# Tropical Poincaré duality spaces

Edvard Aksnes

###### Abstract.

The tropical fundamental class of a rational balanced polyhedral fan induces
cap products between tropical cohomology and tropical Borel–Moore homology.
When all these cap products are isomorphisms, the fan is said to be a
_tropical Poincaré duality space_. If all the stars of faces also are such
spaces, such as for fans of matroids, the fan is called a _local tropical
Poincaré duality space_.

In this article, we first give some necessary conditions for fans to be
tropical Poincaré duality spaces and a classification in dimension one. Next,
we prove that tropical Poincaré duality for the stars of all faces of
dimension greater than zero and a vanishing condition implies tropical
Poincaré duality of the fan. This leads to necessary and sufficient conditions
for a fan to be a local tropical Poincaré duality space. Finally, we use such
fans to show that certain abstract balanced polyhedral spaces satisfy tropical
Poincaré duality.

##  1\. Introduction

Tropical geometry forms a bridge between algebraic geometry and combinatorics.
Through the process of _tropicalizing_ subvarieties of tori, one obtains
polyhedral complexes. In general, an _abstract tropical cycle_ is a
topological space which is locally homeomorphic to rational polyhedral fans
which are _balanced_ : each codimension one face satisfies a geometric
balancing condition. Tropical cycles can be equipped with tropical homology
and cohomology groups, and come equipped with a fundamental class. This
fundamental class can be used to construct a cap product map from cohomology
to homology. In this paper, we study when this cap product is an isomorphism.
We begin by considering the case of fans.

Let $\Sigma$ be a rational polyhedral fan (Definition 2.2). For an integer
$p\geq 0$ and a commutative ring $R$, [IKMZ19] introduced the _tropical
homology_ $H_{\bullet}(\Sigma,\mathcal{F}_{p}^{R})$ and _tropical Borel–Moore
homology_ $H_{\bullet}^{BM}(\Sigma,\mathcal{F}_{p}^{R})$, along with dual
constructions of _tropical cohomology_
$H^{\bullet}(\Sigma,\mathcal{F}_{R}^{p})$ and _tropical cohomology with
compact support_ $H_{c}^{\bullet}(\Sigma,\mathcal{F}_{R}^{p})$, see Definition
3.5. These can be defined through singular chains with local coefficients,
their simplicial or cellular approximation, or through (co)sheaf (co)homology
as in [IKMZ19, MZ14]. Moreover, the tropical cohomology can be computed
through the Dolbeault cohomology of superforms [JSS19]. The sheaf-theoretic
approach was further investigated in [GS19], giving a formulation in terms of
dualizing objects in the derived category of abelian sheaves.

Given the cellular structure on a fan $\Sigma$, the set of $k$-dimensional
faces is denoted $\Sigma^{k}$. One can assign a _weight function_
$w\colon\Sigma^{d}\to R$ to the $d$-dimensional faces, where $d=\dim(\Sigma)$,
see Definition 3.10. The _balancing condition_ for fans [BIMS15, Definition
5.8] is the same as $w$ being a Minkowski weight in toric geometry [FS97],
generalizing to coefficients in the ring $R$. The balancing condition can be
formulated homologically as the existence of a particular _fundamental class_
$[\Sigma,w]\in H_{d}^{BM}(\Sigma,\mathcal{F}_{d}^{R})$ in tropical Borel–Moore
homology ([MZ14, Proposition 4.3], [JRS18, Remark 4.9] and Definition 3.11).
One can use the fundamental class to define a _cap product_

$\frown[\Sigma,w]\colon H^{q}(\Sigma,\mathcal{F}_{R}^{p})\to
H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$

for all $p,q\in\\{0,\dots,d\\}$, see [JRS18, Definition 4.11] and Definition
3.17. If these maps are isomorphisms for all $p,q\in\\{0,\dots,d\\}$, one says
that the fan satisfies _tropical Poincaré duality over $R$_ or is a _tropical
Poincaré duality space over $R$_, see Definition 4.1. We use the abbreviation
TPD for tropical Poincaré duality.

This paper, which extends the results from the author’s master’s thesis
[Aks19], studies two questions related to tropical Poincaré duality over a
given commutative ring $R$.

######  Question 1.1.

Which fans satisfy TPD over $R$?

The fan of a matroid is a TPD space over $\mathbb{R}$ by [JSS19, Proposition
4.27]. This was taken further in [JRS18], showing that these are also TPD
spaces over $\mathbb{Z}$. Moreover, motivating the question, there are fans
satisfying TPD which are not fans of matroids, see Example 4.2.

A useful property of the cap product is that, for any commutative ring R, when
it is non-zero, it is injective (Proposition 3.20), which is also remarked in
[AP20, Section 3.2.2] in the case $R=\mathbb{Z}$ and in [Aks19] for the
$R=\mathbb{R}$ case. Using this in the case where $R$ is a field, we can work
with Euler characteristics and dimensions of homology groups to give a
criterion for a fan to have TPD, under some vanishing assumptions (Proposition
4.6). Furthermore, we completely classify one-dimensional TPD spaces over an
arbitrary commutative ring $R$.

######  Proposition 4.7.

Let $R$ be a commutative ring, and $(\Sigma,w)$ an $R$-balanced fan of
dimension one. Then $(\Sigma,w)$ satisfies tropical Poincaré duality over $R$
if and only if it is uniquely $R$-balanced and all the weights are units in
$R$.

In particular, this implies that over a field, a one-dimensional fan is a TPD
space if and only if it is uniquely balanced (Corollary 4.8). In Proposition
4.10, we show that fan tropical hypersurfaces in $\mathbb{R}^{n}$ must have
simplexes as Newton polytopes.

######  Question 1.2.

Which fans satisfy TPD over $R$ at each of its faces?

By this, we mean that for each face $\gamma\in\Sigma$, the _star fan_
$\gamma_{\succeq}$ (Definition 2.5) should be a TPD space over $R$. We will
call this type of fans _local tropical Poincaré duality spaces over R_
(Definition 5.9), which is equivalent to the notion of tropical smoothness
defined by Amini and Piquerez [AP21] for $R=\mathbb{Z}$. Due to a self-
similarity property of the fans of matroids, they can also be shown to be
local TPD spaces.

Straddling the space between 1.1 and 1.2, we prove the following theorem,
which shows that when the stars of the faces of a fan are TPD spaces, so is
the whole fan, under some vanishing conditions on Borel–-Moore homology.

######  Theorem 5.4.

Let $R$ be a principal ideal domain, and $(\Sigma,w)$ be an $R$-balanced fan
of dimension $d\geq 2$, with $H_{q}^{BM}(\Sigma,\mathcal{F}_{p}^{R})=0$ for
$q\not=d$, for all $p$. If $(\gamma_{\succeq},w)$ satisfies TPD over $R$, for
each $\gamma\in\Sigma$ with $\gamma_{\succeq}\neq\Sigma$, then $(\Sigma,w)$
satisfies TPD over $R$.

Noticing the similarity of this result to the conditions for being a local TPD
space, we are led to the following characterization of local TPD spaces.

######  Theorem 5.10.

Let $R$ be a principal ideal domain, and $(\Sigma,w)$ a $d$-dimensional
$R$-balanced fan. Then $\Sigma$ is a local TPD space over $R$ if and only if
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$ for all $\gamma\in\Sigma$
and $q\neq d$, and for all faces $\beta$ of codimension 1, the star fans
$\beta_{\succeq}$ are TPD spaces over $R$.

Theorem 5.10 places conditions on codimension 1 faces, while Proposition 4.7
classifies the dimension 1 cases. When working over $\mathbb{Z}$, the Künneth
formula of [GS19] leads to the following corollary.

######  Corollary 5.11.

Let $(\Sigma,w)$ a $d$-dimensional $\mathbb{Z}$-balanced fan. Then $\Sigma$ is
a local TPD space over $\mathbb{Z}$ if and only if
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{\mathbb{Z}})=0$ for $q\neq d$
and all $\gamma\in\Sigma$, all the weights are $\pm 1$, and for all faces
$\beta$ of codimension 1, the star fans $\beta_{\succeq}$ are uniquely
$\mathbb{Z}$-balanced in the sense of Definition 3.11.

In the two-dimensional case, we use Theorem 5.4 to show that, assuming the
vanishing of parts of Borel–Moore homology, a fan is a TPD space if and only
if it is a local TPD space, see Proposition 5.7. This motivates two questions
for further investigation.

######  5.13 (Geometry of BM homology vanishing).

Let $(\Sigma,w)$ be an $R$-balanced $d$-dimensional fan. Can the fans with
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$ for each face
$\gamma\in\Sigma$, $q\neq d$ and all $p$ be geometrically characterized?

The second question pertains to the relationship between TPD and local TPD. In
the one-dimensional case, they are equal since the stars of edges are linear
spaces, which must satisfy TPD. Furthermore, in the two-dimensional case,
Proposition 5.7 shows that under the BM homology vanishing assumption, the
two-dimensional TPD spaces are exactly the local TPD spaces.

######  5.14 (Global versus Local TPD).

Let $(\Sigma,w)$ be an $R$-balanced fan which satisfies TPD over $R$. Does
$\gamma_{\succeq}$ also satisfy TPD over $R$ for each $\gamma\in\Sigma$?

In other words, can one find a TPD space $\Sigma$ with a face $\gamma$ so that
the star $\gamma_{\succeq}$ is not a TPD space, making $\Sigma$ not a local
TPD space? We have not been able to construct such a fan, which would have to
have dimension $>2$ by Proposition 5.7.

In the final part of this paper, we turn to generalizations for _rational
polyhedral spaces_ (Definition 6.1), for which we adopt the conventions of
[JRS18, JSS19]. These are topological spaces with charts to rational
polyhedral complexes. The cellular structure on the complexes gives a cellular
structure on the space (Definition 6.2). For a commutative ring $R$, an
_abstract tropical $R$-cycle_ (Definition 6.7) is a rational polyhedral space,
equipped with a cellular structure and an _$R$ -weight function_, such that
each codimension one face of the cellular structure is balanced. Abstract
tropical $R$-cycles can be equipped with tropical homology and cohomology
groups, as well as a cap product. _Tropical manifolds_ are spaces equipped
with charts to Bergman fans of matroids. Using their results for such fans,
[JSS19] and [JRS18] show that tropical manifolds are TPD spaces for
$\mathbb{R}$ and $\mathbb{Z}$ respectively, in the sense that the cap products
are isomorphisms. In [GS19], tropical manifolds are shown to be TPD spaces
over $\mathbb{Z}$ through Poincaré–Verdier duality. The local TPD spaces over
$\mathbb{Z}$ are the building blocks of the smooth tropical cycles as defined
in [AP21]. Tropical Poincaré duality on tropical manifolds is necessary for
defining the _tropical cycle class map_ as in [JRS18].

The Mayer–Vietoris arguments used in [JRS18] to show TPD on tropical manifolds
uses that matroidal fans are local TPD spaces. We carry through a similar
construction, assuming that the abstract tropical cycle is glued from local
TPD spaces. We say that an abstract tropical cycle is a _local TPD space over
$R$_ if the cellular structure is such that all the cells are local TPD spaces
over $R$. We then prove the following theorem.

######  Theorem 6.11.

Let $X$ be a local tropical Poincaré duality space over $R$. Then $X$
satisfies tropical Poincaré duality over $R$.

Recently, [AP20] establishes a full “Kähler package” for smooth projective
tropical cycles, working with rational coefficients. Moreover, they relate TPD
of the canonical compactifications of Bergman fans of matroids to the Poincaré
duality on the _Chow ring_ of a matroid established in [AHK18]. This is a ring
satisfying a version of Poincaré duality, the Hodge-Riemann bilinear
relations, and the hard Lefschetz theorem. The Chow ring of a matroid was
introduced in [AHK18] to establish the full Heron–Rota–Welsh conjecture: for
any matroid, the coefficients of the characteristic polynomial form a log-
concave sequence.

It is suggested in [Huh18] that such “Chow rings” satisfying the above three
properties, collectively dubbed the “Hodge package”, should be responsible for
the log-concavity of many sequences which arise in mathematics.

### Organization

In Section 2, we set conventions for fans, stars and integer weights. Then we
define cellular (co)sheaves and cellular (co)sheaf (co)homology.

In Section 3, we define the tropical multi-tangent cosheaves and sheaves,
which we use to define tropical (co)homology. This is used to describe a
generalized version of the balancing condition in tropical geometry, to
generalize beyond integer weights.

In Section 4, we define the TPD over a ring $R$, and give some necessary
conditions. Furthermore, in the case where $R$ is a field, we relate the TPD
over $R$ to an Euler characteristic condition. Moreover, we give a complete
classification in dimension one, and some criteria in codimension one of
$\mathbb{R}^{n}$ for TPD to hold, which forms a first step towards answering
1.1.

In Section 5, we turn to 1.2. We first relate TPD at the stars of faces to TPD
of the whole fan, which is then used to characterize local TPD spaces. We then
use our dimension one result to give a more geometric description of the
characterization.

Finally, in Section 6, we use local TPD spaces to construct abstract tropical
cycles satisfying tropical Poincaré duality.

### Acknowledgments

I wish to thank Kris Shaw for the many comments, ideas and discussions which
have made this article possible, as well as for supervising the master’s
thesis from which it is inspired. Thank you to Cédric Le Texier and Simen Moe
for many conversations and suggestions. I also would like to thank Omid Amini
and Matthieu Piquerez for sharing an early draft of their article [AP21] and
suggesting a new result. This research was supported by the Trond Mohn
Foundation project “Algebraic and Topological Cycles in Complex and Tropical
Geometries”.

##  2\. Preliminaries

In this section, we define and give references to the main objects and
concepts used in the remainder of the article. In Section 2.1, we introduce
some conventions for weighted fans and the balancing condition, and for
cellular sheaves and cosheaves in Section 2.2. Finally, we introduce notions
of homology and cohomology of cosheaves and sheaves in Section 2.3.

###  2.1. Cones, fans and stars

Let $N\cong\mathbb{Z}^{n}$ be a lattice, and
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}\cong\mathbb{R}^{n}$ be the
associated real vector space.

######  Definition 2.1.

A _rational polyhedral cone_ $\sigma$ in a lattice $N$ is a set of the form

$\sigma=\left\\{\textstyle{\sum_{i=1}^{m}}a_{i}v_{i}\mid
a_{i}\in\mathbb{Z}_{\geq 0}\right\\}\subset N$

for vectors $v_{i}\in N$, such that
$\sigma_{\mathbb{R}}=\sigma\otimes_{\mathbb{Z}}\mathbb{R}\subset
N_{\mathbb{R}}$ is closed and strictly convex, hence has a vertex at the
origin.

The _lattice_ $L_{\mathbb{Z}}(\sigma)$ is the saturated sublattice of $N$
generated by $\sigma$, and the _dimension_ of a cone is the rank of
$L_{\mathbb{Z}}(\sigma)$.

A _face_ of a cone is a subcone in the sense that one or several of the
coefficients $a_{i}$ are $0$. For $\tau$ a face of $\sigma$, the set
$L_{\mathbb{Z}}(\tau)\subset L_{\mathbb{Z}}(\sigma)$ is a sublattice. For
$\dim\tau=\dim\sigma-1$, we may select a _primitive integer vector_
$v_{\sigma/\tau}\in N$ such that
$L_{\mathbb{Z}}(\sigma)=L_{\mathbb{Z}}(\tau)+\mathbb{Z}v_{\sigma/\tau}$.

######  Definition 2.2.

A _rational polyhedral fan_ $\Sigma$ is a finite collection of rational
polyhedral cones in $N$ such that:

  * •

For any cone $\sigma\in\Sigma$, if $\tau$ is a face of $\sigma$, then
$\tau\in\Sigma$,

  * •

For $\sigma_{1},\sigma_{2}\in\Sigma$, the intersection
$\sigma_{1}\cap\sigma_{2}$ is a face of $\sigma_{1}$ and $\sigma_{2}$.

The cones in $\Sigma$ are also called _faces_ , and the collection of faces of
dimension $i$ is denoted by $\Sigma^{i}$. The _dimension of $\Sigma$_ is the
supremum of the dimensions of cones of $\Sigma$. We write $\tau\preceq\sigma$
if $\tau$ is a face of $\sigma$ and $\tau\prec\sigma$ if $\tau$ is a proper
face, which gives a partial ordering on $\Sigma$. We say that a face
$\sigma\in\Sigma$ is _maximal_ if it is maximal with respect to the ordering
$\preceq$. We will require that all fans are _pure dimensional_ in the sense
that all maximal by inclusion faces are of equal dimension.

Abusing notation, we also write $\Sigma$ for the category associated to the
partial ordering $\preceq$, whose objects are the cones $\sigma\in\Sigma$,
with a morphism $\tau\to\sigma\in\operatorname{Hom}_{\Sigma}(\tau,\sigma)$ if
and only if $\tau\preceq\sigma$.

Note that all cones intersect in a common minimal cell, and since we required
each cone to have a vertex, this is the unique vertex $v$ in $\Sigma$.
Moreover, a rational polyhedral fan corresponds to a _cell complex_ in the
sense of [She85, Cur14], when considering the fan as glued abstractly from the
interiors of the cones.

######  Example 2.3.

Consider the fan $\Sigma$ displayed in Figure 1, which consists of the rays
$\tau_{1},\tau_{2},\tau_{3},\tau_{4}$ and the vertex $v$. The fan is pure
dimensional, its maximal faces are the $\tau_{i}$ and it has dimension $1$. It
consists of the union of the line $x=0$ and $y=0$ when considered in
$N_{\mathbb{R}}$. We have $v\prec\tau_{i}$ for each $i$.

$v$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\tau_{4}$ Figure 1. The cross

Fans of particular interest in tropical geometry are the _Bergman fans_ of
matroids (see [Zha13, AK06] for definitions). These serve as the local models
of abstract tropical manifolds (see [MZ14, Section 1.6]).

######  Example 2.4.

Let $M$ be a matroid on $E=\\{0,\dots,n\\}$ with lattice of flats
$\mathscr{L}$, and let $\mathbb{Z}\\{e_{0},\dots,e_{n}\\}$ be the lattice of
rank $n+1$ generated by elements $e_{0},\dots,e_{n}$. Let $N$ be the quotient
defined by

${0}$${{\mathbb{Z}\\{e_{0}+\dots+e_{n}\\}}}$${{\mathbb{Z}\\{e_{0},\dots,e_{n}\\}}}$${N}$${0.}$$\scriptstyle{\pi}$

For any subset $S\subseteq E$, let $p_{S}=\sum_{i\in S}\pi(e_{i})$ in $N$, so
that in particular $p_{E}=0$. For any chain $F_{\bullet}$ of flats of the
matroid $M$,

$F_{\bullet}=\\{\emptyset\subsetneq F_{1}\subsetneq\dots\subsetneq
F_{k}\subsetneq E\\}\subseteq\mathscr{L}.$

the cone associated to $F_{\bullet}$ is the non-negative span

$\sigma(F_{\bullet})=\left\\{\sum_{i=1}^{k}a_{i}p_{F_{i}}\mid a_{i}\geq
0,\,i=1,\dots,k\right\\}.$

The Bergman fan of $M$ is the simplicial fan $\Sigma(M)$ consisting of cones
$\sigma(F_{\bullet})$ for all flags of flats $F_{\bullet}$.

The $U_{3,4}$ matroid on the set $E=\\{0,\dots,3\\}$ given by the rank
function $r\colon 2^{E}\to\mathbb{Z}_{\geq 0}$ taking values
$r(S)=\min(|S|,3)$ has the lattice of flats given by Example 2.4. The Bergman
fan of this matroid is shown in Example 2.4.

figureLattice of flats of the $U_{3,4}$ matroid.

figureBergman fan of the $U_{3,4}$ matroid.

######  Definition 2.5.

The _star_ $\gamma_{\succeq}$ at a face $\gamma\in\Sigma$ is the rational
polyhedral fan with underlying set
$\cup_{\gamma\preceq\kappa}\widetilde{\kappa}\subset N$, where
$\widetilde{\kappa}=\\{t(x-y)\mid t\in\mathbb{Z}_{\geq
0},x\in\kappa,y\in\gamma\\}\subseteq N$, subdivided into rational polyhedral
cones with a shared vertex.

The _cone_ at a face $\gamma\in\Sigma$ is the fan $\gamma_{\preceq}$
consisting of the faces $\kappa\in\Sigma$ for each $\kappa\preceq\gamma$. In
particular the vertex $v$ of $\Sigma$ is the minimal cell in each cone.

An _integer weight function_ on a rational polyhedral fan $\Sigma$ of
dimension $d$ is a function $w\colon\Sigma^{d}\to\mathbb{Z}$. We are
interested in weighted fans satisfying the usual tropical _balancing
condition_. This condition is equivalent to being a _Minkowski weight_ in the
sense of [FS97]. For more on the balancing condition, see for instance [MS15,
Definition 3.3.1] or [BIMS15, Definition 5.7].

######  Definition 2.6.

Let $\Sigma$ be a rational polyhedral fan of dimension $d$ with weights
$w\colon\Sigma^{d}\to\mathbb{Z}$. We say that $\Sigma$ together with $w$ is
_balanced_ at a face $\beta\in\Sigma^{d-1}$ if

$\sum_{\beta\prec\alpha}w(\alpha)v_{\alpha/\beta}\in L_{\mathbb{Z}}(\beta),$

using the notation from Definition 2.1. We say $\Sigma$ together with $w$ is
_balanced_ if it is balanced along each face $\beta\in\Sigma^{d-1}$.

######  Example 2.7.

The fan discussed in Example 2.3 and shown in Figure 1 is balanced, for a
given weight function $w\colon\Sigma^{d}\to\mathbb{Z}$, if and only if
$w(\tau_{1})=w(\tau_{3})$ and $w(\tau_{2})=w(\tau_{4})$.

######  Example 2.8.

It follows from [AK06, Proposition 2], that the stars $\gamma_{\succeq}$ of
faces $\gamma$ in the Bergman fans of a matroid are themselves Bergman fans of
matroids. It is shown in [AHK18, Proposition 5.2] that, for the Bergman fan
$\Sigma(M)$ of a matroid $M$, the only weight functions which satisfy the
balancing condition are the constant ones. The uniqueness of such a weight
function follows from tropical Poincaré duality in [JRS18, Proposition 5.5]
and the earlier [Huh14, Theorem 38]. By our later Definition 3.11, this will
mean that these fans are _uniquely $\mathbb{Z}$-balanced_.

###  2.2. Cellular sheaves and cellular cosheaves

One can define _cellular sheaves_ and _cellular cosheaves_ of modules on a
polyhedral fan:

######  Definition 2.9.

Let $R$ be a commutative ring, $\Sigma$ a rational polyhedral fan. Then:

  * •

A _cellular $R$-sheaf_ $\mathcal{G}$ is a functor
$\mathcal{G}\colon\Sigma\to\mathbf{Mod}_{R}$.

  * •

A _cellular $R$-cosheaf_ $\mathcal{F}$ is a functor
$\mathcal{F}\colon\Sigma^{\mathrm{op}}\to\mathbf{Mod}_{R}$.

A morphism of sheaves or cosheaves is simply a natural transformation of
functors or contravariant functors, respectively.

######  Remark 2.10.

The category $\Sigma$, when viewed as a set, can be given the _Alexandrov
topology_ , such that cellular sheaves and cosheaves in fact are sheaves and
cosheaves with respect to this topology. For more on cellular sheaves and
cosheaves, see [Cur14].

We have considered the fan $\Sigma$ as a category with morphisms
$\tau\to\sigma$ whenever $\tau$ is a face of $\sigma$, so that a sheaf
$\mathcal{G}$ induces a map $\mathcal{G}(\tau)\to\mathcal{G}(\sigma)$, and a
cosheaf $\mathcal{F}$ induces a map $\mathcal{F}(\sigma)\to\mathcal{F}(\tau)$.
This convention is in agreement with [Cur14, She85], but reversed from [Bri97,
How08] in the sense that their _sheaves_ are our _cosheaves_ , and vice versa.

######  Example 2.11.

Let $\Sigma$ be a rational polyhedral fan. For a module $M$ over a ring $R$,
the _constant cosheaf $M^{\Sigma}$ with values in $M$_ is the cosheaf defined
as a functor $M^{\Sigma}\colon\Sigma^{\mathrm{op}}\to\mathbf{Mod}_{R}$ taking
all objects to $M$ and all morphisms to $\operatorname{id}_{M}$.

Similarly, the _constant sheaf $M_{\Sigma}$ with values in $M$_ is the sheaf
defined as a functor $M_{\Sigma}\colon\Sigma\to\mathbf{Mod}_{R}$ taking all
objects to $M$ and all morphisms to $\operatorname{id}_{M}$.

###  2.3. Cellular homology and cohomology

Considering the fan $\Sigma$ as a subset of $N_{\mathbb{R}}$, we select an
orientation for each cone $\sigma\in\Sigma$. For each $\tau\prec\sigma$ such
that $\dim(\tau)=\dim(\sigma)-1$, we keep track of the relative orientations
by writing $\mathcal{O}(\tau,\sigma)=1$ if the restriction of the orientation
of $\sigma$ to $\tau$ coincides with the orientation of $\tau$, and
$\mathcal{O}(\tau,\sigma)=-1$ if it reverses it. In the two next definitions,
we use the orientation $\mathcal{O}(\tau,\sigma)=\pm 1$ to construct certain
(co)chain complexes for a given (co)sheaf. These definitions are equal to the
ones in [Cur14, She85, KSW17], and reversed from [Bri97, How08], who index by
codimension.

######  Definition 2.12.

Given a cellular sheaf $\mathcal{G}$, the _cellular cochain groups_ and
_cellular cochain groups with compact support_ are defined, respectively, by

$C^{q}(\Sigma,\mathcal{G})\coloneqq\bigoplus_{\begin{subarray}{c}\sigma\in\Sigma^{q}\\\
\sigma_{\mathbb{R}}\;\text{compact}\end{subarray}}\mathcal{G}(\sigma)\quad\text{and}\quad
C_{c}^{q}(\Sigma,\mathcal{G})\coloneqq\bigoplus_{\sigma\in\Sigma^{q}}\mathcal{G}(\sigma),$

for $q\geq 0$, where $\sigma_{\mathbb{R}}$ is as in Definition 2.1. The
cellular cochain maps

$d^{q}\colon C^{q}(\Sigma,\mathcal{G})\to
C^{q+1}(\Sigma,\mathcal{G})\quad\text{and}\quad d^{q}\colon
C_{c}^{q}(\Sigma,\mathcal{G})\to C_{c}^{q+1}(\Sigma,\mathcal{G})$

are given component-wise for $\tau\in\Sigma^{q}$ and $\sigma\in\Sigma^{q+1}$
with $\tau\prec\sigma$ by
$d_{\tau\sigma}\colon\mathcal{G}(\tau)\to\mathcal{G}(\sigma)$, where

$d_{\tau\sigma}\coloneqq\mathcal{O}(\tau,\sigma)\mathcal{G}(\tau\to\sigma).$

If $\tau\not\preceq\sigma$, we let the map $d_{\tau\sigma}$ be $0$.

The cohomology groups $H^{\bullet}(\Sigma,\mathcal{G})$ and
$H_{c}^{\bullet}(\Sigma,\mathcal{G})$ of these complexes are the _cellular
sheaf cohomology_ and _cellular sheaf cohomology with compact support_ with
respect to the sheaf $\mathcal{G}$.

######  Definition 2.13.

Given a cellular cosheaf $\mathcal{F}$, the _cellular chain group_ and
_Borel–Moore cellular chain groups_ are defined, respectively, by

$C_{q}(\Sigma,\mathcal{F})\coloneqq\bigoplus_{\begin{subarray}{c}\sigma\in\Sigma^{q}\\\
\sigma_{\mathbb{R}}\;\text{compact}\end{subarray}}\mathcal{F}(\sigma)\quad\text{and}\quad
C_{q}^{BM}(\Sigma,\mathcal{F})\coloneqq\bigoplus_{\begin{subarray}{c}\sigma\in\Sigma^{q}\end{subarray}}\mathcal{F}(\sigma),$

for $q\geq 0$, where $\sigma_{\mathbb{R}}$ is as in Definition 2.1. The
cellular chain maps

$\partial_{q}\colon C_{q}(\Sigma,\mathcal{F})\to
C_{q-1}(\Sigma,\mathcal{F})\quad\text{and}\quad\partial_{q}\colon
C_{q}^{BM}(\Sigma,\mathcal{F})\to C_{q-1}^{BM}(\Sigma,\mathcal{F})$

are given component-wise for $\sigma\in\Sigma^{q}$ and $\tau\in\Sigma^{q-1}$
by $\partial_{\sigma\tau}\colon\mathcal{F}(\sigma)\to\mathcal{F}(\tau)$, where

$\partial_{\sigma\tau}\coloneqq\mathcal{O}(\tau,\sigma)\mathcal{F}(\sigma\to\tau).$

If $\tau\not\preceq\sigma$, we let the map $\partial_{\sigma\tau}$ be $0$.

The homology groups $H_{\bullet}(\Sigma,\mathcal{F})$ and
$H_{\bullet}^{BM}(\Sigma,\mathcal{F})$ of these complexes are the _cellular
cosheaf homology_ and _cellular Borel–Moore cosheaf homology_ with respect to
$\mathcal{F}$.

Proofs that the cellular (co)chain groups and maps defined above form
(co)chain complexes can be found in [Cur14, Definitions 6.2.6-7] and [She85,
Theorem 1.1.3].

######  Remark 2.14.

The above definitions of cellular cohomology work in the more general setting
of polyhedral complexes. Since we are working with pointed polyhedral fans,
the unique compact cell is the vertex $v$. Then, for any sheaf $\mathcal{G}$
on a fan $\Sigma$, the cellular cochain groups $C^{q}(\Sigma,\mathcal{G})$ are
trivial for $q>0$, and therefore:

$\displaystyle H^{q}(\Sigma,\mathcal{G})=\begin{cases*}\mathcal{G}(v)&for
$q=0$,\\\ 0&otherwise.\end{cases*}$

Similarly, for any cosheaf $\mathcal{F}$, the cellular chain groups
$C_{q}(\Sigma,\mathcal{F})$ are trivial for $q>0$, thus:

$\displaystyle H_{q}(\Sigma,\mathcal{F})=\begin{cases*}\mathcal{F}(v)&for
$q=0$,\\\ 0&otherwise.\end{cases*}$

######  Example 2.15.

Consider the fan from Example 2.3, with orientations chosen so that
$\mathcal{O}(v,\tau_{i})=1$ for all $i$. The Borel–Moore homology of the
constant cosheaf $\mathbb{Z}^{\Sigma}$ is the homology of the complex

${0}$${{\mathbb{Z}^{4}}}$${\mathbb{Z}}$${0,}$$\scriptstyle{\partial_{1}}$

where the matrix $\partial_{1}$ is indexed by the $\tau_{i}$ and given by

$\partial_{1}=(\mathcal{O}(v,\tau_{i})\operatorname{id}_{\mathbb{Z}})_{\tau_{i}\in\Sigma^{1}}=(1\;,1\;,1\;,1).$

The Borel–Moore homology becomes
$H_{1}^{BM}(\Sigma,\mathbb{Z}^{\Sigma})=\mathbb{Z}^{3}$ and
$H_{0}^{BM}(\Sigma,\mathbb{Z}^{\Sigma})=0$.

######  Lemma 2.16.

Let $R$ be a ring, $M$ an $R$-module. For any face $\tau\in\Sigma$ distinct
from the unique vertex $v$, the cochain complex with compact support of the
polyhedral sub-fan $\tau_{\preceq}$ from Definition 2.5 in a constant sheaf
$M_{\Sigma}$ has no cohomology, i.e. $H_{c}^{q}(\tau_{\preceq},M_{\Sigma})=0$
for all $q$.

###### Proof.

Whenever the dimension of $\tau$ is greater than or equal to 1, the compact
support cochain complex $C_{c}^{\bullet}(\tau_{\preceq},M_{\Sigma})$ is
equivalent to a complex whose cohomology is the reduced $M$-cohomology (up to
a shift) of a polytope $P$ over which $\tau$ is a cone. Since $P$ is
contractible, all these cohomology groups are 0. ∎

##  3\. Tropical geometry of fans

In this section, we introduce particular cellular (co)sheaves on fans which
are of interest in tropical geometry. After examining some properties of the
resulting _tropical (co)homology_ , we use this to define the _balancing_
condition in tropical geometry. Finally, we define the _tropical cap product_
associated to a balancing of the fan. We then introduce particular sheaves of
interest in tropical geometry. Next, we generalize the balancing condition on
fans to weights in arbitrary rings, which finally leads to a treatment of
tropical Poincaré duality over arbitrary commutative rings.

###  3.1. Tropical sheaves and cosheaves

For tropical (co)homology, the following sheaves and cosheaves are of
interest.

######  Definition 3.1 ([IKMZ19, Definition 13]).

Let $\Sigma$ be a fan of dimension $d$ in $N$. For $\sigma\in\Sigma$, let
$L_{\mathbb{Z}}(\sigma)$ be the lattice of integer points parallel to the cone
$\sigma$. For $p=1,\dots,d$, the _$p$ -th multi-tangent cosheaf_
$\mathcal{F}_{p}^{\mathbb{Z}}$ is the cellular $\mathbb{Z}$-cosheaf defined by
the data:

  * •

For $\sigma\in\Sigma$, one has
$\mathcal{F}_{p}^{\mathbb{Z}}(\sigma)\coloneqq\sum_{\sigma\preceq\gamma}\bigwedge^{p}L_{\mathbb{Z}}(\gamma)\subseteq\bigwedge^{p}N$.

  * •

For $\tau\preceq\sigma$, the morphism
$(\sigma\to\tau)\in\operatorname{Hom}_{\Sigma^{\mathrm{op}}}(\sigma,\tau)$
becomes the map
$\iota_{\sigma,\tau}\colon\mathcal{F}_{p}^{\mathbb{Z}}(\sigma)\to\mathcal{F}_{p}^{\mathbb{Z}}(\tau)$,
which is induced by the natural inclusion.

In the $p=0$ case, we define
$\mathcal{F}_{0}^{\mathbb{Z}}=\mathbb{Z}^{\Sigma}$, with all maps being the
identity.

Furthermore, the cellular cosheaf $\mathcal{F}_{p}^{\mathbb{Z}}$ also gives
rise to a cellular sheaf $\mathcal{F}_{\mathbb{Z}}^{p}$ which is defined by
$\mathcal{F}_{\mathbb{Z}}^{p}(\sigma)\coloneqq\mathcal{F}_{p}^{\mathbb{Z}}(\sigma)^{*}$,
with morphisms
$\rho_{\tau,\sigma}\colon\mathcal{F}_{\mathbb{Z}}^{p}(\tau)\to\mathcal{F}_{\mathbb{Z}}^{p}(\sigma)$
defined by dualizing
$\iota_{\sigma,\tau}\colon\mathcal{F}_{p}^{\mathbb{Z}}(\sigma)\to\mathcal{F}_{p}^{\mathbb{Z}}(\tau)$.

######  Definition 3.2.

Following [JRS18], for any ring $R$, we define a cosheaf $\mathcal{F}_{p}^{R}$
by taking the tensor product
$\mathcal{F}_{p}^{R}(\sigma)=\mathcal{F}_{p}^{\mathbb{Z}}(\sigma)\otimes_{\mathbb{Z}}R$,
giving an $R$-module, and tensoring the maps as well. Dualizing yields a sheaf
$\mathcal{F}_{R}^{p}$.

######  Remark 3.3.

For any $\mathbb{Z}$-module $M$ and commutative ring $R$, the product
$M_{R}\coloneqq M\otimes_{\mathbb{Z}}R$ is an $R$-module. Moreover, by [Bou98,
Proposition III.7.5.8], we have
$\bigwedge^{p}M_{R}\cong(\bigwedge^{p}M)\otimes_{\mathbb{Z}}R$. In particular,
for $\sigma\in\Sigma$ a maximal face of a $d$-dimensional fan,
$L_{R}(\sigma)\coloneqq L_{\mathbb{Z}}(\sigma)\otimes_{\mathbb{Z}}R$ is a free
$R$-module of dimension $d$, and
$\mathcal{F}_{R}^{p}(\sigma)=(\bigwedge^{p}L_{\mathbb{Z}}(\sigma))\otimes_{\mathbb{Z}}R\cong\bigwedge^{p}L_{R}(\sigma)$.

######  Remark 3.4.

Let $\Sigma$ be a fan of dimension $d$. For any $\alpha\in\Sigma^{d}$, by
Remark 3.3 we have
$\mathcal{F}_{d}^{R}(\alpha)=\bigwedge^{d}L_{R}(\alpha)\cong R$. Given a
choice of orientation for $\alpha$, we can select the unique generator
$\Lambda_{\alpha}\in\mathcal{F}_{d}^{\mathbb{Z}}(\alpha)=\bigwedge^{p}L_{\mathbb{Z}}(\sigma)$
compatible with the chosen orientation, and abusing notation, we let
$\Lambda_{\alpha}\in\mathcal{F}_{d}^{R}(\alpha)\cong(\bigwedge^{p}L_{\mathbb{Z}}(\sigma))\otimes_{\mathbb{Z}}R$
denote the corresponding element $\Lambda_{\alpha}\otimes
1_{R}\in\mathcal{F}_{d}^{R}(\alpha)$.

######  Definition 3.5.

The cochain complex $(C^{\bullet}(\Sigma,\mathcal{F}_{R}^{p}),\delta)$ from
Definition 2.12 has cohomology groups $H^{q}(\Sigma,\mathcal{F}_{R}^{p})$
which are called the (cellular) _tropical cohomology groups with
$R$-coefficients_ of $\Sigma$. Moreover, the cohomology groups
$H_{c}^{q}(\Sigma,\mathcal{F}_{R}^{p})$ of the complex
$(C_{c}^{\bullet}(\Sigma,\mathcal{F}_{R}^{p}),\delta)$ are called the
(cellular) _compact support tropical cohomology groups with $R$-coefficients_
of $\Sigma$.

Similarly, the chain complex
$(C_{\bullet}(\Sigma,\mathcal{F}_{p}^{R}),\partial)$ from Definition 2.13 has
homology groups $H_{q}(\Sigma,\mathcal{F}_{p}^{R})$ which are called the
(cellular) _tropical homology groups with $R$-coefficients_ of $\Sigma$.
Finally, the (cellular) _tropical Borel–Moore homology groups with
$R$-coefficients_ $H_{q}^{BM}(\Sigma,\mathcal{F}_{p}^{R})$ are the homology
groups of the chain complex
$(C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{p}^{R}),\partial)$.

######  Proposition 3.6.

The tropical cohomology with $R$-coefficients of any fan $\Sigma$ is

$\displaystyle
H^{q}(\Sigma,\mathcal{F}_{R}^{p})=\begin{cases*}\mathcal{F}_{R}^{p}(v)&for
$q=0$,\\\ 0&otherwise,\end{cases*}$

where $v\in\Sigma$ is the vertex.

###### Proof.

This follows from Remark 2.14. ∎

######  Example 3.7.

Consider again the 1-dimensional fan from Example 2.3. Since it is of
dimension $1$, the only $\mathcal{F}_{R}^{p}$ sheaves are
$\mathcal{F}_{R}^{0}\cong R_{\Sigma}$ and $\mathcal{F}_{R}^{1}$. By
Proposition 3.6, the only non-zero cohomology groups are
$H^{0}(\Sigma,\mathcal{F}_{R}^{0})=R$ and
$H^{0}(\Sigma,\mathcal{F}_{R}^{1})=\mathcal{F}_{R}^{1}(v)=R^{2}$.

Similarly, the only $\mathcal{F}_{p}^{R}$ cosheaves are
$\mathcal{F}_{0}^{R}\cong R^{\Sigma}$ and $\mathcal{F}_{1}^{R}$. The
computation of the homology with the constant cosheaf $\mathbb{Z}^{\Sigma}$
given in Example 2.15 carries through to $R^{\Sigma}$, giving
$H_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R})=R^{3}$ and
$H_{0}^{BM}(\Sigma,\mathcal{F}_{0}^{R})=0$. Finally, to compute the
Borel–Moore homology for $\mathcal{F}_{1}^{R}$, we have the chain complex

${0}$${\oplus_{\tau_{i}\in\Sigma^{1}}\mathcal{F}_{1}^{R}(\tau_{i})}$${\mathcal{F}_{1}^{R}(v)}$${0.}$$\scriptstyle{\partial_{1}}$

Selecting the $\mathbb{Z}$-basis $e_{1}=(1,0)$, $e_{2}=(0,1)$ for $N$, we can
write this complex as

${0}$${\langle(1,0)\rangle\oplus\langle(0,1)\rangle\oplus\langle(-1,0)\rangle\oplus\langle(0,-1)\rangle}$${\langle(1,0),(0,1)\rangle}$${0,}$$\scriptstyle{\partial_{1}}$

where $\partial_{1}$ is now the direct sum of the inclusion maps, and
everything is suitably tensored with $R$. The Borel–Moore homology can then be
shown to be given by $H_{0}^{BM}(\Sigma,\mathcal{F}_{R}^{1})=0$ and
$H_{1}^{BM}(\Sigma,\mathcal{F}_{R}^{1})=\langle(a,b,a,b)\mid a,b\in
R\rangle\cong R^{2}$.

Recall from Definition 2.5 that one must subdivide the stars
$\gamma_{\succeq}$ of faces $\gamma\in\Sigma$ to obtain a fan structure. The
next proposition shows that the tropical cohomology of $\gamma_{\succeq}$ is
determined directly by $\mathcal{F}_{p}^{R}(\gamma)$, and that the tropical
Borel–Moore homology can be computed using a simpler complex than the one
coming from the subdivision.

######  Proposition 3.8.

Let $\Sigma$ be a fan and $\gamma\in\Sigma$ a face of dimension $r$. Let
$\mathcal{F}_{R,\Sigma}^{p}$ and $\mathcal{F}_{R,\gamma}^{p}$ denote the
$p$-th multi-tangent sheaves on $\Sigma$ and $\gamma_{\succeq}$ respectively.
Then

$H^{0}(\gamma_{\succeq},\mathcal{F}_{R,\gamma}^{p})\cong\mathcal{F}_{R,\Sigma}^{p}(\gamma).$

Similarly, let $\mathcal{F}_{p}^{R,\Sigma}$ and $\mathcal{F}_{p}^{R,\gamma}$
denote the $p$-th multi-tangent cosheaves on $\Sigma$ and $\gamma_{\succeq}$
respectively. Then the Borel–Moore homology
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R,\gamma})$ is isomorphic to the
homology of the complex

${0}$${{\bigoplus\limits_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\gamma\end{subarray}}\mathcal{F}_{p}^{R,\Sigma}(\alpha)}}$${\cdots}$${{\bigoplus\limits_{\begin{subarray}{c}\kappa\in\Sigma^{r+1}\\\
\kappa\succ\gamma\end{subarray}}\mathcal{F}_{p}^{R,\Sigma}(\kappa)}}$${{\mathcal{F}_{p}^{R,\Sigma}(\gamma)}}$${0,}$$\scriptstyle{\partial_{q}|_{\kappa\succ\gamma}}$$\scriptstyle{\partial_{q}|_{\alpha\succ\gamma}}$

where we define ${\partial_{q}^{\gamma}}=\oplus\partial_{\sigma\tau}$ with the
sum taken over all $\sigma,\tau\succeq\gamma$, $\sigma\in\Sigma^{q}$ and
$\tau\in\Sigma^{q-1}$.

###### Proof.

First, by Definition 2.5, we must choose a subdivision of the space with
support given by the cones $\widetilde{\kappa}=\\{t(x-y)\mid t\geq
0,x\in\kappa,y\in\gamma\\}$ for each $\kappa\succeq\gamma$, and we will have
only one compact cell given by the created vertex
$\widetilde{v}\in\gamma_{\succeq}$. By Remark 2.14, we have
$H^{0}(\gamma_{\succeq},\mathcal{F}_{R,\gamma}^{p})=\mathcal{F}_{R,\gamma}^{p}(\widetilde{v})$.
Then, observe that for each $\kappa\succ\gamma$, the lattice is unchanged in
the sense that $L_{\mathbb{Z}}(\widetilde{\kappa})=L_{\mathbb{Z}}(\kappa)$ as
subspaces of $N$, and each maximal dimensional face $\alpha$ in the
subdivision of $\gamma_{\succeq}$ is a subspace of a $\widetilde{\kappa}$. In
particular, this implies that
$\mathcal{F}_{R,\gamma}^{p}(\widetilde{v})\cong\mathcal{F}_{R,\Sigma}^{p}(\gamma)$.

Next, observe that the Borel–Moore homology is the equal to the regular
homology of the fan when seen as a subset of the one-point compactification
$N\cup\\{\infty\\}$ of the ambient lattice $N$. Then every cone
$\sigma\in\gamma_{\succeq}$ becomes a disk $\sigma_{\infty}$ in
$N\cup\\{\infty\\}$, and we have a CW complex structure on the underlying set
of $\gamma_{\succeq}\cup\\{\infty\\}$. Then, similarly to [MZ14, Section 2.2]
and [JRS18, Remark 2.8], note that:

$C_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=\oplus_{\sigma\in\gamma_{\succeq}^{q}}\mathcal{F}_{p}^{R}(\sigma)=\oplus_{\sigma\in\gamma_{\succeq}^{q}}H_{q}(\sigma_{\infty},\partial(\sigma_{\infty}),\mathcal{F}_{p}^{R}(\sigma)),$

where the right hand side becomes the cellular homology with coefficients in
the local system induced by $\mathcal{F}_{p}^{R}$ of the CW complex
$\gamma_{\succeq}\cup\\{\infty\\}$. This can be computed with an arbitrary CW
structure. Therefore, the given complex, which is the local system homology of
the CW structure induced on $|\gamma_{\succeq}\cup\\{\infty\\}|$ by taking the
non-subdivided cell structure of $\gamma_{\succeq}$ will compute the
Borel–Moore homology of the $\mathcal{F}_{p}^{R}$ cosheaf. ∎

######  Remark 3.9.

The above proposition shows that, when working with stars of faces in a fan,
the particular cellular structure does not change the tropical (co)homology.
This is not the case for general (co)sheaves, see for instance the _wave
tangent sheaves_ defined in [MZ14, Section 3].

###  3.2. Balancing in tropical geometry

It was observed in [JRS18, Remark 4.9] and [MZ14, Proposition 4.3] that the
balancing condition from Definition 2.6 can be equivalently formulated as the
condition that a particular tropical Borel–Moore chain is closed. In this
subsection, we use this observation to form a generalization of the balancing
condition to arbitrary commutative rings.

######  Definition 3.10.

Let $R$ be a ring. An _$R$ -weight function_ $w\colon\Sigma^{d}\to R$ on a
$d$-dimensional fan $\Sigma$ is a function such that for all
$\alpha\in\Sigma^{d}$, the weight $w(\alpha)$ is not a zero-divisor. A pair
$(\Sigma,w)$ will be called an _$R$ -weighted fan_.

######  Definition 3.11.

An $R$-weighted fan $(\Sigma,w)$ is _$R$ -balanced_ if _the fundamental chain
$\mathrm{Ch}(\Sigma,w)$_ given by

$\mathrm{Ch}(\Sigma,w)\coloneqq(w(\alpha)\Lambda_{\alpha})_{\alpha\in\Sigma^{d}}\in
C_{d}^{BM}(\Sigma,\mathcal{F}_{d})$

is a cycle, where the $\Lambda_{\alpha}$ are chosen as in Remark 3.4. In this
case, we have $H_{d}^{BM}(\Sigma,\mathcal{F}_{d}^{R})\not=0$, together with an
induced _fundamental class_

$[\Sigma,w]=[\mathrm{Ch}(\Sigma,w)]\in
H_{d}^{BM}(\Sigma,\mathcal{F}_{d}^{R}).$

If $H_{d}^{BM}(\Sigma,\mathcal{F}_{d}^{R})=\langle[\Sigma,w]\rangle\cong R$,
we say that $\Sigma$ is _uniquely $R$-balanced_ by $w$.

The above definition, which is equivalent to the usual balancing condition
[BIMS15, Definition 5.8], is similar in flavor to the description given by
[BH17, Theorem 2.9]. We illustrate this with the following example.

######  Example 3.12.

In this example, we explicitly relate the above definition of balancing to the
one given in Definition 2.6. Let $(\Sigma,w)$ be a $\mathbb{Z}$-balanced fan
of dimension $d$ in the sense of Definition 3.11. Then, for each
$\beta\in\Sigma^{d-1}$, we pick a generator $\Lambda_{\beta}\in
L_{\mathbb{Z}}(\beta)$ respecting the orientation. Now for each
$\alpha\in\Sigma^{d}$ with $\beta\prec\alpha$, the vector $v_{\alpha/\beta}$
from Definition 2.6 is such that $\Lambda_{\alpha}=\Lambda_{\beta}\wedge
v_{\alpha/\beta}$.

Looking at the $\beta$-component of $\partial\colon
C_{d}^{BM}(\Sigma,\mathcal{F}_{d}^{\mathbb{Z}})\to
C_{d-1}^{BM}(\Sigma,\mathcal{F}_{d}^{\mathbb{Z}})$, we have

$\partial((w(\alpha)\Lambda_{\alpha}))_{\beta}=\sum_{\beta\prec\alpha}w(\alpha)\Lambda_{\alpha}=\Lambda_{\beta}\wedge\sum_{\beta\prec\alpha}w(\alpha)v_{\alpha/\beta}.$

Therefore, the chain $\mathrm{Ch}(\Sigma,w)$ is closed if and only if each of
the faces $\beta$ are balanced in the sense of Definition 2.6. Thus the
definitions are equivalent.

######  Proposition 3.13.

Let $(\Sigma,w)$ be an $R$-balanced fan and $\gamma\in\Sigma$ a face. Then
$(\gamma_{\succeq},w)$ is $R$-balanced, where $w$ is understood to be the
weight function induced on $\gamma_{\succeq}$ by $w$.

###### Proof.

By Proposition 3.8, we have that
$H_{d}^{BM}(\gamma_{\succeq},\mathcal{F}_{d}^{R})$ can be viewed as the kernel
of the map

${\bigoplus_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\gamma\end{subarray}}\mathcal{F}_{d}^{R}(\alpha)}\xrightarrow{\partial_{d}^{\gamma}}{\bigoplus_{\begin{subarray}{c}\beta\in\Sigma^{d-1}\\\
\beta\succ\gamma\end{subarray}}\mathcal{F}_{d}^{R}(\beta)}.$

Moreover, the class

$\mathrm{Ch}(\gamma_{\succeq},w)=(w(\alpha)\Lambda_{\alpha})_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\gamma\end{subarray}}$

is a cycle since $\mathrm{Ch}(\Sigma,w)$ is a cycle in
$C_{d}^{BM}(\Sigma,\mathcal{F}_{d}^{R})$. Thus $(\gamma_{\succeq},w)$ is
$R$-balanced and we have a fundamental class $[\gamma_{\succeq},w]\in
H_{d}^{BM}(\gamma_{\succeq},\mathcal{F}_{d}^{R})$. ∎

###  3.3. Tropical cap product

There is a cap product relating $H^{q}(\Sigma,\mathcal{F}_{R}^{p})$ to
$H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$, which will be at the core of
tropical Poincaré duality. We extend the version given in [JRS18, Definition
4.10] for $R=\mathbb{Z}$ to arbitrary commutative rings $R$, using the
contraction map from multilinear algebra for a general ring $R$, as developed
in [Bou98, Section III.11].

######  Definition 3.14 ([Bou98, Section III.11.9]).

Let $M_{R}$ be a rank $d$ free $R$-module, $M_{R}^{*}$ the dual module, and
$0\leq p_{1}\leq p_{2}\leq d$. The _interior product_ or _contraction_ defined
by $y=y_{1}\wedge\dots\wedge y_{p^{\prime}}\in\bigwedge^{p_{2}}M_{R}$ is the
map

$\displaystyle\lrcorner\,y\colon\bigwedge^{p_{1}}M_{R}^{*}$
$\displaystyle\to\bigwedge^{p_{2}-p_{1}}M_{R},$

which is defined on $x=x_{1}\wedge\dots\wedge
x_{p}\in\bigwedge^{p_{1}}M_{R}^{*}$ to be

$x\,\lrcorner\,y=(-1)^{p_{1}(p_{1}-1)/2}\sum_{a}\mathrm{sign}(a)\left(\prod_{i=1}^{p_{1}}x_{i}(y_{a(i)})\right)y_{a(p+1)}\wedge\dots\wedge
y_{a(d)},$

where the sum is taken over all permutations $a\in S_{d}$ which are increasing
on $1,\dots,p$ and $p+1,\dots,d$, and is extended linearly.

######  Remark 3.15.

In [Bou98, Section III.11.10], an explicit formula for this contraction map is
given in terms of bases. Letting $e_{1},\dots,e_{m}$ be a basis of $M_{R}$,
the elements $e_{I}\coloneqq e_{i_{1}}\wedge\dots\wedge
e_{i_{p_{2}}}\in\bigwedge^{p_{2}}M_{R}$, for all
$I=\\{i_{1}<\dots<i_{p_{2}}\\}\subseteq[m]$ ordered strictly increasing
subsets of size $p$, form a basis of $\bigwedge^{p_{2}}M_{R}$. Letting
$f_{1},\dots,f_{m}$ be the dual basis to the $e_{i}$ for $M_{R}^{*}$, the
elements $f_{J}$, for all $J=\\{j_{1}<\dots<j_{p_{1}}\\}\subseteq[m]$, form a
basis of $\bigwedge^{p_{1}}M_{R}^{*}$. Then the contraction map defined by
$e_{J}$ is given by

$\begin{cases*}f_{K}\,\lrcorner\,e_{J}=0&if $K\not\subset J$,\\\
f_{K}\,\lrcorner\,e_{J}=(-1)^{v+p_{1}(p_{1}-1)/2}\;e_{J\setminus K}&if
$K\subseteq J$ and $p_{1}=|K|$,\end{cases*}$

where $v$ is the number of ordered pairs $(\lambda,\mu)\in K\times(J\setminus
K)$ such that $\lambda>\mu$.

A proof that these contraction maps are the same as the formulation in terms
of compositions given in [AP21, Aks19] follows from the arguments given in
[Aks19, Lemma 4.1.4].

######  Definition 3.16.

For each facet $\alpha\in\Sigma^{d}$ of a $d$-dimensional $R$-balanced fan
$\Sigma$, we have chosen a generator
$\Lambda_{\alpha}\in\mathcal{F}_{d}^{R}(\alpha)=\bigwedge^{d}L_{R}(\alpha)\cong
R$ by Remark 3.4, and a weight $w(\alpha)\in R$ which is not a zero-divisor.
The _contraction_ defined by $w(\alpha)\Lambda_{\alpha}$ is the map

$\lrcorner\,w(\alpha)\Lambda_{\alpha}\colon\bigwedge^{p}L_{R}(\alpha)^{*}\to\bigwedge^{d-p}L_{R}(\alpha).$

Since $w(\alpha)$ is not a zero-divisor, Remark 3.15 shows that this map is
injective. It is an isomorphism if and only if $w(\alpha)$ is a unit.

######  Definition 3.17.

Let the weighted fan $(\Sigma,w)$ be an $R$-balanced fan of dimension $d$. The
_cap product $\frown\mathrm{Ch}(\Sigma,w)$_ with the fundamental chain of
$\Sigma$ is the map given by:

$\displaystyle\frown\mathrm{Ch}(\Sigma,w)\colon
C^{q}(\Sigma,\mathcal{F}_{R}^{p})$ $\displaystyle\to
C_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$
$\displaystyle(u_{\gamma})_{\gamma\in\Sigma^{q}}$
$\displaystyle\mapsto\left(\sum_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\gamma,\tau\preceq\alpha\end{subarray}}w(\alpha)\iota_{\alpha,\tau}\left(\rho_{\gamma,\alpha}(u_{\gamma})\,\lrcorner\,\Lambda_{\alpha}\right)\right)_{\tau\in\Sigma^{d-q}}$

where $\Lambda_{\alpha}$ is as in Remark 3.4.

######  Remark 3.18.

For any chain $\sigma\in C_{q}^{BM}(\Sigma,\mathcal{F}_{p}^{R})$, a cap
product $\frown\sigma$ can be similarly defined. It is noted in [JRS18, p. 13]
that the Leibniz formula holds for these cap product, such that
$\partial(\alpha\frown\sigma)=(-1)^{q+1}(d(\alpha)\frown\sigma-\alpha\frown\partial(\sigma))$.
In the case where $R=\mathbb{R}$, the Leibniz formula also follows from
[JSS19, Remark 2.2, Definition 4.11]. Therefore the above defined map descends
to tropical (co)homology, and we have the _cap product with the fundamental
class $\frown[\Sigma,w]$_

$\displaystyle\frown[\Sigma,w]\colon H^{q}(\Sigma,\mathcal{F}_{R}^{p})$
$\displaystyle\to H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R}).$

######  Proposition 3.19.

Let $(\Sigma,w)$ be an $R$-balanced fan of dimension $d$. The cap product with
the fundamental class $\frown[\Sigma,w]$ in tropical cohomology

$\displaystyle\frown[\Sigma,w]\colon H^{q}(\Sigma,\mathcal{F}_{R}^{p})$
$\displaystyle\to H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$

is the $0$-map for $q\not=0$.

###### Proof.

By Proposition 3.6, we have that $H^{q}(\Sigma,\mathcal{F}_{p}^{R})=0$ for
$q\neq 0$, hence this cap product is only non-trivial when $q\not=0$, ∎

The above proposition shows that in the fan-case, the only interesting cap
products are of the form $\frown[\Sigma,w]\colon
H^{0}(\Sigma,\mathcal{F}_{R}^{p})\to
H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$, for $p=0,\dots,d$. Moreover, in
Proposition 3.20 below, we show that these are injective for any commutative
ring $R$. In the case where $R=\mathbb{R}$, this was shown in [Aks19, Theorem
4.3.1], and for $R=\mathbb{Z}$, it is stated in [AP21, Section 3.2.2].

######  Proposition 3.20.

For an $R$-balanced fan $(\Sigma,w)$ of dimension $d$, the map

$\displaystyle\frown[\Sigma,w]\colon H^{0}(\Sigma,\mathcal{F}_{R}^{p})$
$\displaystyle\to H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$

is injective.

###### Proof.

We have that $H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})=\ker(\partial_{d})$,
and $H^{0}(\Sigma,\mathcal{F}_{R}^{p})=\mathcal{F}_{R}^{p}(v)$, so that
$\frown[\Sigma,w]$ is exactly

$\displaystyle\frown\mathrm{Ch}(\Sigma,w)\colon\mathcal{F}_{R}^{p}(v)$
$\displaystyle\to\bigoplus_{\alpha\in\Sigma^{d}}\mathcal{F}_{d-p}^{R}(\alpha)$
$\displaystyle u$
$\displaystyle\mapsto(\rho_{v,\alpha}(u)\,\lrcorner\,w(\alpha)\Lambda_{\alpha})_{\alpha\in\Sigma^{d}}$

where the image lies in
$H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})\subseteq\bigoplus_{\alpha\in\Sigma^{d}}\mathcal{F}_{d-p}^{R}(\alpha)$.
This is the composition of the map
$\oplus_{\alpha}\rho_{v,\alpha}\colon\mathcal{F}_{R}^{p}(v)\to\oplus_{\alpha\in\Sigma^{d}}\mathcal{F}_{R}^{p}(\alpha)$,
which is injective, since it is dual to the surjection
$\oplus_{\alpha\in\Sigma^{d}}\mathcal{F}_{p}^{R}(\alpha)\to\mathcal{F}_{p}^{R}(v)$,
and the direct sum of the contractions
$\oplus_{\alpha\in\Sigma^{d}}\,\lrcorner\,w(\alpha)\Lambda_{\alpha}$, which
are injective (Definition 3.16). Thus this cap product is the composition of
injective maps and is therefore injective. ∎

######  Proposition 3.21.

Let $(\Sigma,w)$ be an $R$-balanced fan and $\gamma\in\Sigma$ a face. Then the
cap product map on the star fan

$\frown[\gamma_{\succeq},w]\colon
H^{0}(\gamma_{\succeq},\mathcal{F}_{R}^{p})\to
H_{d}^{BM}(\gamma_{\succeq},\mathcal{F}_{d-p}^{R})$

is given by

$u\mapsto(u\,\lrcorner\,w(\alpha)\Lambda_{\alpha})_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\gamma\end{subarray}},$

where we identify
$H^{0}(\gamma_{\succeq},\mathcal{F}_{R}^{p})\cong\mathcal{F}_{p}^{R}(\gamma)$,
and $H_{d}^{BM}(\gamma_{\succeq},\mathcal{F}_{d-p}^{R})$ as the kernel of the
first map in the complex from Proposition 3.8.

###### Proof.

The identifications are justified by Proposition 3.8, and the existence of
this fundamental class by Proposition 3.13. It remains to show that the stated
formula corresponds to the cap product.

Consider a subdivision making $\gamma_{\succeq}$ a pointed fan. Each $d$-cell
$\widetilde{\alpha}$ of the subdivision maps to a $d$-cell $\alpha\succ\gamma$
of $\Sigma$, similarly to the proof of Proposition 3.8. The formula then
follows from the induced map in homology. ∎

##  4\. Tropical Poincaré duality

In Section 4.1, we define TPD over a ring $R$, and give an example of a non-
matroidal fan satisfying the duality. In Section 4.2, we give some necessary
conditions for the duality to hold, along with a characterization by an Euler
characteristic condition. Finally, in Section 4.3, we turn to the problem of
determining which fans are TPD spaces. We classify all the one-dimensional
fans satisfying TPD over a ring $R$ and study tropical fan hypersurfaces in
$\mathbb{R}^{n}$ satisfying TPD. This forms a first step towards answering
1.1.

###  4.1. Definition and examples

In this subsection, we define what it means for a fan to satisfy TPD over a
commutative ring $R$. When $R=\mathbb{Z}$, this is the definition from [JRS18,
Definition 5.2], and when $R=\mathbb{R}$, our definition can be shown to be
equivalent to [JSS19, Definition 4.12].

######  Definition 4.1.

We say that an $R$-balanced rational polyhedral fan $\Sigma$ of dimension $d$
with weights $w$ satisfies _tropical Poincaré duality over $R$_ if the cap
product with the fundamental class

$\frown[\Sigma,w]\colon H^{q}(\Sigma,\mathcal{F}_{R}^{p})\to
H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$

is an isomorphism for all $p,q=0,\dots,d$.

As mentioned in the introduction, the Bergman fans of matroids satisfy TPD
over $\mathbb{R}$ and $\mathbb{Z}$ [JSS19, JRS18], however these are not the
only such fans, as can be seen from the next example.

######  Example 4.2.

[width=0.7]figures/section_2/example_2_tikz

Figure 2. The graph of cones for Example 4.2.

Let $f_{1}\coloneqq(0,1,1,1)$, $f_{2}\coloneqq(1,0,-1,1)$,
$f_{3}\coloneqq(1,1,0,-1)$ and $f_{4}\coloneqq(1,-1,1,0)$ be vectors in
$\mathbb{R}^{4}$ and let $e_{1},e_{2},e_{3}$ and $e_{4}$ be the standard
basis. Consider the fan generated by the cones of vertices connected by an
edge in Figure 2, so that for instance the cone of $e_{1}$ and $f_{2}$ is
included. This fan was used in [BH17] to construct a counter-example to the
strongly positive Hodge conjecture. It is not matroidal, since it does not
satisfy the Hard Lefschetz property of [AHK18].

We compute its cellular tropical homology and cohomology over $\mathbb{Q}$
using the cellular sheaves package [KSW17] for [polymake], we have:

$\begin{aligned} H^{0}(\Sigma,\mathcal{F}_{\mathbb{Q}}^{0})&\cong\mathbb{Q}\\\
H^{0}(\Sigma,\mathcal{F}_{\mathbb{Q}}^{1})&\cong\mathbb{Q}^{4}\\\
H^{0}(\Sigma,\mathcal{F}_{\mathbb{Q}}^{2})&\cong\mathbb{Q}^{5}\end{aligned}\qquad\text{
and }\qquad\begin{aligned}
H_{2}^{BM}(\Sigma,\mathcal{F}_{2}^{\mathbb{Q}})&\cong\mathbb{Q}\\\
H_{2}^{BM}(\Sigma,\mathcal{F}_{1}^{\mathbb{Q}})&\cong\mathbb{Q}^{4}\\\
H_{2}^{BM}(\Sigma,\mathcal{F}_{0}^{\mathbb{Q}})&\cong\mathbb{Q}^{5}\end{aligned}$

with all other groups being zero. By Proposition 3.20, the cap product is
injective, and since the dimensions agree, the cap products are isomorphisms
when they are nonzero. Hence the fan satisfies TPD over $\mathbb{Q}$, where
the weights for the fundamental class are chosen so as to form a generator of
$H_{2}^{BM}(\Sigma,\mathcal{F}_{2}^{\mathbb{Q}})=\mathbb{Q}$.

###  4.2. Necessary conditions for tropical Poincaré duality

We now turn to giving some necessary conditions for TPD to hold. First, in
light of Proposition 3.6, the Borel–Moore homology of fans satisfying TPD is
concentrated in degree $d$:

######  Lemma 4.3.

Let $(\Sigma,w)$ be an $R$-balanced fan satisfying TPD over $R$. Then the
Borel–Moore homology of $\Sigma$ satisfies

$H_{q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})=0$

for $q\not=d$.

###### Proof.

By Proposition 3.6, $H^{q}(\Sigma,\mathcal{F}_{R}^{p})=0$ for $q\not=0$, hence
the isomorphism $\frown[\Sigma,w]\colon H^{q}(\Sigma,\mathcal{F}_{R}^{p})\cong
H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$ gives
$H_{q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})=0$ for $q\not=d$. ∎

######  Lemma 4.4.

Let $(\Sigma,w)$ be an $R$-balanced fan satisfying TPD over $R$. Then $\Sigma$
must be uniquely $R$-balanced by $w$.

###### Proof.

Since $\Sigma$ satisfies TPD, the cap product maps $1\in R\cong
H^{0}(\Sigma,\mathcal{F}_{R}^{0})$ (see Proposition 3.6) to
$1\frown[\Sigma,w]=[\Sigma,w]\in H_{d}^{BM}(\Sigma,\mathcal{F}_{d}^{R})$,
which must be a generator. Then by Definition 3.11, the fan $\Sigma$ is
uniquely $R$-balanced. ∎

######  Example 4.5.

For any ring $R$, the fan in Figure 1 is $R$-balanced, but not uniquely
$R$-balanced by Example 3.7, hence it cannot satisfy TPD over $R$.

Now, assuming that we are working over a field $k$, and that the Borel–Moore
homology of the fan vanishes in an appropriate way, we can determine that the
fan satisfies TPD through an Euler characteristic argument.

######  Proposition 4.6.

Let $k$ be a field, and $(\Sigma,w)$ be a $k$-balanced fan of dimension $d$.
Suppose $H_{q}^{BM}(\Sigma,\mathcal{F}_{p}^{k})=0$ for $q\neq d$. Then, for a
given $p$, the cap product

$\frown[\Sigma,w]\colon H^{q}(\Sigma,\mathcal{F}_{k}^{p})\to
H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{k})$

is an isomorphism for all $q$ if and only if

(1)
$(-1)^{d}\chi(C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{d-p}^{k}))=\dim_{k}\mathcal{F}_{k}^{p}(v).$

Moreover, $(\Sigma,w)$ satisfies TPD over $k$ if and only if Equation 1 holds
for all $p$.

###### Proof.

Since the only compact cell in $\Sigma$ is the vertex $v$, we have

$H^{q}(\Sigma,\mathcal{F}_{k}^{p})=\begin{cases*}\mathcal{F}_{k}^{p}(v)&if
$q=0$,\\\ 0&otherwise.\end{cases*}$

By the vanishing condition on tropical Borel–Moore homology, the cap product
$\frown[\Sigma,w]\colon H^{q}(\Sigma,\mathcal{F}_{k}^{p})\to
H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{k})$ is then immediately an isomorphism
for $q\neq 0$, and

(2)
$\chi(C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{d-p}^{k}))=\chi(H_{\bullet}^{BM}(\Sigma,\mathcal{F}_{d-p}^{k}))=(-1)^{d}\dim_{k}H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{k}).$

Since the cap product is injective by Proposition 3.20 and we are working over
a field, the maps

$\frown[\Sigma,w]\colon H^{0}(\Sigma,\mathcal{F}_{k}^{p})\to
H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{k})$

are isomorphisms if and only if
$\dim_{k}H^{0}(\Sigma,\mathcal{F}_{k}^{p})=\dim_{k}\mathcal{F}_{k}^{p}(v)$ is
equal to $\dim_{k}H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{k})$. By Equation 2,
this is exactly the claimed result. ∎

###  4.3. Dimension one and codimension one

We completely classify rational polyhedral fans of dimension 1 satisfying TPD
over an arbitrary commutative ring:

######  Proposition 4.7.

Let $R$ be a commutative ring, and $(\Sigma,w)$ an $R$-balanced fan of
dimension one. Then $(\Sigma,w)$ satisfies tropical Poincaré duality over $R$
if and only if it is uniquely $R$-balanced and all the weights are units in
$R$.

###### Proof.

Consider the cellular structure on $\Sigma$ with one vertex $v$ and edges
$\epsilon\in\Sigma^{1}$ for each ray of the fan, all of which we assume are
oriented outward from $v$. We begin by numbering the edges
$\epsilon\in\Sigma^{1}$ as $\epsilon_{1},\dots,\epsilon_{m}$, with weights
$w_{i}=w(\epsilon_{i})$. Since the only compact cell is $v$, we have:

$H^{0}(\Sigma,\mathcal{F}_{R}^{0})=\mathcal{F}_{R}^{0}(v)=R,\quad\text{and}\quad
H^{1}(\Sigma,\mathcal{F}_{R}^{0})=0,$

for the cohomology of the $\mathcal{F}_{R}^{0}$ sheaf, and

$H^{0}(\Sigma,\mathcal{F}_{R}^{1})=\mathcal{F}_{R}^{1}(v),\quad\text{and}\quad
H^{1}(\Sigma,\mathcal{F}_{R}^{1})=0,$

for the cohomology of the $\mathcal{F}_{R}^{1}$ sheaf. Moreover, the tropical
Borel–Moore cochain complexes are:

$\displaystyle C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{0}^{R})$
$\displaystyle\colon\quad\bigoplus_{i=1}^{m}R\xrightarrow{\partial_{1}^{0}}R\to
0,\quad\text{and}$ $\displaystyle
C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{1}^{R})$
$\displaystyle\colon\quad\bigoplus_{i=1}^{m}\mathcal{F}_{1}^{R}(\epsilon_{i})\xrightarrow{\partial_{1}^{1}}\mathcal{F}_{1}^{R}(v)\to
0.$

Here $\partial_{1}^{0}$ is the map given by the matrix $(1\;1\;\dots\;1)$. It
is surjective and thus $H_{0}^{BM}(\Sigma,\mathcal{F}_{0}^{R})=0$. Similarly,
$\mathcal{F}_{1}^{R}(v)=\sum_{i=1}^{m}L_{R}(\epsilon_{i})=\sum_{i=1}^{m}\mathcal{F}_{1}^{R}(\epsilon_{i})$,
and $\partial_{1}^{1}$ is surjective, hence
$H_{0}^{BM}(\Sigma,\mathcal{F}_{1}^{R})=0$. Therefore, both of the cap
products

$\displaystyle\frown[\Sigma,w]\colon H^{1}(\Sigma,\mathcal{F}_{R}^{1})$
$\displaystyle\to H_{0}^{BM}(\Sigma,\mathcal{F}_{0}^{R}),\quad\text{and}$
$\displaystyle\frown[\Sigma,w]\colon H^{1}(\Sigma,\mathcal{F}_{R}^{0})$
$\displaystyle\to H_{0}^{BM}(\Sigma,\mathcal{F}_{1}^{R}),$

are maps between zero modules, hence isomorphisms.

The map $\frown[\Sigma,w]\colon H^{0}(\Sigma,\mathcal{F}_{R}^{0})\to
H_{1}^{BM}(\Sigma,\mathcal{F}_{1}^{R})$ is given by sending a scalar
$\alpha\in H^{0}(\Sigma,\mathcal{F}_{R}^{0})\cong R$ to
$\alpha\frown[\Sigma,w]$. The $0$-contraction of a scalar is multiplication by
this scalar, so that $\alpha\frown[\Sigma,w]=\alpha\cdot[\Sigma,w]$. It is
therefore an isomorphism if and only if $\langle[\Sigma,w]\rangle$ generates
$H_{1}^{BM}(\Sigma,\mathcal{F}_{1}^{R})$, which is the definition of uniquely
$R$-balanced (Definition 3.11).

Finally, we consider $\frown[\Sigma,w]\colon
H^{0}(\Sigma,\mathcal{F}_{R}^{1})\to H_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R})$.
The Borel–Moore cochain group is
$C_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R})=\oplus_{i=1}^{m}R$, which has a basis
$x_{1},\dots,x_{m}$, with $x_{i}$ corresponding to $\epsilon_{i}$. The
elements $x_{i}-x_{m}\in C_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R})$, for
$i=1,\dots,m-1$ form a basis for
$H_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R})=\ker(1,1,\dots,1)$. For each
$\epsilon_{i}$, we select the generator $\Lambda_{i}\in L_{R}(\epsilon_{i})$
compatible with the orientation of $\epsilon_{i}$, and let
$\Theta_{i}\coloneqq\iota_{\epsilon_{i},v}(\Lambda_{i})$ be its image under
the inclusion
$\iota_{\epsilon_{i},v}\colon\mathcal{F}_{1}^{R}(\epsilon_{i})\to\mathcal{F}_{1}^{R}(v)$.
The cap product map $\frown[\Sigma,w]\colon
H^{0}(\Sigma,\mathcal{F}_{R}^{1})\to H_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R})$
takes a covector $\phi\in
H^{0}(\Sigma,\mathcal{F}_{R}^{1})=\mathcal{F}_{R}^{1}(v)$ to the element

$(w_{i}\phi(\Theta_{i}))_{i=1}^{m}\in H_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R}).$

Suppose all the weights $w_{i}$ are units in $R$. Then by the balancing
condition, the elements $w_{i}\Theta_{i}$, for $i=1,\dots,m-1$, form a basis
for $\mathcal{F}_{1}^{R}(v)$, with the corresponding dual basis
$w_{i}^{-1}\Theta_{i}^{*}$ for $\mathcal{F}_{R}^{1}(v)$. Then, for each
$j=1,\dots,m-1$,

$w_{j}^{-1}\Theta_{j}^{*}\frown[\Sigma,w]=(w_{i}w_{j}^{-1}\Theta_{j}^{*}(\Theta_{i}))_{i=1}^{m}=(0,\dots,0,1,0,\dots,w_{m}w_{j}^{-1}\Theta_{j}^{*}(\Theta_{m})),$

where the only two non-zero entries are in the $j$-th and $m$-th positions.
Since this is a cycle in $C_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R})$, we must
have $1+w_{m}w_{j}^{-1}\Theta_{j}^{*}(\Theta_{m})=0,$ so that

$w_{j}^{-1}\Theta_{j}^{*}\frown[\Sigma,w]=x_{i}-x_{m}.$

Hence the images of the basis elements $w_{j}^{-1}\Theta_{j}^{*}$ of
$\mathcal{F}_{1}^{R}(v)$ form a basis of
$H_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R})$, hence $\frown[\Sigma,w]$ is an
isomorphism.

Suppose some weight $w_{k}$ is not a unit in $R$. Then for any $\phi\in
H^{0}(\Sigma,\mathcal{F}_{R}^{1})=\mathcal{F}_{R}^{1}(v)$, the $k$-th
component of $\phi\frown[\Sigma,w]$ is contained in the ideal $\langle
w_{k}\rangle\subset R$, which does not contain $1$. Hence the element
$x_{k}-x_{m}$ of $H_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{R})$ cannot be in the
image of $\frown[\Sigma,w]$, which is therefore not surjective and hence not
an isomorphism. ∎

######  Corollary 4.8.

Let $k$ be a field, $(\Sigma,w)$ a $k$-balanced fan of dimension one. Then
$(\Sigma,w)$ satisfies TPD over $k$ if and only if it is uniquely
$k$-balanced.

###### Proof.

By Proposition 4.7, $(\Sigma,w)$ satisfies TPD if and only if it is uniquely
$k$-balanced, and all the weights are units in $k$. The weights are non-zero
by Definition 3.10, hence must be units since $k$ is a field. ∎

######  Example 4.9.

Let $\Sigma\subset\mathbb{Z}^{3}$ be the 1-dimensional fan with a vertex at
the origin, and the four cones $e_{1},e_{2},e_{3}$ and $e_{4}$ generated by
the vectors
$\nu_{1}=(1,0,2),\,\nu_{2}=(-1,0,0),\,\nu_{3}=(0,-1,0),\,\nu_{4}=(0,1,-2)$
respectively. This is a balanced fan with the constant unit weight function
$w(e_{i})=1$. The Borel–Moore chain complex
$C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{1}^{\mathbb{Z}})$ can be written as

${0}$${{\langle\Lambda_{1}\rangle_{\mathbb{Z}}\oplus\langle\Lambda_{2}\rangle_{\mathbb{Z}}\oplus\langle\Lambda_{3}\rangle_{\mathbb{Z}}\oplus\langle\Lambda_{4}\rangle_{\mathbb{Z}}}}$${{\mathcal{F}_{1}^{\mathbb{Z}}(v)}}$${0.}$$\scriptstyle{(\iota_{e_{i},v})}$

Since $\iota_{e_{i},v}(\Lambda_{i})=\nu_{i}$, we see in fact that
$H_{1}^{BM}(\Sigma,\mathcal{F}_{1}^{\mathbb{Z}})=\langle[\Sigma,w]\rangle$,
where $[\Sigma,w]=(\Lambda_{1},\Lambda_{2},\Lambda_{3},\Lambda_{4})$. Thus
$(\Sigma,w)$ is also uniquely $\mathbb{Z}$-balanced. Moreover, the complex
$C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{0}^{\mathbb{Z}})$ is

${0}$${{\mathbb{Z}^{4}}}$${\mathbb{Z}}$${0,}$$\scriptstyle{(1\,1\,1\,1)}$

so that $H_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{\mathbb{Z}})\cong\mathbb{Z}^{3}$.
We pick the basis $\nu_{1},\nu_{2},\nu_{3}$ for
$\mathcal{F}_{\mathbb{Z}}^{1}(v)$, and balancing gives
$\nu_{4}=-\nu_{1}-\nu_{2}-\nu_{3}$. The dual basis for
$\mathcal{F}_{\mathbb{Z}}^{1}(v)$ is $\nu_{1}^{*},\nu_{2}^{*},\nu_{3}^{*}$. We
see that

$\displaystyle\nu_{1}^{*}\frown[\Sigma,w]$
$\displaystyle=(\oplus\lrcorner\,\Lambda_{e_{i}})(\oplus\rho_{v,e_{i}})(\nu_{1}^{*})$
$\displaystyle=(\oplus\lrcorner\,\Lambda_{e_{i}})(\oplus(\nu_{1}^{*}\circ\iota_{e_{i},v}))$
$\displaystyle=(\nu_{1}^{*}(\iota_{e_{1},v}(\Lambda_{1})),\;\nu_{1}^{*}(\iota_{e_{2},v}(\Lambda_{2})),\;\nu_{1}^{*}(\iota_{e_{3},v}(\Lambda_{3})),\;\nu_{1}^{*}(\iota_{e_{4},v}(\Lambda_{4})))$
$\displaystyle=(\nu_{1}^{*}(\nu_{1}),\nu_{1}^{*}(\nu_{2}),\nu_{1}^{*}(\nu_{3}),\nu_{1}^{*}(\nu_{4}))$
$\displaystyle=(1,0,0,-1).$

Similarly, $\nu_{2}^{*}\frown[\Sigma,w]=(0,1,0,-1)$, and
$\nu_{3}^{*}\frown[\Sigma,w]=(0,0,1,-1)$. Since the images of the generating
set $y_{1}^{*},y_{2}^{*},y_{3}^{*}$ for $\mathcal{F}_{\mathbb{Z}}^{1}(v)$ is a
generating set for
$H_{1}^{BM}(\Sigma,\mathcal{F}_{0}^{\mathbb{Z}})\cong\mathbb{Z}^{3}$, the cap
product is an isomorphism.

For codimension 1 fan tropical cycles in $\mathbb{R}^{n}$, which are fan
tropical hypersurfaces, we can characterize the Newton polytopes of the
hypersurfaces having TPD. We refer to [MR, Chap. 2] for background on tropical
hypersurfaces in $\mathbb{R}^{n}$, which they call _very affine tropical
hypersurfaces_.

######  Proposition 4.10.

Let $f\in\mathbb{T}[x_{0}^{\pm 1},\dots,x_{d}^{\pm 1}]$ be a tropical Laurent
polynomial such that the very affine tropical cycle
$X=V(f)\subset\mathbb{R}^{d+1}$ is supported on a pointed fan. If $X$
satisfies TPD over a commutative ring $R$, then the Newton polytope
$\Delta(f)$ of $f$ is a simplex.

###### Proof.

By assumption, the very affine tropical cycle $X$ is a pointed $d$-dimensional
rational polyhedral fan ([MR, Cor 2.3.2]), thus
$H^{q}(X,\mathcal{F}_{R}^{p})=0$ for all $q>0$ and all $p$, and the
isomorphisms $\frown[X]\colon H^{q}(X,\mathcal{F}_{R}^{p})\to
H_{d-q}^{BM}(X,\mathcal{F}_{d-p}^{R})$ give in particular that
$H_{d-q}^{BM}(X,\mathcal{F}_{0}^{R})=0$ for all $q>0$.

Since $X$ is the $d$-skeleton of the dual fan to $\Delta(f)$ by [MR, Thm
2.3.7, Cor 2.3.2], $\dim_{R}H_{d}^{BM}(X,\mathcal{F}_{0})$ is
$\\#\mathrm{Vert}(\Delta(f))-1$, the number of vertices of the polytope
$\Delta(f)$, minus $1$.

Since $X$ is $d$-dimensional,
$\dim_{R}H^{0}(X,\mathcal{F}_{R}^{p})=\dim_{R}\mathcal{F}_{R}^{p}(v)=\binom{d+1}{p}$,
thus by Poincaré duality we have

$\\#\mathrm{Vert}(\Delta(f))-1=\dim_{R}H_{d}^{BM}(X,\mathcal{F}_{0}^{R})=\dim_{R}H^{0}(X,\mathcal{F}_{R}^{d})=\binom{d+1}{d}=d+1,$

and so $\Delta(f)$, being $(d+1)$-dimensional and having $d+2$ vertices, is a
simplex. ∎

##  5\. Local tropical Poincaré duality spaces

In this section, we study 1.2. In Section 5.1, we prove Theorem 5.4. This
theorem implies that TPD on faces of a fan, along with vanishing of its
tropical BM homology, gives TPD on the whole fan. A version of the proof gives
a partial classification of TPD spaces of dimension two. Using Theorem 5.4, we
prove Theorem 5.10 in Section 5.2, which states that local TPD spaces are
exactly fans whose codimension one faces are local TPD spaces, and all of
whose faces have vanishing tropical BM homology. Finally, we use the dimension
one classification from Proposition 4.7 to give a more geometric
characterization of local TPD spaces in Corollary 5.11.

###  5.1. TPD from faces

We fix a principal ideal domain $R$, and we use the following shortened
notation $H_{d,d-p}^{BM}(\Sigma;R)\coloneqq
H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$. We prove Theorem 5.4 in two steps:
The first step will be to show Proposition 5.2, which relates the cellular
chain complex $C_{c}^{\bullet}(\Sigma,\mathcal{F}_{R}^{p})$ to a complex
involving the Borel–Moore homology groups
$H_{d}^{BM}(\gamma_{\succeq},\mathcal{F}_{d-p}^{R})$ for faces
$\gamma\in\Sigma$ by using the cap product, which we show is exact. We then
prove the theorem by showing that TPD on the faces, along with exactness in
the mentioned complex, imply Poincaré duality for the whole fan.

Let $\Sigma$ be a $d$-dimensional rational polyhedral fan. For each maximal
face $\alpha\in\Sigma^{d}$, the constant sheaf
$\mathcal{F}_{d-p}^{R}(\alpha)_{\alpha_{\preceq}}$ gives a cochain complex
$(C_{c}^{\bullet}(\alpha_{\preceq},\mathcal{F}_{d-p}^{R}(\alpha)_{\alpha_{\preceq}}),d_{\alpha}^{\bullet})$.
Taking the direct sum of these for all $\alpha\in\Sigma^{d}$, we obtain a
complex

(3)
$(A^{\bullet},d^{\bullet})\coloneqq(\oplus_{\alpha}C_{c}^{\bullet}(\alpha_{\preceq},\mathcal{F}_{d-p}^{R}(\alpha)_{\alpha_{\preceq}}),\oplus_{\alpha}d_{\alpha}^{\bullet}).$

The $i$-th term of this complex is given by

$A^{i}=\oplus_{\alpha}C_{c}^{i}(\alpha_{\preceq},\mathcal{F}_{d-p}^{R}(\alpha)_{\alpha_{\preceq}})=\oplus_{\alpha\in\Sigma^{d}}\oplus_{\begin{subarray}{c}\gamma\in\Sigma^{i}\\\
\gamma\prec\alpha\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha).$

Rearranging terms, we may use Proposition 3.8 to obtain an inclusion

$\oplus_{\gamma\in\Sigma^{i}}H_{d,d-p}^{BM}(\gamma_{\succeq};R)\subseteq
A^{i}.$

######  Proposition 5.1.

There is a cochain complex
$(\oplus_{\gamma\in\Sigma^{\bullet}}H_{d,d-p}^{BM}(\gamma_{\succeq};R),\overline{d^{\bullet}}),$
which is the restriction of the cochain complex $(A^{\bullet},d^{\bullet})$
from Equation 3.

###### Proof.

It suffices to show that, for each $i\geq 0$,

$d^{i}\left(\oplus_{\gamma\in\Sigma^{i}}H_{d,d-p}^{BM}(\gamma_{\succeq};R)\right)\subseteq\oplus_{\kappa\in\Sigma^{i+1}}H_{d,d-p}^{BM}(\kappa_{\succeq};R).$

This follows from a direct computation. ∎

######  Proposition 5.2.

For $(\Sigma,w)$ an $R$-balanced fan of dimension $d\geq 2$, there is a
commutative diagram

(4)
${0}$${{\mathcal{F}_{R}^{p}(v)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}\mathcal{F}_{R}^{p}(\tau)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}\mathcal{F}_{R}^{p}(\sigma)}}$${\cdots}$${0}$${{H_{d,d-p}^{BM}(\Sigma;R)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}H_{d,d-p}^{BM}(\tau_{\succeq};R)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}H_{d,d-p}^{BM}(\sigma_{\succeq};R)}}$${\cdots}$$\scriptstyle{\oplus_{\alpha}\overline{d_{\alpha}^{0}}}$$\scriptstyle{\oplus_{\alpha}\overline{d_{\alpha}^{1}}}$$\scriptstyle{\frown[\Sigma,w]}$$\scriptstyle{\delta^{0}}$$\scriptstyle{\oplus_{\tau}\frown[\tau_{\succeq},w]}$$\scriptstyle{\oplus_{\alpha}\overline{d_{\alpha}^{2}}}$$\scriptstyle{\delta^{1}}$$\scriptstyle{\delta^{2}}$$\scriptstyle{\oplus_{\sigma}\frown[\sigma_{\succeq},w]}$

with all the vertical maps being injective, where upper row is given by the
complex $(C_{c}^{\bullet}(\Sigma,\mathcal{F}_{p}^{R}),\delta^{\bullet})$, and
the lower row is the complex
$(\oplus_{\gamma\in\Sigma^{\bullet}}H_{d,d-p}^{BM}(\gamma_{\succeq};R),\overline{d^{\bullet}})$
from Proposition 5.1.

###### Proof.

First, we wish to show that the following diagram is commutative:

${0}$${{\mathcal{F}_{R}^{p}(v)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}\mathcal{F}_{R}^{p}(\tau)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}\mathcal{F}_{R}^{p}(\sigma)}}$${\cdots}$${0}$${{\bigoplus_{\alpha\in\Sigma^{d}}\mathcal{F}_{d-p}^{R}(\alpha)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}\bigoplus\limits_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\tau\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}\bigoplus\limits_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\sigma\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha)}}$${\cdots.}$$\scriptstyle{\oplus_{\alpha}d_{\alpha}^{0}}$$\scriptstyle{\oplus_{\alpha}d_{\alpha}^{1}}$$\scriptstyle{\oplus_{\alpha}d_{\alpha}^{2}}$$\scriptstyle{\frown\mathrm{Ch}(\Sigma,w)}$$\scriptstyle{\delta^{0}}$$\scriptstyle{\delta^{1}}$$\scriptstyle{\delta^{2}}$$\scriptstyle{\oplus_{\sigma}\frown\mathrm{Ch}(\sigma_{\succeq},w)}$$\scriptstyle{\oplus_{\tau}\frown\mathrm{Ch}(\tau_{\succeq},w)}$

The upper row is the compact support complex
$(C_{c}^{\bullet}(\Sigma,\mathcal{F}_{R}^{p}),\delta^{\bullet})$ for the
$\mathcal{F}_{R}^{p}$ cohomology (see Definition 2.12). The lower row is the
complex $(A^{\bullet},d^{\bullet})$ from (3), where the order of indexing is
changed for clarity in relation to the cap morphism.

The first vertical map in diagram (4) is given by the cap product on the chain
level of $(\Sigma,w)$, as in Definition 3.17. For the $r$-th column, the
vertical map is given as the direct sum over all $\gamma\in\Sigma^{r}$ of the
maps:

$\displaystyle\frown\mathrm{Ch}(\gamma_{\succeq},w)\colon\mathcal{F}_{R}^{p}(\gamma)$
$\displaystyle\to\oplus_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\gamma\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha)$ $\displaystyle
v$
$\displaystyle\mapsto\left(v_{\gamma}\,\lrcorner\,w(\alpha)\Lambda_{\alpha}\right)_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\gamma\end{subarray}}.$

To obtain commutativity of the described diagram, we select one square and
show commutativity there:

(5)
${{\bigoplus\limits_{\gamma\in\Sigma^{r}}\mathcal{F}_{R}^{p}(\gamma)}}$${{\bigoplus\limits_{\kappa\in\Sigma^{r+1}}\mathcal{F}_{R}^{p}(\kappa)}}$${{\bigoplus\limits_{\gamma\in\Sigma^{r}}\bigoplus\limits_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\gamma\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha)}}$${{\bigoplus\limits_{\kappa\in\Sigma^{r+1}}\bigoplus\limits_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\kappa\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha)}}$$\scriptstyle{\oplus_{\alpha}d_{\alpha}^{r}}$$\scriptstyle{\oplus_{\gamma}\frown\mathrm{Ch}(\gamma_{\succeq},w)}$$\scriptstyle{\oplus_{\kappa}\frown\mathrm{Ch}(\kappa_{\succeq},w)}$$\scriptstyle{\delta^{r}}$

For
$v=(v_{\gamma})_{\gamma\in\Sigma^{r}}\in\bigoplus_{\gamma\in\Sigma^{r}}\mathcal{F}_{R}^{p}(\gamma)$,
we can expand the definitions for the right then down composition to get

$\displaystyle((\oplus_{\kappa}\frown\mathrm{Ch}(\kappa_{\succeq},w)\circ\delta^{r})(v)$
$\displaystyle=(\oplus_{\kappa}\frown\mathrm{Ch}(\kappa_{\succeq},w))\left(\left(\textstyle\sum_{\gamma\prec\kappa}\mathcal{O}(\gamma,\kappa)v_{\gamma}\right)_{\kappa\in\Sigma^{r+1}}\right)$
$\displaystyle=\left((\textstyle\sum_{\gamma\prec\kappa}\mathcal{O}(\gamma,\kappa)v_{\gamma})\,\lrcorner\,w(\alpha)\Lambda_{\alpha}\right)_{\begin{subarray}{c}\kappa\in\Sigma^{r+1}\\\
\alpha\in\Sigma^{d}\\\ \alpha\succ\kappa\end{subarray}}.$

For the down then right composition, we get

$\displaystyle((\oplus_{\alpha}d_{\alpha}^{r})\circ({\oplus_{\gamma}\frown\mathrm{Ch}(\gamma_{\succeq}},w)))(v)$
$\displaystyle=(\oplus_{\alpha}d_{\alpha}^{r})\left(\left(w(\alpha)v_{\gamma}\,\lrcorner\,\Lambda_{\alpha}\right)_{\begin{subarray}{c}\gamma\in\Sigma^{r+1},\;\alpha\in\Sigma^{d}\\\
\alpha\succ\gamma\end{subarray}}\right)$
$\displaystyle=\left(\textstyle\sum_{\gamma\prec\kappa}\mathcal{O}(\gamma,\kappa)(v_{\gamma}\,\lrcorner\,w(\alpha)\Lambda_{\alpha})\right)_{\begin{subarray}{c}\kappa\in\Sigma^{r+1}\\\
\alpha\in\Sigma^{d}\\\ \alpha\succ\kappa\end{subarray}}.$

Comparing the two above equations, diagram (5) is commutative since the
contraction $\lrcorner\,w(\alpha)\Lambda_{\alpha}$ is $R$-linear.

Now by Proposition 3.21 and Proposition 3.8, for each $\kappa\in\Sigma$, we
have that

$H_{d}^{BM}(\kappa_{\succeq},\mathcal{F}_{d-p}^{R})\cong\ker\left(\oplus_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\kappa\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha)\to\oplus_{\begin{subarray}{c}\beta\in\Sigma^{d-1}\\\
\beta\succ\kappa\end{subarray}}\mathcal{F}_{d-p}^{R}(\beta)\right),$

and the given formulas for the maps
$\oplus_{\kappa\in\Sigma^{r+1}}\frown\mathrm{Ch}(\kappa_{\succeq},w)$
correspond exactly to the cap products in homology

$\oplus_{\kappa}\frown[\kappa_{\succeq},w]\colon\bigoplus_{\kappa\in\Sigma^{r+1}}\mathcal{F}_{R}^{p}(\kappa)\to\bigoplus\limits_{\kappa\in\Sigma^{r+1}}H_{d}^{BM}(\kappa_{\succeq},\mathcal{F}_{d-p}^{R}).$

We have the following diagram when only considering the images

(6)
${0}$${{\mathcal{F}_{R}^{p}(v)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}\mathcal{F}_{R}^{p}(\tau)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}\mathcal{F}_{R}^{p}(\sigma)}}$${\cdots}$${0}$${{H_{d,d-p}^{BM}(\Sigma;R)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}H_{d,d-p}^{BM}(\tau_{\succeq};R)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}H_{d,d-p}^{BM}(\sigma_{\succeq};R)}}$${\cdots}$$\scriptstyle{\oplus_{\alpha}\overline{d_{\alpha}^{0}}}$$\scriptstyle{\oplus_{\alpha}\overline{d_{\alpha}^{1}}}$$\scriptstyle{\frown[\Sigma,w]}$$\scriptstyle{\delta^{0}}$$\scriptstyle{\oplus_{\tau}\frown[\tau_{\succeq},w]}$$\scriptstyle{\oplus_{\alpha}\overline{d_{\alpha}^{2}}}$$\scriptstyle{\delta^{1}}$$\scriptstyle{\delta^{2}}$$\scriptstyle{\oplus_{\sigma}\frown[\sigma_{\succeq},w]}$

where the cochain differentials in the lower row have been restricted.

These vertical maps are the direct sum of cap products, so by Proposition
3.20, are injective. ∎

######  Proposition 5.3.

Let $\Sigma$ be a fan of dimension $d\geq 2$ such that
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$, for $q\not=d$ and all
$p$, for each face $\gamma\in\Sigma$. Then the complex
$(\oplus_{\gamma\in\Sigma^{\bullet}}H_{d,d-p}^{BM}(\gamma_{\succeq};R),\overline{d^{\bullet}})$
in Proposition 5.1 is exact except in the rightmost position.

###### Proof.

We will now construct a double complex $K_{\bullet,\bullet}$, which
corresponds to the Cartan-Eilenberg resolution of diagram (4), using that
these are the homology groups of the complexes
$C_{\bullet}^{BM}(\gamma_{\succeq},\mathcal{F}_{d-p}^{R})$. Then, we will use
the two spectral sequences converging to the homology of the total complex
$H_{\bullet}(\mathrm{Tot}(K_{\bullet,\bullet}))$ to deduce that the complex is
exact except in the rightmost position.

Let $(K_{\bullet,\bullet},d_{\wedge}^{0},d_{>}^{0})$ be the first-quadrant
double complex given by

$K_{r,s}=\bigoplus_{\kappa\in\Sigma^{r}}\bigoplus_{\begin{subarray}{c}\gamma\in\Sigma^{d-s}\\\
\gamma\succeq\kappa\end{subarray}}\mathcal{F}_{d-p}^{R}(\gamma),$

for $r\geq 0$, $s\geq 0$, see Figure 3. Since all the indices used are
relating to the dimensions of particular faces of the fan $\Sigma$, this is a
first-quadrant double complex.

${{\mathcal{F}_{d-p}^{R}(v)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}\mathcal{F}_{d-p}^{R}(\tau)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}\mathcal{F}_{d-p}^{R}(\tau)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}\mathcal{F}_{d-p}^{R}(\sigma)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}\bigoplus\limits_{\begin{subarray}{c}\sigma\in\Sigma^{2}\\\
\sigma\succ\tau\end{subarray}}\mathcal{F}_{d-p}^{R}(\sigma)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}\mathcal{F}_{d-p}^{R}(\sigma)}}$${\vdots}$${\vdots}$${\vdots}$${{\bigoplus\limits_{\beta\in\Sigma^{d-1}}\mathcal{F}_{d-p}^{R}(\beta)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}\bigoplus\limits_{\begin{subarray}{c}\beta\in\Sigma^{d-1}\\\
\beta\succ\tau\end{subarray}}\mathcal{F}_{d-p}^{R}(\beta)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}\bigoplus\limits_{\begin{subarray}{c}\beta\in\Sigma^{d-1}\\\
\beta\succ\sigma\end{subarray}}\mathcal{F}_{d-p}^{R}(\beta)}}$${\dots}$${{\bigoplus_{\alpha\in\Sigma^{d}}\mathcal{F}_{d-p}^{R}(\alpha)}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}\bigoplus\limits_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\tau\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha)}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}\bigoplus\limits_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\sigma\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha)}}$${\dots}$$\scriptstyle{d_{>}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$$\scriptstyle{d_{>}^{0}}$$\scriptstyle{d_{>}^{0}}$$\scriptstyle{d_{>}^{0}}$$\scriptstyle{d_{>}^{0}}$$\scriptstyle{d_{>}^{0}}$$\scriptstyle{d_{>}^{0}}$$\scriptstyle{d_{>}^{0}}$$\scriptstyle{d_{>}^{0}}$$\scriptstyle{d_{\wedge}^{0}}$
Figure 3. The double complex $(K_{\bullet,\bullet},d_{\wedge}^{0},d_{>}^{0})$.

The vertical differential $(d_{\wedge}^{0})_{r,s}\colon K_{r,s}\to K_{r,s+1}$
is the direct sum over the differentials $\partial_{\bullet}^{\kappa}$ of the
chain complex for tropical Borel–Moore homology on the star $\kappa_{\succeq}$
for each face $\kappa\in\Sigma^{r}$, i.e.
$(d_{\wedge}^{0})_{r,s}=\oplus_{\kappa\in\Sigma^{r}}\partial_{d-s}^{\kappa}$
from Proposition 3.8.

The horizontal differential $(d_{>}^{0})_{r,s}\colon K_{r,s}\to K_{r+1,s}$ is
the direct sum over the differentials $d_{\gamma}^{\bullet}$ in the complex of
cochains of compact support for the constant sheaf taking value
$\mathcal{F}_{d-p}^{R}(\gamma)$ on the cone $\gamma_{\preceq}$, truncated in
degree $1$, for each face $\gamma\in\Sigma^{d-s}$. Explicitly,
$(d_{>}^{0})_{r,s}=\oplus_{\gamma\in\Sigma^{d-s}}d_{\gamma}^{r}$, where
$d_{\gamma}^{\bullet}$ comes from the complex:

${0}$${{\mathcal{F}_{d-p}^{R}(\gamma)}}$${{\bigoplus\limits_{\begin{subarray}{c}\tau\in\Sigma^{1}\\\
\gamma\succ\tau\end{subarray}}\mathcal{F}_{d-p}^{R}(\gamma)}}$${\dots}$${{\bigoplus\limits_{\begin{subarray}{c}\kappa\in\Sigma^{s-1}\\\
\gamma\succ\kappa\end{subarray}}\mathcal{F}_{d-p}^{R}(\gamma)}}$${{\mathcal{F}_{d-p}(\gamma)}}$${0,}$$\scriptstyle{d_{\gamma}^{0}}$$\scriptstyle{d_{\gamma}^{s-1}}$$\scriptstyle{d_{\gamma}^{1}}$$\scriptstyle{d_{\gamma}^{s-2}}$

from Proposition 3.8.

We have that $d_{\wedge}^{0}\circ d_{\wedge}^{0}=0$ and $d_{>}^{0}\circ
d_{>}^{0}=0$ since both are direct sums of differentials of complexes.
Moreover, we have $d_{>}^{0}\circ d_{\wedge}^{0}=d_{\wedge}^{0}\circ
d_{>}^{0}$ which can be checked directly.

Now, since $K_{\bullet,\bullet}$ is a double complex, we can look at the two
associated spectral sequences converging to the homology of the total complex
$(\operatorname{Tot}(K_{\bullet,\bullet}),d_{\text{T}})$ given by
$\operatorname{Tot}(K_{\bullet,\bullet})_{m}=\prod_{r+s=m}K_{r,s}$ with
differential $d_{\text{T}}=d_{>}+d_{\wedge}$. We refer to [Wei94, Chapter 5.6]
for details.

First, we take the spectral sequence $E^{r}$, with $E^{0}=K_{\bullet,\bullet}$
and the first differential $d^{0}$ being the horizontal differential
$d_{>}^{0}$ of $K_{\bullet,\bullet}$, which is equivalent to computing the
homology row by row. Since the rows $K_{\bullet,s}$ are the complexes
$\bigoplus_{\gamma\in\Sigma^{d-s}}C_{c}^{\bullet}(\gamma_{\preceq},F(\gamma)_{\gamma_{\preceq}})$
with $F(\gamma)\coloneqq\mathcal{F}_{d-p}^{R}(\gamma)$, Lemma 2.16 gives

$H_{k}(K_{\bullet,s},d_{>}^{0})\cong 0$

for each $s\neq d$. In degree $d$, we have
$H_{k}(K_{\bullet,d},d_{>}^{0})\cong\mathcal{F}_{d-p}^{R}(v)$ for $k=0$ and
$H_{k}(K_{\bullet,d},d_{>}^{0})=0$ otherwise. Thus, the $E^{1}$ page becomes

$E_{r,s}^{1}\cong\begin{cases*}\mathcal{F}_{d-p}^{R}(v)&$r=0$ and $s=d,$\\\
0&otherwise.\end{cases*}$

There are now no further non-zero differentials of the spectral sequence, so
the $E^{1}$ page is the $E^{\infty}$ page. In particular, we conclude that

(7)
$H_{q}(\operatorname{Tot}(K_{\bullet,\bullet})_{\bullet})=\begin{cases*}\mathcal{F}_{d-p}^{R}(v)&for
$q=d$,\\\ 0&otherwise.\end{cases*}$

Next, we consider the spectral sequence $\overline{E^{r}}$, with
$\overline{E^{0}}=K_{\bullet,\bullet}$ and the first differential $d^{0}$
being the vertical differential $d_{\wedge}^{0}$ of $K_{\bullet,\bullet}$.
Taking this differential is therefore equivalent to computing the homology
column by column. The $r$-th column is the direct sum over each
$\gamma\in\Sigma^{r}$ of the complex

${0}$${{\bigoplus\limits_{\begin{subarray}{c}\alpha\in\Sigma^{d}\\\
\alpha\succ\gamma\end{subarray}}\mathcal{F}_{d-p}^{R}(\alpha)}}$${{\bigoplus\limits_{\begin{subarray}{c}\beta\in\Sigma^{d-1}\\\
\beta\succ\gamma\end{subarray}}\mathcal{F}_{d-p}^{R}(\beta)}}$${\dots}$${{\bigoplus\limits_{\begin{subarray}{c}\kappa\in\Sigma^{l+1}\\\
\kappa\succ\gamma\end{subarray}}\mathcal{F}_{d-p}^{R}(\kappa)}}$${{\mathcal{F}_{d-p}^{R}(\gamma)}}$${{0,}}$$\scriptstyle{\partial_{n}^{\gamma}}$$\scriptstyle{\partial_{n-1}^{\gamma}}$$\scriptstyle{\partial_{l+2}^{\gamma}}$$\scriptstyle{\partial_{l+1}^{\gamma}}$

each of which has the tropical Borel–Moore homology of the star
$\gamma_{\succeq}$ by Proposition 3.8. Since by assumption,
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$ for $q\not=d$ and all
$p$, for each face $\gamma\in\Sigma$, we find:

$H_{k}(K_{r,\bullet},d_{\wedge}^{\bullet})=\begin{cases}\oplus_{\kappa\in\Sigma^{r}}H_{d}^{BM}(\kappa_{\succeq},\mathcal{F}_{d-p}^{R})&\text{if
$k=0$,}\\\ 0&\text{otherwise.}\end{cases}$

Thus the $\overline{E^{1}}$ page has only the bottom row:

${0}$${{H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})}}$${{\bigoplus\limits_{\tau\in\Sigma^{1}}H_{d}^{BM}(\tau_{\succeq},\mathcal{F}_{d-p}^{R})}}$${{\bigoplus\limits_{\sigma\in\Sigma^{2}}H_{d}^{BM}(\sigma_{\succeq},\mathcal{F}_{d-p}^{R})}}$${\dots}$$\scriptstyle{\oplus_{\alpha}\overline{d_{\alpha}^{0}}}$$\scriptstyle{\oplus_{\alpha}\overline{d_{\alpha}^{1}}}$$\scriptstyle{\oplus_{\alpha}\overline{d_{\alpha}^{2}}}$

The $\overline{E^{2}}$ page is then merely the homology of this complex, and
since it is concentrated in one row, this must be the $\overline{E^{\infty}}$
page. In particular, by (7), the complex only has homology in the rightmost
position. ∎

######  Theorem 5.4.

Let $R$ be a principal ideal domain, and $(\Sigma,w)$ be an $R$-balanced fan
of dimension $d\geq 2$, with $H_{q}^{BM}(\Sigma,\mathcal{F}_{p}^{R})=0$ for
$q\not=d$, for all $p$. If $(\gamma_{\succeq},w)$ satisfies TPD over $R$, for
each $\gamma\in\Sigma$ with $\gamma_{\succeq}\neq\Sigma$, then $(\Sigma,w)$
satisfies TPD over $R$.

###### Proof.

By assumption $H_{q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})=0$ for $q\neq d$, and
$H^{q}(\Sigma,\mathcal{F}_{R}^{p})=0$ for $q\neq 0$ by Remark 2.14, for all
$p$. Thus the cap product map

$\frown[\Sigma,w]\colon H^{q}(\Sigma,\mathcal{F}_{R}^{p})\to
H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$

is an isomorphism for all $q=1,\dots,d$, for all $p$, and it remains to show
that

$\frown[\Sigma,w]\colon H^{0}(\Sigma,\mathcal{F}_{R}^{p})\to
H_{d}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})$

is an isomorphism for all $p$.

Since $H_{q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})=0$ for $q\neq d$ and
$(\gamma_{\succeq},w)$ satisfies TPD over $R$ for each $\gamma\in\Sigma$ with
$\gamma_{\succeq}\neq\Sigma$, Lemma 4.3 gives that
$H_{d-q}^{BM}(\gamma_{\succeq},\mathcal{F}_{d-p}^{R})=0$ for all
$\gamma\in\Sigma$. Thus, by Proposition 5.3, the lower row in diagram (4) is
exact in all degrees except $d$.

Moreover, the upper row of diagram (4) is the complex
$C_{c}^{\bullet}(\Sigma,\mathcal{F}_{R}^{p})$, which can be seen to be the
dual complex to $C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{p}^{R})$ by the
definitions (Definitions 2.12, 2.13 and 3.1).

The complex $C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{p}^{R})$ consists only of
free $R$-modules, since $\mathcal{F}_{p}^{R}(\gamma)$ is a sublattice of
$N\otimes_{\mathbb{Z}}R$ for all $\gamma\in\Sigma$, and the ring $R$ is a
principal ideal domain, hence we may apply the Universal Coefficient Theorem
for cohomology [Wei94, Theorem 3.6.5]. Thus, for each $q$, we have:

${0}$${{\operatorname{Ext}_{R}(H_{q-1}^{BM}(\Sigma,\mathcal{F}_{p}^{R}),R)}}$${{H_{c}^{q}(\Sigma,\mathcal{F}_{p}^{R})}}$${{\operatorname{Hom}_{R}(H_{q-1}^{BM}(\Sigma,\mathcal{F}_{p}^{R}),R)}}$${0.}$

Since we assumed $H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})=0$ for $q\not=0$,
for all $p$, one has $H_{c}^{q}(\Sigma,\mathcal{F}_{p}^{R})=0$ for $q\not=d$,
for all $p$. Hence the upper row of diagram (4) is exact except in the last
position.

The cokernel complex to the chain complex map in (4) gives following short
exact sequence of chain complexes:

$0\to(C_{c}^{\bullet}(\Sigma,\mathcal{F}_{p}^{R}),\delta^{\bullet})\to\left(\oplus_{\gamma\in\Sigma^{\bullet}}H_{d}^{BM}(\gamma_{\succeq},\mathcal{F}_{d-p}^{R}),\;\oplus_{\alpha}\overline{d_{\alpha}^{\bullet}}\right)\to\operatorname{coker}(\frown)\to
0.$

This gives a long exact sequence in homology, and since the two first chain
complexes are exact in all but the last position, so is the cokernel chain
complex. Thus we have the following exact sequence:

${0}$${{\operatorname{coker}(\frown[\Sigma,w])}}$${{\operatorname{coker}(\oplus_{\tau}\frown[\tau_{\succeq},w])}}$${{\operatorname{coker}(\oplus_{\sigma}\frown[\sigma_{\succeq},w])}}$${\cdots}$

Since each of the stars $(\gamma_{\succeq},w)$ satisfies TPD over $R$, we have
$\operatorname{coker}(\oplus_{\gamma}\frown[\gamma_{\succeq},w])=0$ and so
exactness gives ${\operatorname{coker}(\frown[\Sigma,w])}=0$. Thus
$\frown[\Sigma,w]$ is both injective by Proposition 3.20 and surjective, hence
is an isomorphism. ∎

######  Remark 5.5.

In the proof of Theorem 5.4, the condition that $R$ is a PID is only used to
show that $H_{q}^{BM}(\Sigma,\mathcal{F}_{p}^{R})=0$ for all $q\neq d$ implies
that $H_{c}^{q}(\Sigma,\mathcal{F}_{R}^{p})=0$ for all $q\neq d$. One can let
$R$ be an arbitrary commutative ring if we instead assume this latter
condition, giving another variant of the theorem.

It is not sufficient that all the star fans $\gamma_{\succeq}$ of faces
$\gamma\in\Sigma$ with $\gamma\neq v$ satisfy TPD. The assumption
$H_{d-q}^{BM}(\Sigma,\mathcal{F}_{d-p}^{R})=0$ for $q\not=0$, for all $p$, is
necessary and not implied by TPD of the faces. This is shown by the following
example, which is also studied in [Aks19] and in [AP21, Section 11.1].

######  Example 5.6.

Figure 4. A fan whose faces excluding the vertex satisfy TPD, but does not
itself satisfy it.

Let $\Sigma$ be the fan shown in Figure 4, where the rays are
$e_{1},e_{2},-e_{1},-e_{2},-e_{1}+e_{2}+e_{3},-e_{1}+e_{2}-e_{3},e_{1}-e_{2}+e_{3},e_{1}-e_{2}-e_{3}$.
Each of its two-dimensional faces is maximal, so that the star at these faces
is just a two-dimensional linear space, which satisfies TPD. Moreover, each
ray has exactly three faces meeting in it, so that the stars are uniquely
balanced, and satisfy TPD. We could therefore expect Theorem 5.4 to give us
that the whole fan $\Sigma$ has TPD.

However, observe that
$\dim_{\mathbb{Q}}\mathcal{F}_{2}^{\mathbb{Q}}(\sigma)=1$ for each
$\sigma\in\Sigma^{2}$, while
$\dim_{\mathbb{Q}}\mathcal{F}_{2}^{\mathbb{Q}}(\tau)=2$ for each
$\tau\in\Sigma^{1}$ and $\dim_{\mathbb{Q}}\mathcal{F}_{2}^{\mathbb{Q}}(v)=3$.
There are $12$ two-dimensional faces and $8$ one-dimensional faces, so that

$\chi(C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{2}^{\mathbb{Q}}))=12-8\cdot
2+3=-1.$

Since
$\chi(C_{\bullet}^{BM}(\Sigma,\mathcal{F}_{2}^{\mathbb{Q}}))=\chi(H_{\bullet}^{BM}(\Sigma,\mathcal{F}_{2}^{\mathbb{Q}}))$,
we must have that $H_{1}^{BM}(\Sigma,\mathcal{F}_{2}^{\mathbb{Q}})\not\cong
0$. Finally, since $H^{1}(\Sigma,\mathcal{F}_{\mathbb{Q}}^{0})=0$, the cap
product $\frown[\Sigma,w]\colon H^{1}(\Sigma,\mathcal{F}_{\mathbb{Q}}^{0})\to
H_{1}^{BM}(\Sigma,\mathcal{F}_{2}^{\mathbb{Q}})$ cannot be an isomorphism.

######  Proposition 5.7.

Let $k$ be a field, and $(\Sigma,w)$ a $k$-balanced fan of dimension $2$.
Suppose $H_{q}^{BM}(\Sigma,\mathcal{F}_{p}^{k})=0$ for $q\neq 2$, for all $p$.
Then $\Sigma$ satisfies TPD over $k$ if and only if each of the stars
$(\tau_{\succeq},w)$, for $\tau\in\Sigma^{1}$ satisfies TPD over $k$.

###### Proof.

First, we show that for each $\sigma\in\Sigma^{2}$, the star
$\sigma_{\succeq}$ satisfies TPD over $k$. For each $\sigma\in\Sigma^{2}$, we
have from Proposition 3.8 that

$\displaystyle H^{0}(\sigma_{\succeq},\mathcal{F}_{k}^{p})$
$\displaystyle=\mathcal{F}_{k}^{p}(\sigma)=\left(\textstyle{\bigwedge^{p}}L_{\mathbb{Z}}(\sigma)\otimes_{\mathbb{Z}}k\right)^{*},$
$\displaystyle H_{d}^{BM}(\sigma_{\succeq},\mathcal{F}_{d-p}^{k})$
$\displaystyle=\mathcal{F}_{d-p}^{k}(\sigma)=\textstyle{\bigwedge^{d-p}}L_{\mathbb{Z}}(\sigma)\otimes_{\mathbb{Z}}k.$

Moreover, the cap product is an injective map by Proposition 3.20. These two
vector spaces have the same dimension, hence the cap product is an
isomorphism.

Next, we consider again the sequence

${0}$${{\operatorname{coker}(\frown[\Sigma,w])}}$${{\operatorname{coker}(\oplus_{\tau}\frown[\tau_{\succeq},w])}}$${{\operatorname{coker}(\oplus_{\sigma}\frown[\sigma_{\succeq},w])}}$${\cdots}$

from the proof of Theorem 5.4. Since $\sigma_{\succeq}$ satisfies TPD over
$k$, $\operatorname{coker}(\oplus_{\sigma}\frown[\sigma_{\succeq},w])=0$, and
so
$\operatorname{coker}(\frown[\Sigma,w])\cong\operatorname{coker}(\oplus_{\tau}\frown[\tau_{\succeq},w])$.
Since both these maps are injective by Proposition 3.20, the result follows. ∎

######  Remark 5.8.

Theorem 5.4 shows that under the assumption of the vanishing of Borel–Moore
homology, TPD on a fan $\Sigma$ can be deduced from TPD on its faces. In fact,
it is not necessary to assume that all the faces satisfy TPD: In general, the
“vertical first” spectral sequence in Proposition 5.3 degenerates on page 2
when $H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$, for $q\not=d$ and
all $p$, for each face $\gamma\in\Sigma$. However, the exactness of the lower
row in diagram (4) in positions 0 and 1 follow from the weaker assumption that
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$, for $q\not=d$ and all
$p$, for each face $\gamma\in\Sigma^{i}$, with $i=0,1$. One can then show, in
a restricted version of the proof of Theorem 5.4, that TPD for all
$\tau\in\Sigma^{1}$ implies that $\Sigma$ satisfies TPD.

###  5.2. A characterization of local TPD spaces

We now turn to studying fans which satisfy TPD at every face. Using Theorem
5.4, we characterize such fans as the ones for which the condition holds in
codimension 1 along with a vanishing condition on Borel–Moore homology, which
was suggested to us by Amini and Piquerez [AP21].

######  Definition 5.9.

Let $R$ be a ring, and $(\Sigma,w)$ an $R$-balanced fan. If, for each face
$\gamma\in\Sigma$, the star fan $\gamma_{\succeq}$ satisfies tropical Poincaré
duality over $R$, we say that $\Sigma$ is a _local tropical Poincaré duality
space over $R$_.

In particular, this implies that $\Sigma$ satisfies TPD over $R$. In the case
where $R=\mathbb{Z}$ or $\mathbb{Q}$, being a local TPD space is equivalent to
the _tropical smoothness_ introduced by Amini and Piquerez in [AP21]. We use a
different notion of the star of a face, but the equivalence of the definitions
can be seen from [AP21, Proposition 3.17], which in turn follows from the
tropical Künneth theorem [GS19, Theorem B].

######  Theorem 5.10.

Let $R$ be a principal ideal domain, and $(\Sigma,w)$ a $d$-dimensional
$R$-balanced fan. Then $\Sigma$ is a local TPD space over $R$ if and only if
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$ for all $\gamma\in\Sigma$
and $q\neq d$, and for all faces $\beta$ of codimension 1, the star fans
$\beta_{\succeq}$ are TPD spaces over $R$.

###### Proof.

If $\Sigma$ is a local TPD space over $R$, each of the star fans
$\gamma_{\succeq}$ for $\gamma\in\Sigma$, in particular the codimension 1
faces are TPD spaces over $R$. Moreover, this implies that the Borel–Moore
homology groups $H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})$ vanish for
$q\neq d$ and all $\gamma\in\Sigma$ by Lemma 4.3.

Next, assume that the star fans $\beta_{\succeq}$ for $\beta\in\Sigma^{d-1}$
are local TPD spaces over $R$ and
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$ for all
$\gamma\in\Sigma$. First, we apply Theorem 5.4 to all faces of codimension two
$\mu$ in $\Sigma$. For a given $\mu\in\Sigma^{d-2}$, we have
$H_{q}^{BM}(\mu_{\succeq},\mathcal{F}_{p}^{R})=0$ for all $p$ by assumption.
Moreover, each face $\widetilde{\beta}\in\mu_{\succeq}$ is a subdivision of a
face $\beta\in\Sigma$ with $\beta\prec\mu$. By assumption all these faces of
$\Sigma$ are TPD spaces, and therefore the subdivided faces
$\widetilde{\beta}$ of $\mu_{\succeq}$ are as well. Hence we may apply Theorem
5.4, and conclude that $\mu_{\succeq}$ is a TPD space. Thus, all of
codimension 2 of $\Sigma$ are TPD spaces. Proceeding inductively, we can apply
Theorem 5.4 to all the stars $\gamma_{\succeq}$ of faces $\gamma\in\Sigma$.
Thus $\Sigma$ is a local TPD space. ∎

######  Corollary 5.11.

Let $(\Sigma,w)$ a $d$-dimensional $\mathbb{Z}$-balanced fan. Then $\Sigma$ is
a local TPD space over $\mathbb{Z}$ if and only if
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{\mathbb{Z}})=0$ for $q\neq d$
and all $\gamma\in\Sigma$, all the weights are $\pm 1$, and for all faces
$\beta$ of codimension 1, the star fans $\beta_{\succeq}$ are uniquely
$\mathbb{Z}$-balanced in the sense of Definition 3.11.

###### Proof.

For each face $\beta$ of codimension 1 of $\Sigma$, observe that each face of
dimension $d$ of the star fan $\beta_{\succeq}$ is a linear space, hence is a
TPD space over $\mathbb{Z}$. Furthermore, each star fan $\beta_{\succeq}$ has
a $(d-1)$-dimensional lineality space, and we may write
$\beta_{\succeq}=\Sigma_{\beta}\times\mathbb{R}^{d-1}$ for some “reduced star”
$\Sigma_{\beta}$ of dimension 1. Since $\mathbb{R}^{d-1}$ satisfies TPD over
$\mathbb{Z}$, by [AP21, Proposition 3.18], the star fan $\beta_{\succeq}$ is a
TPD space over $\mathbb{Z}$ if and only if $\Sigma_{\beta}$ is a a TPD space
over $\mathbb{Z}$. By Proposition 4.7, this is the case if and only if
$\Sigma_{\beta}$ is uniquely $\mathbb{Z}$-balanced with $\pm 1$-weights.
Therefore $\beta_{\succeq}$ is a local TPD space over $\mathbb{Z}$ if and only
if it is uniquely $\mathbb{Z}$-balanced with $\pm 1$-weights. Finally, the
equivalence follows from comparing with Theorem 5.10. ∎

######  Remark 5.12.

Passing from Theorem 5.10 to Corollary 5.11 is mostly dependent on the Künneth
formula for the $\mathcal{F}_{p}^{\mathbb{Z}}$ cosheaves from [GS19]. A
generalization of the Künneth formula to $\mathcal{F}_{p}^{R}$ for another
ring $R$ would also lead to a generalization of Corollary 5.11.

Theorem 5.10 illustrates that it would be desirable to obtain a geometric
understanding of the vanishing condition for the tropical Borel–Moore
homology.

######  Question 5.13 (Geometry of BM homology vanishing?).

Let $(\Sigma,w)$ be an $R$-balanced $d$-dimensional fan. Can the fans with
$H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$ for each face
$\gamma\in\Sigma$, $q\neq d$ and all $p$ be geometrically characterized?

We note that it is not clear whether TPD of the whole fan implies local TPD.
We have not been able to construct a fan satisfying TPD such that the star of
one of its faces does not.

######  Question 5.14 (Global versus Local TPD).

Let $(\Sigma,w)$ be an $R$-balanced fan which satisfies TPD over $R$. Does
$\gamma_{\succeq}$ also satisfy TPD over $R$ for each $\gamma\in\Sigma$?

Even assuming that $H_{q}^{BM}(\gamma_{\succeq},\mathcal{F}_{p}^{R})=0$ for
$q\neq d$, along with TPD on the whole fan $(\Sigma,w)$, the proof of Theorem
5.4 does not directly imply that $\Sigma$ is a local TPD space generally. In
algebraic topology, the question of going from Poincaré duality globally on a
CW complex to Poincaré duality locally has been investigated using techniques
from surgery and homotopy theory (see [Ran11] for an overview).

##  6\. Tropical Poincaré duality for polyhedral spaces

In this section, we use the results of Section 5 to prove that abstract
tropical cycles which have charts to local TPD spaces satisfy tropical
Poincaré duality. In [JRS18, Theorem 5.3], the Mayer–Vietoris argument that
shows that tropical manifolds satisfy tropical Poincaré duality over
$\mathbb{Z}$ is predicated on the existence of charts to fans of matroids,
which are local TPD spaces over $\mathbb{Z}$. This suggests that the local TPD
spaces characterized in Theorem 5.10 are useful as building blocks for general
spaces satisfying TPD. We show this in Theorem 6.11.

We begin by recalling some of the main definitions needed, and refer to
[JRS18] for a more detailed treatment. Let
$\mathbb{T}^{n}=[-\infty,\infty)^{n}$ be the _tropical affine $n$-space_,
equipped with the product topology of the order topology on the half open
intervals. A _rational polyhedron_ in $\mathbb{T}^{n}$ is the closure of a
rational polyhedron in $\mathbb{R}^{n}\subset\mathbb{T}^{n}$. A _rational
polyhedral complex_ in $\mathbb{T}^{n}$ is a collection of rational polyhedra,
containing the faces of each polyhedron, such that all the polyhedra either do
not intersect or meet in a common face. A map
$F\colon\mathbb{T}^{n}\to\mathbb{T}^{m}$ is said to be _extended affine
$\mathbb{Z}$-linear_ if $F(\mathbf{x})=A\mathbf{x}+\mathbf{b}$, for
$A\in\mathbb{Z}^{n\times m}$ and $\mathbf{b}\in\mathbb{R}^{m}$.

######  Definition 6.1 ([JRS18, Definition 2.1]).

A _rational polyhedral space $X$ of pure dimension $d$_ is a paracompact,
second countable Hausdorff topological space with an atlas of charts
$(\phi_{\alpha}\colon U_{\alpha}\to\Omega_{\alpha}\subseteq X_{\alpha})$ such
that:

  1. (1)

The $U_{\alpha}$ are open subsets of $X$, the $\Omega_{\alpha}$ are open
subsets of rational polyhedral complexes
$X_{\alpha}\subseteq\mathbb{T}^{r_{\alpha}}$ of dimension $d$, and the maps
$\phi_{\alpha}~\colon U_{\alpha}\to\Omega_{\alpha}$ are homeomorphisms;

  2. (2)

for all $\alpha,\beta\in A$, the transition maps

$\phi_{\alpha}\circ\phi_{\beta}^{-1}\colon\phi_{\beta}(U_{\alpha}\cap
U_{\beta})\to\phi_{\alpha}(U_{\alpha}\cap U_{\beta})$

are extended affine $\mathbb{Z}$-linear maps.

######  Definition 6.2 ([JRS18, Definition 2.2]).

Let $X$ be a rational polyhedral space of pure dimension $d$. A _rational
polyhedral structure_ on $X$ is a finite family of closed subsets
$\mathcal{C}$ such that the following conditions hold:

  1. (1)

$X=\cup_{\sigma\in\mathcal{C}}\sigma$,

  2. (2)

for each $\sigma$, there exists a chart $\phi_{\sigma}\colon
U\to\Omega\subseteq X$ such that the star $\sigma_{\succeq}\subseteq U$ and
the set $\\{\phi_{\sigma}(\tau)\mid\tau\in\sigma_{\succeq}\\}$ is a rational
polyhedral complex in $\mathbb{T}^{s}\times\mathbb{R}^{r-s}$ of pure dimension
less than or equal to $d$.

The dimension of a cell $\sigma\in\mathcal{C}$ is the dimension of
$\phi_{\sigma}(\sigma)$. For each $k\geq 0$, we denote by $\mathcal{C}^{k}$
the set of $k$-dimensional cells of $\mathcal{C}$.

######  Definition 6.3.

Let $X$ be a rational polyhedral space, with a rational polyhedral structure
$\mathcal{C}$. Let $R$ be a commutative ring, and $\sigma\in\Sigma$ a cell. We
define the _$p$ -th multi-tangent space_ of $X$ at $\sigma$ with respect to
$\mathcal{C}$ to be the $R$-module
$\mathcal{F}_{p}^{R}(\sigma)=\mathcal{F}_{p}^{R}(\phi_{\sigma}(\sigma))$,
where $\phi_{\sigma}$ is the chart in Definition 6.2, and we
$\mathcal{F}_{p}^{R}(\phi_{\sigma}(\sigma))$ is given as Definition 3.1. For
$\tau\prec\sigma\in\mathcal{C}$, there is an inclusion map
$\iota_{\sigma,\tau}\colon\mathcal{F}_{p}^{R}(\sigma)\to\mathcal{F}_{p}^{R}(\tau)$,
which is equal to
$\iota_{\sigma,\tau}\colon\mathcal{F}_{p}^{R}(\phi_{\tau}(\sigma))\to\mathcal{F}_{p}^{R}(\phi_{\tau}(\tau))$.
This data defines the _$p$ -th multi-tangent cosheaf_ $\mathcal{F}_{p}^{R}$.

Similarly, the _$p$ -th multi-cotangent space_ of $X$ at $\sigma$ with respect
to $\mathcal{C}$ is the $R$-module
$\mathcal{F}_{R}^{p}(\sigma)=\mathcal{F}_{R}^{p}(\phi_{\sigma}(\sigma))$. For
$\tau\prec\sigma\in\mathcal{C}$, there is a restriction map
$\rho_{\sigma,\tau}\colon\mathcal{F}_{R}^{p}(\sigma)\to\mathcal{F}_{R}^{p}(\tau)$,
which is equal to
$\rho_{\sigma,\tau}\colon\mathcal{F}_{R}^{p}(\phi_{\tau}(\tau))\to\mathcal{F}_{R}^{p}(\phi_{\tau}(\sigma))$.
This data defines the _$p$ -th multi-tangent sheaf_ $\mathcal{F}_{R}^{p}$.

######  Definition 6.4 ([JRS18, Definitions 2.6-2.7]).

Let $X$ be a rational polyhedral space, with a rational polyhedral structure
$\mathcal{C}$. Replacing each instance of $\Sigma$ by $\mathcal{C}$ in
Definitions 3.5, 2.12 and 2.13, $X$ can be given _tropical cohomology groups
with $R$-coefficients_ $H^{q}(X,\mathcal{F}_{R}^{p})$, as well as _tropical
Borel–Moore homology groups with $R$-coefficients_
$H_{q}^{BM}(X,\mathcal{F}_{p}^{R})$.

######  Remark 6.5.

A priori, Definition 6.4 is dependent on the chosen polyhedral structure
$\mathcal{C}$ on $X$. However, it can be shown ([MZ14, Section 2.2], [JRS18,
Remark 2.8]) that any two choices of cellular structures give the same
$R$-modules $H^{q}(X,\mathcal{F}_{R}^{p})$ and
$H_{q}^{BM}(X,\mathcal{F}_{p}^{R})$.

######  Definition 6.6 ([JRS18, Definitions 4.7-4.8]).

An _$R$ -weight function_ on a rational polyhedral space $X$ of pure dimension
$d$ with a polyhedral structure $\mathcal{C}$ is a function
$w\colon\mathcal{C}^{d}\to R$ such that for all $\sigma\in\mathcal{C}^{d}$,
the weight $w(\sigma)$ is not a zero-divisor. Each $R$-weight function induces
a _fundamental chain_ $\mathrm{Ch}(X,w)\in C_{d}^{BM}(X,\mathcal{F}_{d}^{R})$,
defined by

$(w(\sigma)\Lambda_{\sigma})_{\sigma}\in\bigoplus_{\sigma\in\mathcal{C}^{d}}\mathcal{F}_{d}^{R}(\sigma)=C_{d}^{BM}(X,\mathcal{F}_{d}^{R}),$

where $\Lambda_{\sigma}$ is the generator of $\mathcal{F}_{d}^{R}(\sigma)\cong
R$ as in Remark 3.4.

######  Definition 6.7.

A rational polyhedral space $X$ of pure dimension $d$ with a rational
polyhedral structure $\mathcal{C}$ and a weight function $w$ is _balanced_ if
the fundamental chain $\mathrm{Ch}(X,w)\in C_{d}^{BM}(X,\mathcal{F}_{d}^{R})$
is closed, inducing a _fundamental class_ $[X,w]\in
H_{d}^{BM}(X,\mathcal{F}_{d}^{R})$ in tropical Borel–Moore homology. We call
the triple $(X,\mathcal{C},w)$ an _abstract tropical $R$-cycle_.

######  Definition 6.8 ([JRS18, Definiton 4.11]).

Let $X$ be an abstract tropical $R$-cycle of dimension $d$. The fundamental
class $[X,w]$ induces a _cap product_

$\frown[X,w]\colon H^{q}(X,\mathcal{F}_{R}^{p})\to
H_{d-p}^{BM}(X,\mathcal{F}_{d-p}^{R})$

between tropical cohomology and tropical Borel–Moore homology. If
$\frown[X,w]$ is an isomorphism for all $p$ and $q$, we say that $X$ is a
_tropical Poincaré duality space over $R$_.

######  Definition 6.9.

Let $(X,\mathcal{C},w)$ be an abstract tropical $R$-cycle over a commutative
ring $R$. We say that $(X,\mathcal{C},w)$ is a _local tropical Poincaré
duality space_ if for each $\sigma\in\mathcal{C}$, the rational polyhedral
complexes $\\{\phi_{\sigma}(\tau)\mid\tau\in\sigma_{\succeq}\\}$ are local TPD
spaces over $R$.

######  Example 6.10.

Tropical manifolds, which have charts to Bergman fans of matroids, are
examples of local TPD spaces over $\mathbb{Z}$ and $\mathbb{R}$, see [JRS18,
JSS19].

######  Theorem 6.11.

Let $X$ be a local tropical Poincaré duality space over a principal ideal
domain $R$. Then $X$ satisfies tropical Poincaré duality over $R$.

###### Proof.

The two steps of the proof given in [JRS18, Proof of Theorem 5.3] carry
through. Since the open stars of faces satisfy TPD over $R$, the first step is
identical, noting that the same arguments carry through working in the
category of $R$-modules. The induction argument given in the second step also
carries through, as the same sequence of complexes can be constructed in the
category of $R$-modules. ∎

######  Remark 6.12.

Note that Definition 6.9 in the case where $R=\mathbb{Z}$ is equivalent to the
definition of _smooth tropical variety_ given in [AP21, Definition 3.22], such
that the case $R=\mathbb{Z}$ of Theorem 6.11 is equivalent to [AP21, Theorem
3.23].

Theorem 6.11 justifies the naming in Definition 6.9, generalizing the
relationship between local TPD spaces and TPD spaces as defined in Definition
5.9 and Definition 4.1. Moreover, 5.14 about the relationship between local
TPD and TPD are also applicable in this more general setting.

######  Question 6.13 (Global versus Local TPD for abstract tropical cycles).

Let $(X,\mathcal{C},w)$ be an abstract tropical $R$-cycle satisfying TPD over
$R$. Does $\gamma_{\succeq}$ also satisfy TPD over $R$ for each
$\gamma\in\mathcal{C}$?

## References

  * [polymake] Ewgenij Gawrilow and Michael Joswig  “polymake: a framework for analyzing convex polytopes”  In _Polytopes—combinatorics and computation (Oberwolfach, 1997)_ 29, DMV Sem.  Birkhäuser, Basel, 2000, pp. 43–73
  * [AHK18] Karim Adiprasito, June Huh and Eric Katz  “Hodge theory for combinatorial geometries”  In _Ann. of Math. (2)_ 188.2, 2018, pp. 381–452  DOI: 10.4007/annals.2018.188.2.1
  * [AK06] Federico Ardila and Caroline J. Klivans  “The Bergman complex of a matroid and phylogenetic trees”  In _J. Combin. Theory Ser. B_ 96.1, 2006, pp. 38–49  DOI: 10.1016/j.jctb.2005.06.004
  * [Aks19] Edvard Aksnes  “Tropical Poincaré duality spaces”, 2019
  * [AP20] Omid Amini and Matthieu Piquerez  “Hodge theory for tropical varieties”, 2020  arXiv:2007.07826 [math.AG]
  * [AP21] Omid Amini and Matthieu Piquerez  “Homology of tropical fans”, 2021  arXiv:2105.01504 [math.AG]
  * [BH17] Farhad Babaee and June Huh  “A tropical approach to a generalized Hodge conjecture for positive currents”  In _Duke Math. J._ 166.14, 2017, pp. 2749–2813  DOI: 10.1215/00127094-2017-0017
  * [BIMS15] Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin and Kristin Shaw  “Brief introduction to tropical geometry”  In _Proceedings of the Gökova Geometry-Topology Conference 2014_ Gökova Geometry/Topology Conference (GGT), Gökova, 2015, pp. 1–75
  * [Bou98] Nicolas Bourbaki  “Algebra I. Chapters 1–3” Translated from the French, Reprint of the 1989 English translation [ MR0979982 (90d:00002)], Elements of Mathematics (Berlin)  Springer-Verlag, Berlin, 1998, pp. xxiv+709
  * [Bri97] Michel Brion  “The structure of the polytope algebra”  In _Tohoku Math. J. (2)_ 49.1, 1997, pp. 1–32  DOI: 10.2748/tmj/1178225183
  * [Cur14] Justin Michael Curry  “Sheaves, cosheaves and applications”  ProQuest LLC, Ann Arbor, MI, 2014
  * [FS97] William Fulton and Bernd Sturmfels  “Intersection theory on toric varieties”  In _Topology_ 36.2, 1997, pp. 335–353  DOI: 10.1016/0040-9383(96)00016-X
  * [GS19] Andreas Gross and Farbod Shokrieh  “A sheaf-theoretic approach to tropical homology”, 2019  arXiv:1906.09245 [math.AG]
  * [How08] Valerie Hower  “Hodge spaces of real toric varieties”  In _Collect. Math._ 59.2, 2008, pp. 215–237  DOI: 10.1007/BF03191368
  * [Huh14] June Huh  “Rota’s conjecture and positivity of algebraic cycles in permutohedral varieties” Thesis (Ph.D.)–University of Michigan  ProQuest LLC, Ann Arbor, MI, 2014, pp. 73
  * [Huh18] June Huh  “Tropical geometry of matroids”  In _Current developments in mathematics 2016_ Int. Press, Somerville, MA, 2018, pp. 1–46
  * [IKMZ19] Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin and Ilia Zharkov  “Tropical homology”  In _Math. Ann._ 374.1-2, 2019, pp. 963–1006  DOI: 10.1007/s00208-018-1685-9
  * [JRS18] Philipp Jell, Johannes Rau and Kristin Shaw  “Lefschetz $(1,1)$-theorem in tropical geometry”  In _Épijournal Geom. Algébrique_ 2, 2018, pp. Art. 1127
  * [JSS19] Philipp Jell, Kristin Shaw and Jascha Smacka  “Superforms, tropical cohomology, and Poincaré duality”  In _Adv. Geom._ 19.1, 2019, pp. 101–130  DOI: 10.1515/advgeom-2018-0006
  * [KSW17] Lars Kastner, Kristin Shaw and Anna-Lena Winz  “Cellular sheaf cohomology of polymake”  In _Combinatorial algebraic geometry_ 80, Fields Inst. Commun.  Fields Inst. Res. Math. Sci., Toronto, ON, 2017, pp. 369–385
  * [MR] Grigory Mikhalkin and Johannes Rau  “Tropical Geometry” Draft at https://math.uniandes.edu.co/~j.rau/downloads/main.pdf
  * [MS15] Diane Maclagan and Bernd Sturmfels  “Introduction to tropical geometry” 161, Graduate Studies in Mathematics  American Mathematical Society, Providence, RI, 2015, pp. xii+363
  * [MZ14] Grigory Mikhalkin and Ilia Zharkov  “Tropical eigenwave and intermediate Jacobians”  In _Homological mirror symmetry and tropical geometry_ 15, Lect. Notes Unione Mat. Ital.  Springer, Cham, 2014, pp. 309–349  DOI: 10.1007/978-3-319-06514-4_7
  * [Ran11] Andrew Ranicki  “The Poincaré Duality Theorem and its converse” Presentation available at https://www.maths.ed.ac.uk/~v1ranick/surgery/poincareconverse.pdf, 2011
  * [She85] Allen Dudley Shepard  “A cellular description of the derived category of a stratified space” Thesis (Ph.D.)–Brown University  ProQuest LLC, Ann Arbor, MI, 1985, pp. 161  URL: https://search.proquest.com/openview/ca196f7bbe67f464b8da5c5930e20635/1?pq-origsite=gscholar&cbl=18750&diss=y
  * [Wei94] Charles A. Weibel  “An introduction to homological algebra” 38, Cambridge Studies in Advanced Mathematics  Cambridge University Press, Cambridge, 1994
  * [Zha13] Ilia Zharkov  “The Orlik-Solomon algebra and the Bergman fan of a matroid”  In _J. Gökova Geom. Topol. GGT_ 7, 2013, pp. 25–31