as
the product of the $0$-section and a geodesic representative of $\alpha$
passing through the origin in the fiber; this is a trivial $S^{1}$-bundle over
the $0$-section, hence a $\xi$-cycle. We make the following observation:

_The only global tropical $1$-form in $K$ are $\alpha=dx$ and its multiples,
and the corresponding homology class in the fibers intersects
$L_{\eta_{s}^{i}}$ along the $0$-section. Hence
$\omega_{\mathcal{K}\times\C}(W_{2}\pitchfork([K]\times\alpha\times\C))=0$._

With these in mind, one can compute as in [SS21, Lemma 2.10] to obtain

$\operatorname{\mathrm{alb}}(\sum_{i}(b_{i}^{+}-b_{i}^{-}))(\alpha)=\int_{(\pi_{\mathcal{K}})_{*}(V+W_{1}+W_{2})\pitchfork([K]\times\alpha)}\omega_{\mathcal{K}}$

(the key being that the extra $2$-chain
$W_{2}\pitchfork([K]\times\alpha\times\C)$ that appears with respect to their
proof is contained in the zero-section, which is Lagrangian).

The last step is to obtain a $1$-cycle $\beta\in H_{1}(K;\Z)$ from the
$3$-cycle $(\pi_{\mathcal{K}})_{*}(V+W_{1}+W_{2})\in H_{3}(\mathcal{K};\xi)$.
This is not as straightforward as in [SS21, Lemma 2.10], since (i) the cycle
lives in $H_{2}(\mathcal{K};\xi)$ (instead of $H_{2}(\mathcal{K};\Z)$) and
(ii) the fibration is not trivial, hence there is not a Künneth decomposition
or a projection to $H_{1}(K;\Z)$. Nonetheless, by analyzing the Leray spectral
sequence $E^{2}_{pq}=H_{p}(K;\mathcal{H}_{q}(F;\Z)\otimes\xi)\Rightarrow
H_{p+q}(\mathcal{K};\xi)$ we show in 32 that there is a projection map
$pr:H_{3}(\mathcal{K};\xi)\to H_{1}(K;\Z)$. We define the cycle

$\beta:=pr((\pi_{\mathcal{K}})_{*}(V+W_{1}+W_{2}))\in H_{1}(K;\Z).$

Lastly, consider the SES

$0\to H_{0}(K;\mathcal{H}_{2}(F))\to H_{2}(\mathcal{K};\Z)\to
H_{1}(K;\mathcal{H}_{1}(F))\to 0$

coming from the Leray spectral sequence. It shows that the class
$(\pi_{\mathcal{K}})_{*}(V+W_{1}+W_{2})\pitchfork([K]\times\alpha)\in
H_{2}(\mathcal{K};\Z)$ is of the form

$(\pi_{\mathcal{K}})_{*}(V+W_{1}+W_{2})\pitchfork([K]\times\alpha)=\beta\otimes\alpha+\gamma,$

where $\gamma\in H_{0}(K;\mathcal{H}_{2}(F))\subset H_{2}(\mathcal{K};\Z)$.
The class $\gamma$ is Lagrangian (it is supported on a fiber), hence one
concludes

$\operatorname{\mathrm{alb}}(\sum_{i}(b_{i}^{+}-b_{i}^{-}))(\alpha)=\int_{(\pi_{\mathcal{K}})_{*}(V+W_{1}+W_{2})\pitchfork([K]\times\alpha)}\omega_{\mathcal{K}}=\int_{\beta\times\alpha}\omega_{\mathcal{K}}=\int_{\beta}\alpha,$

completing the proof.

######  Lemma 32.

There is a projection map $H_{3}(\mathcal{K};\xi)\to H_{1}(K;\Z)$.

######  Proof 4.9.

Let us consider the Leray spectral sequence

$E^{2}_{pq}=H_{p}(K;\mathcal{H}_{q}(F;\Z)\otimes\xi)\Rightarrow
H_{p+q}(\mathcal{K};\xi).$

For degree reasons one has $E^{\infty}_{11}\cong
E^{2}_{11}=H_{1}(K;\mathcal{H}_{1}(F;\Z)\otimes\xi)$, hence there is a short
exact sequence

(29) $0\to H_{1}(K;\mathcal{H}_{2}(F;\Z)\otimes\xi)\to
H_{3}(\mathcal{K};\xi)\to E^{\infty}_{21}\to 0.$

It follows from the computations in Appendix A that $\im(E^{2}_{21}\to
E^{2}_{02})=0$, hence $E^{\infty}_{21}\cong
E^{2}_{21}=H_{2}(K;\mathcal{H}_{1}(F)\otimes\xi)$. This group is isomorphic to
$\Z$ and generated by $[K]\times\alpha$, thus Equation 29 splits canonically
giving a decomposition

$H_{3}(\mathcal{K};\xi)\cong H_{2}(K;\mathcal{H}_{1}(F;\Z)\otimes\xi)\oplus
H_{1}(K;\mathcal{H}_{2}(F;\Z)\otimes\xi).$

The geometry of Lagrangian torus fibrations shows that
$\mathcal{H}_{2}(F;\Z)\cong\wedge^{2}T^{*}_{\Z}K=\xi^{\vee}$, and hence the
local system $\mathcal{H}_{2}(F;\Z)\otimes\xi$ is trivial. Putting everything
together, we get the desired projection $pr:H_{3}(\mathcal{K};\xi)\to
H_{1}(K;\Z)$.

As mentioned in [SS21, Remark 2.11], there is an extension of the fibered
albanese map that incorporates the local systems on the Lagrangians. In the
present case, if $\lambda=\lambda_{x}\otimes\lambda_{y}$ is a local system on
$F$ with monodromies $\lambda_{x},\lambda_{y}\in G$ along the homology classes
corresponding to $dx$ and $dy$, then this map takes the form

(30)
$\displaystyle\begin{split}\operatorname{\mathrm{alb}}^{loc}_{fib}:\operatorname{\mathrm{Cob}}_{fib}(\mathcal{K})_{\hom}&\to\operatorname{\mathrm{Alb}}(K)\oplus
G\\\
\sum((F_{i}^{+},\lambda_{i}^{+})-(F_{i}^{-},\lambda_{i}^{-}))&\mapsto(\operatorname{\mathrm{alb}}(\sum_{i}(F_{i}^{+}-F_{i}^{-})),\prod_{i}\lambda_{i,x}^{+}(\lambda_{i,x}^{-})^{-1}).\end{split}$

The fact that this map is well-defined follows from a similar argument to that
of 31. Namely, given $(V,\lambda)$ a cobordism with local system between a
linear combination of fibers, its intersection with the $5$-cycle
$K\times[\alpha]\times\C$ is a $2$-dimensional surface with boundary inside
$V$. It follows that the product of the monodromies around the boundary
components of this surface of any local system must vanish. In particular, for
the local system $\lambda$—which restricts to $\lambda_{i}^{\pm}$ on the ends
of $V$—we have that $\prod_{i}\lambda_{i,x}^{+}(\lambda_{i,x}^{-})^{-1}=1$.

######  Proposition 33.

The fibered albanese map
$\operatorname{\mathrm{alb}}_{fib}^{loc}:\operatorname{\mathrm{Cob}}_{fib}(\mathcal{K})_{\hom}\to\operatorname{\mathrm{Alb}}(K)\oplus
G$ is an isomorphism.

######  Proof 4.10.

The map is clearly surjective (recall $H^{0}(T^{*}_{\Z}K)\cong\Z\langle
dx\rangle$). To prove injectivity we proceed as follows. Denote by $pr:K\to
S^{1}$ the projection to the line $\\{y=1/2\\}$. Consider a Lagrangian fiber
with local system $(F_{p},\lambda=\lambda_{x}\otimes\lambda_{y})$. We will
show that

(31) $(F_{p},\lambda)\sim(F_{pr(p)},\lambda_{x}).$

Given this, injectivity follows from injectivity of
$\operatorname{\mathrm{alb}}_{fib}^{loc}:\operatorname{\mathrm{Cob}}_{fib}(T^{2})_{\hom}\to
S^{1}\times G$ (a consequence of 15).

Write $\pi_{x}:K\to S^{1}$ and $\pi_{y}:K\to I=[0,1]$ for the projections to
the (equivalence class of the) $x$ and $y$ coordinates of $K$. Note both of
these maps give tropical $S^{1}$-fibrations of $K$, meaning the fibers are
embedded tropical circles. Let $\theta\in S^{1}$ and $t\in I$ be the unique
points with $p\in\pi^{-1}_{x}(\theta)\cap\pi_{y}^{-1}(t)$.

We first claim that if $\iota:K\to K$ is the map induced by
$(x,y)\mapsto(x,-y)$, then the cobordism class

$\mathbb{F}_{1}=(F_{p},\lambda_{x}\otimes\lambda_{y})-(F_{\iota(p)},\lambda_{x}\otimes\lambda_{y}^{-1})$

is $2$-torsion. Indeed, let $p:T^{2}\times T^{2}\to\mathcal{K}$ be the double
cover of $\mathcal{K}$ (cf. 19) and write $T^{2}\times
T^{2}=(\R/\Z\times\R/l_{1}\Z)\times(\R/\Z\times\R/l_{2}\Z)$. We denote by
$F^{i}_{t}$ the fibers of the projection $\R/\Z\times\R/l_{i}\Z\to\R/\Z$. Then
the pull back of the class $\mathbb{F}_{1}$ under $p$ is a linear combination
of Lagrangian torus fibers of the form (see Figure 2):

$p^{-1}(\mathbb{F}_{1})=(F^{1}_{\theta}-F^{1}_{\theta+1/2})\times(F^{2}_{t}+F^{2}_{1-t}).$

Now note that the linear combination $F^{1}_{\theta}-F^{1}_{\theta+1/2}$ is a
null-homologous combination of fibers in $\operatorname{\mathrm{Cob}}(T^{2})$,
and that its image under the map $\Phi$ of 11 is $2$-torsion. It then follows
from 13 that this class is $2$-torsion in
$\operatorname{\mathrm{Cob}}(T^{2})$, and hence—by crossing the cobordism with
the constant Lagrangian $F^{2}_{t}+F^{2}_{1-t}$ in the other
$T^{2}$-factor—that $p^{-1}(\mathbb{F}_{1})$ is $2$-torsion in
$\operatorname{\mathrm{Cob}}(T^{2}\times T^{2})$. The cobordism is furthermore
$p$-invariant, hence descends to give the relation $2\mathbb{F}_{1}\sim
0\in\operatorname{\mathrm{Cob}}(\mathcal{K})$.

$>>$$>>$$>$$<$$p$$\iota(p)$$>>$$>>$$>$$>$$(\theta,t)$$(\theta,1-t)$$(\theta+1/2,t)$$(\theta+1/2,1-t)$$K$$T^{2}$$x$$y$$>>$$>>$$>$$>$$\times$$>>$$>>$$>$$>$$F^{1}_{\theta}$$F^{1}_{\theta+1/2}$$F^{2}_{t}$$>>$$>>$$>$$>$$\times$$>>$$>>$$>$$>$$F^{1}_{\theta}$$F^{1}_{\theta+1/2}$$F^{2}_{1-t}$$x$$dx$$y$$dy$$T^{2}\times
T^{2}$ Figure 2: Schematic picture to show that $2\mathbb{F}_{1}\sim 0$. The
left picture shows the (projection to $K$ of the) class
$\mathbb{F}_{1}\in\operatorname{\mathrm{Cob}}_{fib}(\mathcal{K})_{\hom}$ as
well as its preimage
$p^{-1}(\mathbb{F}_{1})\in\operatorname{\mathrm{Cob}}_{fib}(T^{2}\times
T^{2})_{\hom}$. The right picture shows the decomposition of
$p^{-1}(\mathbb{F}_{1})$ in the product $T^{2}\times T^{2}$; namely, the top
part shows the class $(F_{\theta}^{1}-F_{\theta+1/2}^{1})\times F^{2}_{t}$,
whereas the bottom part shows $(F_{\theta}^{1}-F_{\theta+1/2}^{1})\times
F^{2}_{1-t}$. Note that while the torus in the left picture represents a
tropical affine base, the $2$-tori in the right picture represent symplectic
tori living above the tropical base (that is, the symplectic manifold
$T^{2}\times T^{2}$ in the right picture is the total space of the Lagrangian
torus fibration over the tropical $2$-torus $T^{2}$ in the left picture).

Next, we claim that the cobordism class

$\mathbb{F}_{2}=(F_{p},\lambda_{x}\otimes\lambda_{y})-(F_{pr(p)},\lambda_{x})$

is $4$-torsion. By the previous claim we have that

$4(F_{p},\lambda_{x}\otimes\lambda_{y})\sim
2(F_{p},\lambda_{x}\otimes\lambda_{y})+2(F_{\iota(p)},\lambda_{x}\otimes\lambda_{y}^{-1}).$

As before, we can pull back the class $4\mathbb{F}_{2}$ to $T^{2}\times T^{2}$
and obtain a linear combination of fibers of the form

$p^{-1}(\mathbb{F}_{2})=(F^{1}_{\theta}+F^{1}_{\theta+1/2})\times(2F^{2}_{t}+2F^{2}_{1-t}-4F^{2}_{1/2}).$

The linear combination $2F^{2}_{t}+2F^{2}_{1-t}-4F^{2}_{1/2}$ is null-
homologous and maps to $0$ under $\Phi$, hence vanishes in
$\operatorname{\mathrm{Cob}}(T^{2})$ by 13. Crossing with the constant
Lagrangian $F^{1}_{\theta}+F^{1}_{\theta+1/2}$ in the other $T^{2}$ factor
yields a cobordism in $T^{2}\times T^{2}\times\C$, and descending it to
$\mathcal{K}\times\C$ we get the desired relation $4\mathbb{F}_{2}\sim
0\in\operatorname{\mathrm{Cob}}(\mathcal{K})$.

We have now proved Equation 31 up to torsion. The complete result follows from
noting that

$(F_{p},\lambda_{x}\otimes\lambda_{y})-(F_{pr(p)},\lambda_{x})\sim
4((F_{q},\lambda_{x}\otimes\lambda_{y})-(F_{pr(p)},\lambda_{x}))$

for any $q\in\pi_{x}^{-1}(\theta)$ such that
$4(q-pr(q))=p-pr(p)\in\operatorname{\mathrm{Alb}}(\pi_{x}^{-1}(\theta))$
(again, a consequence of 13).

A similar argument to the proof of 31 shows that
$\operatorname{\mathrm{alb}}^{loc}_{fib}$ extends to the whole tropical
cobordism group, giving a map

(32)
$\operatorname{\mathrm{alb}}^{loc}:\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\to\operatorname{\mathrm{Alb}}(K)\oplus
G.$

###  4.7 Computation and dimensionality

The following is one of the two main results of this paper:

######  Theorem 34.

The short exact sequence

(33)
$0\to\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\to\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\xrightarrow{\widetilde{cyc}}H_{2}(\mathcal{K};\xi)\oplus
G_{(2)}\to 0$

is split, and
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\cong(S^{1}\oplus G)^{2}$.

######  Proof 4.11.

Let $\Psi:\mathcal{K}\to\mathcal{K}$ be the involution swapping the
$y$-coordinate on $K$ with its cotangent coordinate (recall
$\mathcal{K}=T^{\ast}K/T^{*}_{\Z}K$ and $K$ is as in 6). That is, $\Psi$ is
given in local coordinates by

$(x_{1},y_{1},x_{2},y_{2})\mapsto(x_{1},y_{1},y_{2},x_{2}),$

where we are identifying $\mathcal{K}$ with the quotient of
$(\R^{4},dx_{1}\wedge dy_{1}+dx_{2}\wedge dy_{2})$ by the symplectic
transformations

$\displaystyle(x_{1},y_{1},x_{2},y_{2})$
$\displaystyle\mapsto(x_{1},y_{1},x_{2},y_{2})+e_{i},\quad i=1,\dots,4$
$\displaystyle(x_{1},y_{1},x_{2},y_{2})$
$\displaystyle\mapsto(x_{1}+p_{1},y_{1}+p_{2},-x_{2},-y_{2}),\quad
2(p_{1},p_{2})=0.$

The composition $\pi^{\prime}=\pi\circ\Psi:\mathcal{K}\to K$ is still a
Lagrangian torus fibration; let $K^{\prime}$ be the Klein bottle whose
tropical affine structure is that induced by $\pi^{\prime}$.

Consider the map
$\operatorname{\mathrm{alb}}^{loc}:\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\to\operatorname{\mathrm{Alb}}(K)\times
G$ of Equation 32. There is an analogous map
$\operatorname{\mathrm{alb}}^{\prime,loc}:\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\to\operatorname{\mathrm{Alb}}(K^{\prime})\times
G$ arising from the Lagrangian torus fibration $\pi^{\prime}$. We claim the
map

(34)
$\operatorname{\mathrm{alb}}^{loc}\oplus\operatorname{\mathrm{alb}}^{\prime,loc}:\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\to(\operatorname{\mathrm{Alb}}(K)\oplus
G)\oplus(\operatorname{\mathrm{Alb}}(K^{\prime})\oplus G)$

is an isomorphism. Indeed, note that $\Psi$ interchanges fibers with the
tropical Lagrangians $L_{f_{m\theta}}$, so that
$\operatorname{\mathrm{alb}}^{\prime}$ is the map

$\displaystyle\begin{split}\operatorname{\mathrm{alb}}^{\prime}:\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}&\to\operatorname{\mathrm{Alb}}(K^{\prime})\\\
\sum_{i}(F_{b_{i}^{+}}-F_{b_{i}^{-}})&\mapsto 0\\\
\sum_{i}(L_{f_{m_{i}^{+}\theta_{i}^{+}}}-L_{f_{m_{i}^{-}\theta_{i}^{-}}})&\mapsto\operatorname{\mathrm{alb}}^{\prime}(p_{i}^{+}-p_{i}^{-})\end{split}$

where $p_{i}^{\pm}=\pi^{\prime}(L_{f_{m_{i}^{\pm}\theta_{i}^{\pm}}})\in
K^{\prime}$. The fact that (34) is an isomorphism follows now immediately from
33 applied to both $\operatorname{\mathrm{alb}}^{loc}$ and
$\operatorname{\mathrm{alb}}^{\prime,loc}$.

To show that Equation 33 splits, we consider the map
$H_{2}(\mathcal{K};\xi)\oplus G_{(2)}\cong\Z^{4}\oplus\Z_{2}\oplus
G_{(2)}\to\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}$ that sends an
element $(N_{f},N_{0},m,n,[l],g)$ to the cobordism class

$N_{f}F_{(0,1/2)}+N_{0}\Gamma_{0}+mL_{f_{10}}+nL_{f^{10}}+lL_{f^{01}}+((L_{f^{10}},g)-L_{f^{10}}).$

Here, all the Lagrangians are equipped with the preferred Pin structure and
trivial local system, except $(L_{f^{10}},g)$ which carries a local system
whose monodromy around the $dy$-coordinate is $g$. The argument at the end of
30 shows that

$\sum_{i}((L_{f^{10}},g_{i})-L_{f^{10}})\sim(L_{f^{10}},\prod_{i}g_{i})-L_{f^{10}},$

hence this map is a group homomorphism, and it is clearly a section by
Appendix A. It follows that the short exact sequence (33) is split.

Note that, in particular,

$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\cong
H_{2}(\mathcal{K};\xi)\oplus G_{(2)}\oplus(S^{1}\oplus G)^{2}$

is _$2$ -dimensional_ in the sense of 50. As discussed in the introduction,
this should be compared with the work of Sheridan-Smith [SS20], who show that
a symplectic $4$-manifold with trivial canonical bundle containing a
Lagrangian of genus at least 1 has infinite-dimensional cobordism group. The
crucial difference in 34 is that the canonical bundle of a bielliptic surface
is torsion but non-trivial, thus a graded Lagrangian is not necessarily
oriented (see Section 1.2 for more details).

One would expect—based on the analogous result for the mirror bielliptic
surface—that a computation of the cobordism group of a symplectic bielliptic
surface living over $K=K_{2}$ (see Section 2.4) would follow the same
argument. However, in the symplectic setting some difficulties arise. While we
do have a short exact sequence as in Equation 33, the definition of the Abel-
Jacobi type map
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})_{0}\to\operatorname{\mathrm{Alb}}(K)$
is not so clear in this case.

###  4.8 The Lagrangian cobordism group and the Grothendieck group

We now turn to the comparison between the cobordism group
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ and the Grothendieck group
of the Fukaya category of $\mathcal{K}$. We do this via intermediate
comparison with the Chow groups of the mirror $Y$ (see Section 4.2). Thus,
from now on we fix $G=U_{\Lambda}$ to be the unitary group of the Novikov
field.

Recall from Appendix A that there is an isomorphism
$H_{2}(\mathcal{K};\xi)\cong\Z^{4}\oplus\Z_{2}$. Putting this together with
Theorem 34, one obtains an explicit presentation

$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\cong\Z^{4}\oplus\Z_{2}^{2}\oplus
E^{2}$

for $E$ the closed points of an elliptic curve over the Novikov field. We now
sketch how to compute the Chow groups of the mirror $Y$ and exhibit a specific
isomorphism between $\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ and
$\operatorname{\mathrm{CH}}^{*}(Y)$:

  * •

$\operatorname{\mathrm{CH}}_{2}(Y)\cong\Z$ is generated by the bielliptic
surface $Y$ itself. This factor is mirror to a $\Z$-factor in
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ corresponding to the zero-
section.

  * •

For complex bielliptic surfaces
$\operatorname{\mathrm{CH}}_{0}(Y)\cong\Z\oplus\operatorname{\mathrm{Alb}}(Y)$:
as follows from the results in [BKL76]. The proof is based on analyzing the
kernel of the Albanese map
$\operatorname{\mathrm{alb}}:\operatorname{\mathrm{CH}}_{0}(Y)_{\hom}\to\operatorname{\mathrm{Alb}}(Y)$.
The Albanese map exists for fields other than the complex numbers, and one can
run the arguments in [BKL76] to show that $\ker\operatorname{\mathrm{alb}}=0$
still holds. Thus
$\operatorname{\mathrm{CH}}_{0}(Y)\cong\Z\oplus\operatorname{\mathrm{Alb}}(Y)$,
where the $\Z$ factor is generated by a fixed reference point and the
$\operatorname{\mathrm{Alb}}(Y)$ factor is generated by differences
$p^{+}-p^{-}\in\operatorname{\mathrm{CH}}_{0}(Y)_{hom}$ for points $p^{\pm}\in
Y$. This is mirror to
$\operatorname{\mathrm{Cob}}_{\text{fib}}(\mathcal{K})\cong\Z\oplus(\operatorname{\mathrm{Alb}}(K)\oplus
U_{\Lambda})$ (an explicit isomorphism is given by the map
$\deg\oplus\operatorname{\mathrm{alb}}_{\text{fib}}^{loc}:\operatorname{\mathrm{Cob}}_{\text{fib}}(\mathcal{K})\to\Z\oplus(\operatorname{\mathrm{Alb}}(K)\oplus
U_{\Lambda})$, where $\operatorname{\mathrm{alb}}_{\text{fib}}^{loc}$ is
constructed from the map in 33 after choosing a reference fiber).

  * •

$\operatorname{\mathrm{CH}}_{1}(Y)\cong\Pic(Y)$ fits into a short exact
sequence

$0\to\Pic^{0}(Y)\to\Pic(Y)\to NS(Y)\to 0$

where $\Pic^{0}(Y)$ is the Picard variety of $Y$ (the connected component of
the identity in $\Pic(Y)$) and $NS(Y)=\Pic(Y)/\Pic^{0}(Y)$ is the so-called
Neron-Severi group. For a complex bielliptic surface, the group $NS(Y)\cong
H^{2}(Y;\Z)$ has been fully computed in [Ser90], and this computation applies
to bielliptic surfaces over algebraically closed fields of characteristic zero
by the Lefschetz principle. Since
$\Pic^{0}(Y)\cong(\operatorname{\mathrm{Alb}}(Y))^{\vee}\cong E$,111111Recall
that $\operatorname{\mathrm{Alb}}(Y)$ is an abelian variety and that its
dimension is equal to $h^{1}(Y,\mathcal{O}_{Y})=1$. It follows that
$\operatorname{\mathrm{Alb}}(Y)$ is an elliptic curve, hence isomorphic to its
dual. we get

$\operatorname{\mathrm{CH}}_{1}(Y)\cong NS(Y)\oplus E$

where $E$ is an elliptic curve over the Novikov field. The group
$NS(Y)\cong\Z^{2}\oplus\Z_{2}^{2}$ is generated by two general fibers of the
projections $\pi_{1}:Y\to E_{1}/\Z_{2}$ and $\pi_{2}:Y\to
E_{2}/Z_{2}\cong\mathbb{P}^{1}$ (cf. Section 4.1) and differences
$D_{1}-D_{2},D_{1}-D_{3}$ of the four reduced special fibers $D_{i}$ of the
projection $\pi_{2}$ [Ber+19]; this is mirror to the subgroup
$(\Z^{2}\oplus\Z_{2})\oplus\Z_{2}\subset H_{2}(\mathcal{K};\xi)\oplus\Z_{2}$
generated by the tropical Lagrangians
$L_{C_{1}},L_{C_{2}},L_{C_{2}}-L_{C_{3}}$ and $L_{C_{3}}-(L_{C_{3}},\eta)$.
Here we denote by $L_{C_{i}}$ the tropical Lagrangians living over the
tropical subvarieties $C_{i}$ of Equation 19; they all carry the natural Pin
structure and the trivial local system, except $(L_{C_{3}},\eta)$ that carries
a $2$-torsion local system that is non-trivial in the $dy$-direction. The $E$
factor is generated by differences of fibers of the projection $\pi_{1}$; it
is mirror to $\operatorname{\mathrm{Alb}}(K^{\prime})\oplus U_{\Lambda}$ and
is generated by differences of tropical Lagrangians $L_{C_{1}}-L_{\\{x=a\\}}$
equipped with arbitrary local systems.

We now turn to the comparison between the cobordism group
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ and the Grothendieck group
of the Fukaya category. A homomorphism relating the Lagrangian cobordism group
(without the extra data of a brane structure) and the Grothendieck group of
the Fukaya category as in Equation 1 was first considered by Biran-Cornea in
[BC14]. Their work proves the existence of such homomorphism for a version of
the cobordism group whose generators and relations are monotone Lagrangians
and Lagrangian cobordisms, i.e. Lagrangians $L\subset M$ for which the
homomorphisms

$\displaystyle\omega:\pi_{2}(M,L)$ $\displaystyle\to\R$
$\displaystyle\mu:\pi_{2}(M,L)$ $\displaystyle\to\Z$

given by measuring the symplectic area and the Maslov index of topological
discs with boundary on $L$ satisfy $\omega=\lambda_{L}\mu$ for some
$\lambda_{L}>0$. This monotonicity condition is imposed to exclude bubbling of
pseudo-holomorphic disks, so that Floer homology is well-defined.121212In that
paper Biran-Cornea impose an additional condition on the fundamental groups of
all Lagrangians and cobordism. This condition gives a bound on the area of
holomorphic disks and is not needed when working over the Novikov field, so we
will not treat it here. In later work [BCS21], Biran-Cornea-Shelukhin relax
the assumptions on cobordisms to include tautologically unobstructed
cobordisms (recall this means there exists a compatible almost complex
structure such that $V$ bounds no non-constant holomorphic disks, a property
to which they refer as ‘quasi-exactness’). This comes at the cost of
strengthening the monotonicity condition imposed on the ends, where they must
now be weakly exact Lagrangians (recall this means $\omega(\pi_{2}(M,L))=0$).
See their results in [BCS21, Chapter 4].

There is a stronger expectation—which is believed by experts but has not yet
appeared in the literature—that the iterated cone decomposition results of
[BC14, BCS21] hold for the broadest class of Lagrangians and cobordisms for
which Floer theory can be defined, namely (Floer-theoretically) unobstructed
Lagrangians in the sense of [Fuk+09]. These are the most natural Lagrangians
to consider in the context of mirror symmetry (for instance, to compare
Lagrangian cobordism groups with Chow groups). We remark that the group
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ that we compute in this
paper (a group generated by Lagrangian fibers and sections) does not depend on
whether we consider it as a subgroup of
$\operatorname{\mathrm{Cob}}(\mathcal{K})$, as defined in 26, or as a subgroup
of a cobordism group built from Floer-theoretically unobstructed Lagrangians
and cobordisms. However, since the existence of a map from the cobordism group
to the Grothendieck group has only been proved to exist for a cobordism group
of weakly exact Lagrangians and tautologically unobstructed cobordisms, we
think of $\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ as a subgroup of
$\operatorname{\mathrm{Cob}}(\mathcal{K})$ as defined in 26.

Recall from 26 that the generators of
$\operatorname{\mathrm{Cob}}(\mathcal{K})$ are weakly exact. On the other
hand, we imposed the weaker condition that cobordisms are tautologically
unobstructed only when pulling them back under the covering map
$T^{4}\times\C\to\mathcal{K}\times\C$. We argue in Appendix C that cobordisms
with the above property still induce iterated cone decompositions between
their ends, see 48. Hence we have a map

$\operatorname{\mathrm{Cob}}(\mathcal{K})\to
K_{0}(D\mathcal{F}uk(\mathcal{K})).$

The above matching of the tropical cobordism group and the Chow groups of the
mirror together with the HMS statement of Section 4.2 suggest our second main
result:

######  Theorem 35.

The natural homomorphism

(35) $\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\to
K_{0}(D^{b}\mathcal{F}uk(\mathcal{K}))$

is an isomorphism.

######  Proof 4.12.

We first prove an isomorphism
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\cong
K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))$ between the Lagrangian cobordism
group and the Grothendieck group of the split-closed derived Fukaya category.
Recall from Section 4.2 that we have an equivalence of triangulated categories
$D^{\pi}\mathcal{F}uk(\mathcal{K})\simeq D^{b}Coh(\check{\mathcal{K}})$, the
latter category being isomorphic to the derived category of coherent sheaves
$D^{b}Coh(Y)$ of an algebraic bielliptic surface over the Novikov field.
Taking the Grothendieck group on both sides yields an isomorphism of abelian
groups

$K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))\cong K_{0}(D^{b}Coh(Y))$

which reduces the computation of $K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))$ to
algebraic geometry. Now recall that there is a Chern character map

$ch:K_{0}(D^{b}Coh(Y))\to\operatorname{\mathrm{CH}}^{*}(Y)_{\Q}$

which furthermore is a rational isomorphism. In the present case, one can show
that $ch$ is well-defined as a map to $\operatorname{\mathrm{CH}}^{*}(Y)$
(i.e. we can get rid of denominators), and that such map is an isomorphism—see
Appendix B. Putting this together with the Chow group computations at the
beginning of this section, we have

$K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))\cong
K_{0}(D^{b}Coh(Y))\cong\operatorname{\mathrm{CH}}^{*}(Y)\cong\Z^{4}\oplus\Z_{2}^{2}\oplus
E^{2},$

showing that $K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))$ and
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ are, at least abstractly,
isomorphic groups.

To see that the natural homomorphism
$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\to
K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))$ provides an isomorphism we will show
that the composition

$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\to
K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))\xrightarrow{\sim}K_{0}(D^{b}Coh(Y))$

is an isomorphism. For this we use that the mirror functor of [Abo14, Abo21]
sends a tropical Lagrangian $L_{V}$ living over a tropical submanifold
$V\subset K$ to the structure sheaf $\mathcal{O}_{Z_{V}}$ of the corresponding
algebraic subvariety $Z_{V}\subset Y$ living over $V$. Choosing a set of
generators of $\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})$ given by
(linear combinations of) tropical Lagrangians as at the beginning of this
section, the map

$\operatorname{\mathrm{Cob}}^{trop}(\mathcal{K})\cong\Z^{4}\oplus\Z_{2}^{2}\oplus
E^{2}\to K_{0}(D^{b}Coh(Y))$

coincides with the map of 47, hence the result follows.

To conclude the proof, note that the natural embedding
$D^{b}\mathcal{F}uk(\mathcal{K})\hookrightarrow
D^{\pi}\mathcal{F}uk(\mathcal{K})$ induces an isomorphism of Grothendieck
groups: it is always an injection and the generators of
$K_{0}(D^{\pi}\mathcal{F}uk(\mathcal{K}))$ are honest Lagrangians (as opposed
to direct summands), thus it is also surjective.

###  4.9 Admissibility of the surgery cobordism

In this section we upgrade 24 to include all the extra structure that is
present in the cobordism group ($G$-brane structures and unobstructedness
properties). It is mostly technical and can safely be skipped by readers on a
first read.

There are two separate issues to treat. The first is the transformation of
gradings, Pin structures and local systems under fiberwise addition. The
second is the admissibility of the Lagrangian cobordisms obtained as surgery
of a section $\Gamma^{n[l]}$ (or $\Gamma_{m\theta}$) and the zero section
$\Gamma_{0}$—that is, whether these cobordisms admit $G$-brane structures and
whether their pullback to $T^{4}\times\C$ is tautologically unobstructed.

Let us start with the transformation of gradings, Pin structures and local
systems under fiberwise addition. This is part of the more general story of
how these properties transform under geometric composition of Lagrangian
correspondences: transformation of gradings is treated in [WW10, Section 3],
transformation of Pin structures is discussed in [WW15] and transformation of
local systems is detailed in [Sub10]. To apply these results, recall that the
Lagrangian correspondence giving rise to fiberwise addition is given by

$\Sigma=\\{(q_{1},p_{1},q_{2},p_{2},q_{3},p_{3})\in(X(B)\times X(B))^{-}\times
X(B)\,|\,q_{1}=q_{2}=q_{3},\,p_{3}=p_{1}+p_{2}\\}$

for $(q_{i},p_{i})\in X(B)$ canonical coordinates [Sub10]; one then defines

$L_{1}\otimes L_{2}=\Phi_{\Sigma}(L_{1}\times L_{2})=\pi_{13}((L_{1}\times
L_{2}\times X(B))\cap\Sigma).$

######  Lemma 36.

The Lagrangian correspondence $\Sigma$ admits a natural grading. Furthermore,
any Pin structure on $B$ induces a natural Pin structure on $\Sigma$.

######  Proof 4.13.

With respect to a quadratic volume form defined by complexifying a section of
$(\wedge^{3n}T^{*}B^{3})^{\otimes 2}$, a computation shows the Lagrangian
correspondence $\Sigma$ has phase map $\alpha_{\Sigma}:\Sigma\to S^{1}$
constant and equal to $(-1)^{\dim B}\in S^{1}$. We equip $\Sigma$ with the