# Monotone cohomologies and oriented matchings

Luigi Caputi  ,  Daniele Celoria  and  Carlo Collari

###### Abstract.

In this paper, we extend the definition of cohomology associated to monotone
graph properties, to encompass twisted functor coefficients. We introduce
oriented matchings on graphs, and focus on their (twisted) cohomology groups.
We characterise oriented matchings in terms of induced free-flow
pseudoforests, and explicitly determine the homotopy type of the associated
simplicial complexes. Furthermore, we provide a connection between the
cohomology of oriented matchings with certain functor coefficients, and the
recently defined multipath cohomology. Finally, we define a further oriented
homology for graphs and interpret it as a count of free-flow orientations.

##  1\. Introduction

In recent years, the study of graphs by means of techniques rooted in
combinatorial algebraic topology has developed at a very rapid pace. One of
the main avenues of research focuses on the interplay between graphs and
simplicial complexes; indeed, there are many distinct natural ways to
associate a simplicial complex to a graph. Combinatorial, homotopical, and
homological properties of these complexes encode interesting information about
the graph. The study of these simplicial complexes is now an established and
active area of research [Wac03, Koz08, Jon08], with deep connections with
other areas of mathematics – see [Jon08, Chapter 1] for an overview.

In this paper we focus on a special class of simplicial complexes associated
to certain graph matchings. As most complexes related to matchings on graphs,
they arise as a special case of _monotone complexes_ ; that is simplicial
complexes of subgraphs defined by a property which is closed under edge
removal. Such properties are collectively called _monotone properties_.
Monotone complexes have been extensively studied, both from a combinatorial
and topological perspective [Jon08].

In most cases, investigations of monotone complexes have been carried out by
applying standard homological methods. We start our analysis with the
observation that, quite often, more refined invariants arise when considering
twisted coefficients. Paralleling well-known constructions in topology (_cf_.
[Qui73] and [Ric20, Theorem 16.2.3]), our interest lies in extending these
constructions to functors coefficient. To this end, using techniques borrowed
from the recently defined poset homology [Cha19, CCDT21b] – for a given
monotone property $\mathscr{M}$ on a graph ${\tt G}$ and a certain functor
$\mathcal{F}_{A}$ (depending on an algebra $A$) – we construct a cohomology
theory $\mathrm{H}_{\mathscr{M}}({\tt G};A)$ which we call _monotone
cohomology_. This approach provides a novel and comprehensive tool to study
monotone complexes. By varying the choice of monotone property, one can
realise, for example, the chromatic homology of graphs [HGR05] and the
multipath cohomology of directed graphs [CCDT21b] as monotone cohomologies.

We focus our approach on the special case of _oriented matchings_ , and show
that their associated monotone cohomology can be effectively computed. More
precisely, for any oriented graph ${\tt G}$, we define the oriented matching
complex $\mathrm{M}^{o}({\tt G})$; this is obtained by considering a subset of
matchings on the face poset of ${\tt G}$ that are “compatible” with the given
orientation. Oriented matchings are naturally related to pseudoforests endowed
with a specific type of orientations, which we call _free-flow_ (_cf._
Definition 3.6). As a first result, we provide a complete characterisation of
these matchings in terms of the subgraphs they induce, in analogy with [Koz99,
Cha00].

######  Proposition 1.1.

Let ${\tt G}$ be an oriented graph. Simplices in $\mathrm{M}^{o}({\tt G})$ are
in bijection with rooted spanning pseudoforests in ${\tt G}$ whose induced
orientation is free-flow.

The homotopical and combinatorial properties of the oriented matching
complexes, turn out to be tightly intertwined. Indeed, the homotopy type of
$\mathrm{M}^{o}({\tt G})$ is completely determined by simple quantities
associated to the orientation of ${\tt G}$. The explicit description that we
obtain exhibits these complexes as iterated suspensions, whose parameters are
related to the graph’s indegrees (_cf._ Proposition 5.3).

######  Proposition 1.2.

For a connected oriented graph ${\tt G}$, the complex $\mathrm{M}^{o}({\tt
G})$ is either contractible or homotopic to a wedge of spheres. More
precisely, if a vertex in ${\tt G}$ has indegree $1$, then
$\mathrm{M}^{o}({\tt G})$ is contractible. Otherwise, we have the homotopy
equivalence

$\mathrm{M}^{o}({\tt G})\simeq\bigvee^{q}\mathbb{S}^{N-1}\ ,$

where $q$ and $N$ are natural numbers depending on the indegrees of vertices
in ${\tt G}$.

Our proof of the above result makes use of a correspondence between matchings
on graphs and multipaths – some special subgraphs satisfying a monotone
property [TW12]. A similar correspondence between oriented matchings and
multipaths can be used to shed light on the simplicial structure of
$\mathrm{M}^{o}({\tt G})$, allowing us to obtain the following result:

######  Corollary 1.3.

Oriented matching complexes are either strongly shellable or contractible.
In particular, all non-contractible matching complexes are Cohen-Macaulay.

For an oriented graph ${\tt G}$ and a commutative algebra $A$, let
$C_{o}^{*}({\tt G};A)$ and $C_{\mu}^{*}({\tt G};A)$ be the cochain complexes
computing the cohomologies associated to the oriented matching and multipath
complexes, respectively. The relation between these two complexes is not
direct. Rather, to an oriented graph ${\tt G}$ we associate its “source
resolution” ${\tt G}_{sr}$ (Definition 4.6), then the relation is established
as follows:

######  Theorem 1.4.

The association ${\tt G}\mapsto{\tt G}_{sr}$ yields the following isomorphism
of cochain complexes

$C_{o}^{*}({\tt G};A)\cong C_{\mu}^{*}({\tt G}_{sr};A)\otimes A^{\otimes s}$

where $s$ is the number of vertices of indegree $0$ in ${\tt G}$.

Another interesting aspect of the oriented matching complexes
$\mathrm{M}^{o}$, is that they provide a decomposition in subcomplexes of the
matching complex of ${\tt G}$. In the spirit of [CCC22], we combine them in a
single object; given an oriented graph ${\tt G}_{\mathfrak{o}}$, we consider
all possible orientations on the underlying unoriented graph ${\tt G}$. The
choice of $\mathfrak{o}$ provides a natural identification of this set with a
Boolean poset. We decorate the vertices of this poset with the simplices of
the corresponding oriented matching complex. We can then define
$\mathrm{OH}^{*}({\tt G}_{\mathfrak{o}})$, the oriented homology of ${\tt G}$
with respect to the orientation $\mathfrak{o}$. Surprisingly, it turns out
that the structure of this homology is very simple, and tightly related to
free-flow orientations:

######  Theorem 1.5.

Let ${\tt G}$ be a connected unoriented graph, and let $\mathfrak{o}$ be an
orientation on ${\tt G}$. Then, generators of $\mathrm{OH}^{*}({\tt
G}_{\mathfrak{o}})$ are in bijection with free-flow orientations on ${\tt G}$.

Despite the fact that the dimension of ${\rm OH}({\tt G}_{\mathfrak{o}})$
depends only on the underlying unoriented graph, the grading in ${\rm OH}$ can
distinguish between different orientations.

### Acknowledgements

LC acknowledges support from the École Polytechnique Fédérale de Lausanne via
a collaboration agreement with the University of Aberdeen. DC was partially
supported by the European Research Council (ERC) under the EU Horizon 2020
research and innovation programme (grant agreement No 674978), and by Hodgson-
Rubinstein’s ARC grant DP190102363 “Classical And Quantum Invariants Of Low-
Dimensional Manifolds”. CC is supported by the MIUR-PRIN project 2017JZ2SW5.

##  2\. Monotone cohomologies of oriented graphs

Among the most common approaches in defining (co)homology theories of graphs
is to first associate to a graph ${\tt G}$ a simplicial complex $S({\tt G})$,
and then to compute the classical (co)homology groups of $S({\tt G})$.
Similarly, one can associate to $S({\tt G})$ a poset; for instance, one can
consider its face poset ${F}(S({\tt G}))$. Then, one can apply a (co)homology
theory of posets. The simplest approach, given by computing poset homology
groups as in [Wac06], yields the simplicial cohomology groups of $S({\tt G})$.
Since a poset can be seen as a category, we can borrow results and methods
from category theory; the following diagram illustrates the idea:

${\mathbf{Cat}}$${\mathbf{Poset}}$${\mathbf{Simp}}$OGraph${\mathbf{Ab}}$Homologies
of oriented graphs

The unmarked arrows pointing towards ${\bf Ab}$ represent homology theories
for simplicial complexes, posets, and categories, respectively. In this
section we introduce the so-called “poset homology” as in [Cha19, CCDT21b],
and specialise this construction to certain posets associated to oriented
graphs.

###  2.1. Poset homology

We start by reviewing the definition of poset homology for a finite poset $P$
with coefficients in a functor $\mathcal{F}$. We remark here that this
construction is related to, but not the same as, the classical poset homology
(see, _e.g._ [Wac06]) which is defined as the homology of the nerve associated
to the poset. We refer to [Cha19, CCDT21b] for more general expositions on the
topic.

For a poset $(P,<)$, let $\prec$ denote the _covering relation_ associated to
$<$, _i.e._ $x\prec y$ if and only if $x<y$ and there is no $z$ such that
$x<z<y$.

We say that $P$ is _ranked_ if there is a rank function $\ell\colon
P\to\mathbb{N}$ such that $x\prec y$ implies $\ell(y)=\ell(x)+1$. We say that
$P$ is _squared_ if, for each triple $x,y,z\in P$ such that $z\prec y\prec x$,
there is a unique $y^{\prime}\neq y$ such that $z\prec y^{\prime}\prec x$.
Such elements $z,y,y^{\prime},x$, together with their covering relations in
$P$, will be called a _square_. In what follows, we will assume all posets to
be ranked and squared.

######  Example 2.1.

Recall that a _regular_ CW-complex is a CW-complex for which all attaching
maps are homeomorphisms. A _CW-poset_ (_i.e._ the collection of the cells in a
regular CW complex, ordered by containment [Bjo84, Definition 2.1 &
Proposition 3.1]) is ranked and squared; the rank function is given by the
dimension of the cells. In particular, the face poset $F(X)$ of a simplicial
complex $X$ is ranked and squared.

A finite poset $(P,<)$ can be seen as a (small) category $\mathbf{P}$; the set
of objects of P is the set $P$, and there is a unique morphism $x\to y$ if and
only if $x\leq y$. Functors on the category associated to the poset $P$
preserve commutative squares: For each $x,z\in P$, $x\leq z$, there is a
unique mapping $f_{x,z}\colon x\to z$ in the category P. Assume there is a
square between $x$ and $z$; the existence of such a square implies that
$f_{x,z}$ factors as follows

$f_{x,z}=f_{y,z}\circ f_{x,y}=f_{y^{\prime},z}\circ f_{x,y^{\prime}}\ .$

Therefore, given a covariant functor $\mathcal{F}:{\bf P}\to{\bf C}$, we must
have:

$\mathcal{F}(f_{y,z})\circ\mathcal{F}(f_{x,y})=\mathcal{F}(f_{y,z}\circ
f_{x,y})=\mathcal{F}(f_{x,z})=\mathcal{F}(f_{y^{\prime},z}\circ
f_{x,y^{\prime}})=\mathcal{F}(f_{y^{\prime},z})\circ\mathcal{F}(f_{x,y^{\prime}})\
.$

In other words, all functors preserve the commutativity of the squares in $P$.

Let $\mathbb{Z}_{2}$ be the cyclic group on two elements.

######  Definition 2.2.

A _sign assignment_ on a poset $(P,<)$ is an assignment of elements
$\varepsilon({x,y})\in\mathbb{Z}_{2}$ to each pair of elements $x,y\in P$ with
$x\prec y$, such that

$\varepsilon({x,y})+\varepsilon({y,z})\equiv\varepsilon({x,y^{\prime}})+\varepsilon({y^{\prime},z})+1\mod
2$

holds for each square $x\prec y,\leavevmode\nobreak\ y^{\prime}\prec z$.

In general, the existence of a sign assignment on a poset $P$ depends on the
topology of a certain CW-complex associated to $P$ – see, _e.g._ [CCDT21b,
Section 3.2], [Put14, Section 5].

######  Remark 2.3.

Every CW-poset admits a sign assignment, which is unique up to (a suitable
notion of) isomorphism – see, _e.g._ [Cha19, Section 4].

We can now recall the definition of poset homology of a poset $P$ with
coefficients in a functor $\mathcal{F}$.

Let ${\bf A}$ be an Abelian category – such as the category of left modules on
a commutative ring $R$ – $P$ a ranked squared poset with rank function $\ell$,
and $\varepsilon$ a sign assignment on $P$. Given a covariant functor
$\mathcal{F}\colon{\bf P}\to{\bf A}$, we define the cochain groups

(1) $C^{n}_{\mathcal{F}}(P)\coloneqq\bigoplus_{\begin{subarray}{c}x\in P\\\
\ell(x)=n\end{subarray}}\mathcal{F}(x),$

and the differentials

(2) $d^{n}=d^{n}_{\mathcal{F}}\coloneqq\sum_{\begin{subarray}{c}x\in P\\\
\ell(x)=n\end{subarray}}\ \sum_{\begin{subarray}{c}x^{\prime}\in P\\\ x\prec
x^{\prime}\end{subarray}}(-1)^{\varepsilon(x,x^{\prime})}\mathcal{F}(x\prec
x^{\prime})\ .$

With these definitions in place, we can state one of the main results from
[Cha19] and [CCDT21b].

######  Theorem 2.4.

Let ${\bf A}$ be an Abelian category, $P$ be a ranked squared poset, and
$\varepsilon$ be a sign assignment on $P$. Then, for any $n\in\mathbb{N}$ and
any functor $\mathcal{F}\colon{\bf P}\to{\bf A}$ we have $d^{n+1}\circ
d^{n}\equiv 0$. In particular, $(C^{*}_{\mathcal{F}}(P),d^{*})$ is a cochain
complex.

The differentials $d^{n}$, and therefore the cochain complexes, depend _a
priori_ upon the choice of the sign assignment $\varepsilon$. In the cases of
interest to us, that is CW-posets, the choice of the sign assignment does not
affect the isomorphism type of the cochain complexes
$(C^{*}_{\mathcal{F}}(P),d^{*})$ – see, for instance, [CCDT21b, Corollary
3.18].

###  2.2. Monotone cohomologies of oriented graphs

Recall that an _unoriented graph_ G is a pair of finite sets $(V,E)$
consisting of: a set of _vertices_ $V$, and a set $E$ whose elements, called
_edges_ , are unordered pairs of distinct vertices of ${\tt G}$. All graphs
are assumed to be simple. We will also consider oriented graphs, whose
definition we now recall;

######  Definition 2.5.

An _oriented graph_ ${\tt G}$ is a pair of finite sets $(V({\tt G}),E({\tt
G}))$, such that $E({\tt G})$ is a subset of $V({\tt G})\times V({\tt
G})\setminus\\{(v,v)\ |\ v\in V({\tt G})\\}$, and at most one among $(v,w)$
and $(w,v)$ belongs to $E({\tt G})$.

By definition, an edge $e$ of an oriented graph ${\tt G}$ is an ordered set of
two distinct vertices, say $e=(v,w)$. The vertex $v$ is the _source_ ${\tt
s}(e)$ of $e$, while the vertex $w$ is the _target_ ${\tt t}(e)$ of $e$.

A morphism of oriented graphs is a function $\phi\colon V({\tt G}_{1})\to
V({\tt G}_{2})$ sending edges to edges:

$(v,w)\in E({\tt G}_{1})\Rightarrow(\phi(v),\phi(w))\in E({\tt G}_{2})\ ;$

observe that a morphism of oriented graphs does not allow collapsing, meaning
that $(v,w)\in E({\tt G}_{1})\Rightarrow\phi(v)\neq\phi(w)$. We call _regular_
those morphisms of oriented graphs that are also injective as maps of the
vertices111Note that non-collapsing does not imply regular.. Oriented graphs
and regular morphisms of oriented graphs form a category that we denote by
OGraph.

######  Remark 2.6.

The results in this section are stated in the category OGraph, but everything
holds verbatim for other categories of graphs, such as unoriented graphs,
directed graphs, quivers, _etc_. Throughout the rest of the paper, when clear
from the context, we will omit the reference to the category of graphs we are
using.

Let $\mathbf{PosFun}$ be the category of tuples
$(P,\mathcal{F},\mathbf{A},\varepsilon)$ consisting of a poset $P$, a functor
$\mathcal{F}\colon\mathbf{P}\to\mathbf{A}$ with values in an additive category
$\mathbf{A}$, and a sign assignment $\varepsilon$ on $P$. Then, any functor

$\textbf{OGraph}\to\mathbf{PosFun}$

associating to each oriented graph ${\tt G}$ the tuple
$(P,\mathcal{F},\mathbf{A},\varepsilon)$, produces a homology theory of
oriented graphs. We can then apply the poset homology construction detailed in
the previous section to obtain cohomology groups
$\mathrm{H}_{*}(P;\mathcal{F})$. For a general overview on this framework, see
[Cha19, Section 7].

######  Example 2.7.

Let $S\colon\textbf{OGraph}\to\mathbf{Simp}$ a functor from oriented graphs to
the category of simplicial complexes and simplicial maps. Let $R$ be a
commutative unital ring, $R\text{-}\mathbf{Mod}$ the category of $R$-modules,
and $\mathcal{F}$ the constant functor $R$ on the (category associated to the)
face poset $\mathbf{F}(S({\tt G}))$. Then, for each choice of a sign
assignment $\varepsilon$ on $P$, we have an induced functor ${\tt
G}\mapsto(\mathbf{F}(S({\tt G})),R,\mathbf{Ab},\varepsilon)$ whose associated
poset homology is the classical simplicial homology of the simplicial complex
$S({\tt G})$, with coefficient in the ring $R$.

We want to apply this general machinery to the case of a poset $P$, whose
elements are subgraphs of a given graph ${\tt G}$. In such case, we can
specify a functor $\mathcal{F}_{A}\colon{\bf P}\to R\text{-}\mathbf{Mod}$
depending on an algebra $A$.

Let ${\tt G}$ be a graph. Recall that a property $\mathscr{M}$ on the set
$SS({\tt G})$ of spanning222That is, a subgraph whose set of vertices is the
same as the set of vertices of the whole graph. subgraphs of ${\tt G}$ is
called _monotone_ if

$\mathscr{M}({\tt H})\Rightarrow\mathscr{M}({\tt K})$

for any subgraph ${\tt K}$ in $SS({\tt G})$ obtained from ${\tt H}$ by
removing one edge. For a given monotone property $\mathscr{M}$ on ${\tt G}$,
we define the poset $P_{\mathscr{M}}({\tt G})$ whose elements are the spanning
subgraphs of ${\tt G}$ satisfying $\mathscr{M}$, endowed with the inclusion
relation.

######  Example 2.8.

For a (non-necessarily oriented) graph ${\tt G}$, the set of spanning graphs
$SS({\tt G})$, endowed with the inclusion relation, is the poset associated to
the monotone property “being a spanning subgraph”. Note that $SS({\tt G})$ is
isomorphic (as a poset) to the Boolean poset $(E({\tt G}),\subset)$.

######  Example 2.9.

A _multipath_ in an oriented graph ${\tt G}$ is a spanning subgraph whose
vertices have both indegree and outdegree at most one, and does not contain
any cycle – _cf._ [TW12]. The _path poset_ $P({\tt G})$ is the poset given by
all multipaths in ${\tt G}$ ordered by inclusion.

######  Remark 2.10.

For any given monotone property $\mathscr{M}$, the poset $P_{\mathscr{M}}({\tt
G})$ is squared and downward closed in the Boolean poset $SS({\tt G})$.
Moreover, the poset ${P}_{\mathscr{M}}({\tt G})$ is a CW-poset, and in fact
simplicial.

From now on, $R$ denotes a commutative ring with identity, and $A$ is a
commutative unital $R$-algebra. We now define an explicit functor
$\mathcal{F}_{A}\colon{\bf P}_{\mathscr{M}}({\tt G})\to
R\text{-}\mathbf{Mod}$.

Given a subgraph ${\tt H}\subseteq{\tt G}$ such that $\mathscr{M}({\tt H})$
holds, to each connected component of ${\tt H}$, we associate a copy of $A$.
Then we take their ordered tensor product; to do so, we must fix an order of
the connected components of each subgraph in $\mathscr{M}({\tt G})$. It can be
proved that this choice is immaterial, _cf_. [HGR05, CCDT21b]. For the sake of
concreteness, we fix an order on $V({\tt G})$, thus inducing an order of the
components in each ${\tt H}\in SS({\tt G})$, according to the minimal vertex
on each component. If $c_{0}<\dots<c_{k}$ is the set of ordered connected
components of ${\tt H}$, we define:

(3) $\mathcal{F}_{A}({\tt H})\coloneqq
A_{c_{1}}\otimes_{R}\cdots\otimes_{R}A_{c_{k}}\ ,$

where all the modules are labelled by the respective connected component.

Assume now that ${\tt H}^{\prime}={\tt H}\cup e$ and that $\mathscr{M}({\tt
H}^{\prime})$ holds. Denote by $c_{0}$,…,$c_{k}$ the ordered components of
${\tt H}$, and by $c^{\prime}_{0}$,…,$c^{\prime}_{k-1}$ the ordered components
of ${\tt H}^{\prime}$; further assume that the addition of $e$ merges the
components $c_{i}$ and $c_{j}$. Then, for each $h=0,...,k-1$, there is a
natural identification

(4) $c^{\prime}_{h}=\begin{cases}c_{h}&\text{if }0\leq h<i\text{ or }i<h<j;\\\
c_{i}\cup e\cup c_{j}&\text{if }h=i;\\\ c_{h+1}&\text{if }j\leq
h<k.\end{cases}$

for some $0\leq i<j\leq k$. Using this identification, we define $\mu_{{\tt
H}\prec{\tt H}^{\prime}}\colon\mathcal{F}_{A}({\tt
H})\longrightarrow\mathcal{F}_{A}({\tt H}^{\prime})$ as

$\mu_{{\tt H}\prec{\tt H}^{\prime}}(a_{0}\otimes\cdots\otimes
a_{k})=a_{0}\otimes\cdots\otimes a_{i-1}\otimes a_{i}\cdot a_{j}\otimes
a_{i+1}\otimes\cdots\otimes\widehat{a_{j}}\otimes\cdots\otimes a_{k-1}\otimes
a_{k}$

where $\widehat{a_{j}}$ indicates the removal of $a_{j}$. Now, assume that
${\tt H}^{\prime}={\tt H}\cup e$, that $\mathscr{M}({\tt H}^{\prime})$ holds,
and that, unlike the above case, the addition of $e$ does not affect the
number of connected components. In this case, we have a natural identification
between connected components in ${\tt H}$ and ${\tt H}^{\prime}$, which
induces an identification ${\rm I}_{{\tt H}\prec{\tt
H}^{\prime}}:\mathcal{F}_{A}({\tt H})\to\mathcal{F}({\tt H}^{\prime})$. We can
therefore define

(5) $\mathcal{F}_{A}({\tt H}\preceq{\tt
H}^{\prime})\coloneqq\begin{cases}\mu_{{\tt H}\prec{\tt H}^{\prime}}&\text{ if
}{\tt H}\prec{\tt H}^{\prime},\ |\pi_{0}({\tt H}^{\prime})|<|\pi_{0}({\tt
H})|\\\ {\rm I}_{{\tt H}\prec{\tt H}^{\prime}}&\text{ if }{\tt H}\prec{\tt
H}^{\prime},\ |\pi_{0}({\tt H}^{\prime})|=|\pi_{0}({\tt H})|\\\
\mathrm{Id}_{\mathcal{F}_{A}({\tt H})}&\text{ if }{\tt H}={\tt
H}^{\prime}\end{cases}\ .$

Equations (3) and (5) describe a functor

(6) $\mathcal{F}_{A}\colon\mathbf{P}_{\mathscr{M}}({\tt G})\to
R\text{-}\mathbf{Mod}$

from the category $\mathbf{P}_{\mathscr{M}}({\tt G})$ to the additive category
$R$-$\mathbf{Mod}$ of left $R$-modules. In fact, we have the following:

######  Proposition 2.11.

The assignment $\mathcal{F}_{A}\colon\mathbf{P}_{\mathscr{M}}({\tt G})\to
R\text{-}\mathbf{Mod}$ defines a covariant functor.

###### Proof.

The assignment $\mathcal{F}_{A}({\tt H}\prec{\tt
H}^{\prime})\coloneqq\mu_{{\tt H}\prec{\tt H}^{\prime}}$ in Equation (5)
preserves all commutative squares in $\mathbf{P}_{\mathscr{M}}({\tt G})$ –
here we used that the algebra $A$ is commutative (see also [HGR05, Subsection
2.2]). The poset $\mathbf{P}_{\mathscr{M}}({\tt G})$ is a CW-poset, hence the
statement follows from [Cha19, Theorems 6.1 & 5.14]. ∎

We can summarize the results of this section in the following theorem;

######  Theorem 2.12.

Let $\mathscr{M}$ be a monotone graph property. Then the graded $R$-module
$C^{*}_{\mathcal{F}_{A}}(P_{\mathscr{M}}({\tt G}))$, endowed with the
differential $d^{*}_{\mathcal{F}_{A}}$ is a cochain complex.

###### Proof.

By Remark 2.3 there exists a sign assignment on $P_{\mathscr{M}}({\tt G})$,
and $\mathcal{F}_{A}\colon{\bf P}_{\mathscr{M}}({\tt G})\to
R\text{-}\mathbf{Mod}$ is a functor by Proposition 2.11. Therefore, by Theorem
2.4, $(C^{*}_{\mathcal{F}_{A}}(P_{\mathscr{M}}({\tt G})),d^{*})$ is a cochain
complex. ∎

We define the associated cohomology groups:

######  Definition 2.13.

Let $\mathscr{M}$ be a monotone property. The _monotone cohomology_
$\mathrm{H}_{\mathscr{M}}^{*}({\tt G};A)$ (with respect to $\mathscr{M}$ and
$A$) of an oriented graph ${\tt G}$ is the homology of the cochain complex
$(C^{*}_{\mathscr{M}}({\tt G};A),d^{*})$, where $C^{*}_{\mathscr{M}}({\tt
G};A)\coloneqq C^{*}_{\mathcal{F}_{A}}(P_{\mathscr{M}}({\tt G}))$.

Consider the posets of spanning subgraphs and the path poset (_cf._ Examples
2.8 and 2.9). The associated monotone cohomologies are the chromatic homology
[HGR05] and the multipath cohomology [CCDT21b], respectively.

In the definition of $\mathcal{F}_{A}$, one can replace $I_{{\tt H}\prec{\tt
H}^{\prime}}$ with the zero morphism. The replacement yields again a well-
defined cochain complex. When $\mathscr{M}=SS({\tt G})$, this was done by
Przyticki [Prz10] to obtain a variation of the chromatic homology. This theory
was used to provide a connection between Khovanov homology and Hochschild
homology.

##  3\. Oriented matching complexes

The purpose of this section is twofold; first, we introduce several classical
concepts related to graphs, such as matchings and their associated simplicial
complexes. Then, we show how to extend this to the oriented setting by
defining oriented matchings. These matchings turn out to be related to a
special kind of orientations, called free-flow.

###  3.1. Matchings

A (oriented or unoriented) graph ${\tt G}$ can be regarded as a
$1$-dimensional simplicial complex. We denote by $F({\tt G})$ its face poset;
this is the poset whose elements are the non-empty simplices in ${\tt G}$, and
whose order is given by inclusion.

$v_{0}$$v_{1}$$v_{2}$$v_{0}$$v_{1}$$v_{2}$$(v_{0},v_{2})$$(v_{1},v_{2})$$(v_{0},v_{1})$$v_{0}$$v_{1}$$v_{2}$$(v_{0},v_{2})$$(v_{1},v_{2})$$(v_{0},v_{1})$
Figure 1. From left to right: an unoriented graph, its face poset and its
(oriented) barycentric subdivision.

Note that $F({\tt G})$ can be straightforwardly seen as an oriented graph,
denoted by $\mathtt{F}({\tt G})$, as follows; the vertices of $\mathtt{F}({\tt
G})$ are the elements in $F({\tt G})$, and there is an oriented edge $(x,y)$
if $y\prec x$. The graph $\mathtt{F}({\tt G})$ can be identified with the
oriented barycentric subdivision of ${\tt G}$, as shown in Figure 1.

######  Definition 3.1.

A _graph matching_ on a graph ${\tt G}$ is a subset of $E({\tt G})$ consisting
of pairwise disjoint edges. The collection of graph matchings on ${\tt G}$
will be denoted by $\overline{\mathrm{M}}({\tt G})$.

######  Definition 3.2.

For a simplicial complex $X$, a _matching_ on $X$ is a graph matching on the
underlying unoriented graph of $\mathtt{F}(X)$. We denote the set of all
matchings on $X$ by $\mathrm{M}(X)$.

Note that, given a simplicial complex $X$, we have
$\mathrm{M}(X)=\overline{\mathrm{M}}(\mathtt{F}(X))$. In particular, there are
at least two kinds of matchings that can be considered on a simple graph ${\tt
G}$, that is $\overline{\mathrm{M}}({\tt G})$ and $\mathrm{M}({\tt G})$.
Observe that $\mathrm{M}(X)$ and $\overline{\mathrm{M}}({\tt G})$ admit a
natural simplicial structure; the $i$-simplices are the matchings with $i$
edges. To stress the difference between these two, we call the former
simplicial complex _matching complex_ , and the latter the _graph matching
complex_.

In [CY20] a $\mathbb{N}$-valued filtration $J$ on $\mathrm{M}(X)$ was defined.
Roughly speaking, the value of $J$ on a matching $m\in\mathrm{M}(X)$ is the
number of oriented cycles in $\mathtt{F}(X)$ obtained by inverting the
orientation of all the edges in $m$. For $j\in\mathbb{N}$ set
$\mathrm{M}_{j}(X)=J^{-1}([0,j])$. Each $\mathrm{M}_{j}(X)$ is a simplicial
subcomplex of $\mathrm{M}(X)$. The elements of
$\mathrm{M}_{0}(X)\eqqcolon\mathcal{M}(X)$ are called _discrete Morse
matchings_ [Cha00].

Given an unoriented graph ${\tt G}$, denote by ${O}({\tt G})$ the set of all
possible orientations on ${\tt G}$. For each $\mathfrak{o}\in{O}({\tt G})$,
the corresponding oriented graph will be denoted by ${\tt G}_{\mathfrak{o}}$.
It is apparent from Definition 3.2 that, for an oriented graph ${\tt G}$,
$\mathrm{M}({\tt G})$ does not depend on the orientation of the edges of ${\tt
G}$. We include this information as follows:

######  Definition 3.3.

An _oriented matching_ $m$ on an oriented graph ${\tt G}$ is a matching on the
subgraph of $\mathtt{F}({\tt G})$ consisting of the edges connecting the
barycentres of the edges of ${\tt G}$ to their targets.

######  Example 3.4.

Consider the oriented graph ${\tt G}$ depicted on the left side of Figure 2.
The matching $\\{((v_{0},v_{1}),v_{1}),((v_{0},v_{2}),v_{2})\\}$ is an
oriented matching and $\\{((v_{0},v_{1}),v_{1}),((v_{2},v_{1}),v_{1})\\}$ is
not.

We denote by $\mathrm{M}^{o}({\tt G})$ the simplicial complex consisting of
the oriented matchings on an oriented graph ${\tt G}$.

$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$
Figure 2. An oriented graph (left) and the graph associated to its face poset
(right). In the latter, we only kept in red the edges which can be used to
construct an oriented matching, and shaded in gray the remaining ones.

###  3.2. Pseudoforests and free-flow orientations

The combinatorics of the simplicial complex of oriented matchings is related
to certain orientations on graphs. We start by recalling some basic
definitions.

######  Definition 3.5.

A _pseudotree_ is a connected (unoriented) graph containing at most one cycle.
A _pseudoforest_ is a disjoint union of pseudotrees. A pseudotree not
containing any cycle is a _tree_ ; a _rooted tree_ is a tree together with the
choice of a preferred vertex called _root_. We will say that a pseudoforest is
_rooted_ if every tree component is rooted.

By definition, the set $V({\tt G})$ considered as a subgraph, is a spanning
pseudoforest. More in general, “being a spanning pseudoforest” is a monotone
property.

$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{n}$$\cdot$$\cdot$$\cdot$
Figure 3. The coherently oriented cycle ${\tt P}_{n}$.

######  Definition 3.6.

We say that an orientation $\mathfrak{o}$ on a rooted tree is _free-flow_ if
the edges of the tree are all oriented away from the root. If ${\tt G}$ is a
pseudotree which is not a tree, $\mathfrak{o}$ is free-flow if the unique
cycle of ${\tt G}$ is a coherently oriented cycle (see Figure 3), and all the
remaining edges are oriented away from the cycle (see Figure 4).

Figure 4. Free-flow orientations on a pseudo-tree and a tree. The root of the
tree is shown in white.

In particular, it follows that a pseudotree that is not a tree has exactly two
free-flow orientations, obtained from one another by inverting the orientation
on the cycle. On the other hand, a tree with $n$ vertices has exactly $n$
free-flow orientations, each determined by the choice of a vertex acting as
the unique root.

For an oriented graph ${\tt G}$, let $v\in V({\tt G})$ be a vertex. The
_indegree_ (respectively _outdegree_) _of $v$_, denoted by
$\mathrm{indeg}(v)$, is the number of edges of ${\tt G}$ whose target
(respectively source) is $v$. It is straightforward to obtain the following
characterisation of the connected components of a free-flow pseudoforest,
which will be used throughout the rest of this section.

######  Lemma 3.7.

Let ${\tt G}$ be an oriented graph. Then ${\tt G}$ is a free-flow pseudoforest
if and only if the indegree of each vertex is either $0$ or $1$.

###### Proof.

One direction is immediate: if ${\tt G}$ is a free-flow pseudoforest, then by
definition the indegree of its vertices is less or equal to one. Conversely,
for ${\tt G}^{\prime}$ a connected component of ${\tt G}$, let $V=|V({\tt
G}^{\prime})|$ and $E=|E({\tt G}^{\prime})|=\sum_{v\in V({\tt
G}^{\prime})}\mathrm{indeg}(v)$. By assumption $\chi({\tt G}^{\prime})=V-E\geq
0$, but since ${\tt G}^{\prime}$ is connected (hence in particular homotopic
to a wedge of circles), we have that $\chi({\tt G}^{\prime})\leq 1$.
Therefore, as an unoriented graph, ${\tt G}^{\prime}$ is either a tree or a
pseudotree. It is now easy to see that the requirement $\mathrm{indeg}(v)\leq
1$ for all vertices implies that the orientation is free-flow. ∎

######  Proposition 3.8.

Let ${\tt G}$ be an oriented graph. Simplices in $\mathrm{M}^{o}({\tt G})$ are
in bijection with rooted spanning pseudoforests in ${\tt G}$ whose induced
orientation is free-flow. In particular, $\mathrm{M}^{o}({\tt G})$ consists of
a unique simplex (of dimension $|E({\tt G})|-1$) if and only if ${\tt G}$ is a
free-flow pseudotree.

###### Proof.

Let $F$ be a rooted pseudoforest in ${\tt G}$; consider the edges $E_{F}$ in
${\tt F}({\tt G})$ obtained by connecting the barycentres of the edges in $F$
to their targets. Lemma 3.7 can be used to show that $E_{F}$ defines a
matching on ${\tt G}$, thus a simplex in
$\sigma_{F}\subseteq\mathrm{M}^{o}({\tt G})$.

Figure 5. From left to right: an oriented graph ${\tt G}$, $\mathtt{F}({\tt
G})$ with a (maximal) oriented matching $m\in\mathrm{M}^{o}({\tt G})$
highlighted in red, and the rooted free-flow pseudoforest induced by $m$ on
${\tt G}$.

For the converse, let $\sigma\subseteq\mathrm{M}^{o}({\tt G})$ be any simplex.
We want to show that $\sigma$ uniquely determines a pseudoforest
$F_{\sigma}\subseteq{\tt G}$ (see Figure 5), and that moreover the orientation
of $F_{\sigma}$ is free-flow. Each vertex in $\sigma$ uniquely determines a
directed edge in ${\tt G}$, and we take $F_{\sigma}$ to be the union of all of
these edges. We can conclude by noting that, as $\sigma\in\mathrm{M}^{o}({\tt
G})$, the indegree of each vertex in $F_{\sigma}$ must be $\leq 1$. Therefore,
by Lemma 3.7 $F_{\sigma}$ is a free-flow pseudoforest.

The second part of the statement follows readily from the previous one. ∎

Proposition 3.8 implies, in particular, that simplices in $\mathcal{M}({\tt
G})$ are in bijection with free-flow oriented spanning forests in ${\tt G}$.
This was the starting point of [CJ05], where Kozlov’s complex of directed
forests was identified with the discrete Morse complex they defined:

######  Proposition 3.9 ([CJ05, Proposition 3.1]).

The set of discrete Morse matchings on an unoriented graph ${\tt G}$ is in
one-to-one correspondence with the set of rooted forests of ${\tt G}$.

It is easy to see that, for each orientation $\mathfrak{o}\in\mathcal{O}({\tt
G})$ and $j\in\mathbb{N}$, $\mathrm{M}^{o}_{j}({\tt
G}_{\mathfrak{o}})\coloneqq\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}})\cap\mathrm{M}_{j}({\tt G})$ is a simplicial subcomplex of
$\mathrm{M}_{j}({\tt G})$; in particular this holds for the _oriented discrete
Morse matchings_ $\mathcal{M}^{o}({\tt
G}_{\mathfrak{o}})\coloneqq\mathrm{M}^{o}_{0}({\tt G}_{\mathfrak{o}})$. We
remark that a simple application of [CY20, Proposition 2.9] shows that
simplices in $\mathrm{M}^{o}_{j}({\tt G})$ are in bijection with free-flow
pseudoforests with at most $j$ pseudotrees that are not trees. Moreover,

(7) $\mathrm{M}({\tt G})=\bigcup_{\mathfrak{o}\in\mathcal{O}({\tt
G})}\mathrm{M}^{o}({\tt G}_{\mathfrak{o}})\ .$

In other words, all matchings on ${\tt G}$ arise as oriented matchings for
some orientation on ${\tt G}$; see Figure 6 for an example. Furthermore,
Equation (7) provides a decomposition of the matching complex $\mathrm{M}({\tt
G})$ in terms of the oriented ones.

Figure 6. The decomposition of the matching complex $\mathrm{M}({\tt C}_{3})$
for the cycle graph of length three (on the left) into its $8$ pieces
corresponding to the oriented matching complexes (right). The two
$2$-simplices correspond to the cyclic orientations on ${\tt C}_{3}$, while
the other six $1$-dimensional components are induced by all the other possible
orientations.

##  4\. Relations between matchings and multipaths

In this section we prove that, in certain cases, matchings on graphs and path
posets can be identified. First, we study the case of graph matching complexes
on unoriented graphs. Then, we provide an isomorphism between oriented
matching complexes on oriented graphs and path posets.

Call an orientation $\mathfrak{o}\in O({\tt G})$ on an unoriented graph ${\tt
G}$ _alternating_ if there exists a partition $V\sqcup W$ of $V({\tt G})$ such
that all elements of $V$ have indegree $0$ and all elements of $W$ have
outdegree $0$. Note that the existence of an alternating orientation implies
that ${\tt G}$ is a bipartite graph.

Recall that, for an oriented graph ${\tt G}$, $P({\tt G})$ denotes its path
poset (_cf._ Example 2.9) and that $F(S)$ denotes the face poset of a
simplicial complex $S$. Then, we have the following result:

######  Theorem 4.1.

Let ${\tt G}$ be an unoriented graph. Then, we have an isomorphism
$F(\overline{\mathrm{M}}({\tt G}))\cong P({\tt G}_{\mathfrak{o}})$ if and only
if $\mathfrak{o}$ is alternating.

###### Proof.

First, note that every graph matching in $\overline{\mathrm{M}}({\tt G})$ can
be regarded as being a multipath in ${\tt G}_{\mathfrak{o}}$, independently on
the chosen orientation $\mathfrak{o}\in O({\tt G})$. In particular, $|P({\tt
G}_{\mathfrak{o}})|\geq|F(\overline{\mathrm{M}}({\tt G}))|$. We claim that, if
we fix an arbitrary alternating orientation $\mathfrak{o}\in\mathcal{O}({\tt
G})$, then every multipath induces a matching on ${\tt G}_{\mathfrak{o}}$.

Since $\mathfrak{o}$ is alternating, there is a partition $V\sqcup W=V({\tt
G})$ such that all the edges of ${\tt G}_{\mathfrak{o}}$ are of the form
$(v,w)$ with $v\in V,$ and $w\in W$. We only have to observe that the
connected components of any multipath in ${\tt G}_{\mathfrak{o}}$ are either
vertices or single edges. For the sake of contradiction, assume there exists
at least one component $c$ of a certain multipath ${\tt H}$ that is not a
single edge or a vertex. Then, there are at least two edges in $c$ which share
a vertex; in particular the target of one edge, which is a vertex in $W$, must
be the source of another edge, and therefore it is in $V$. This is a
contradiction since $V\cap W$ is empty, and the “if” part of the statement
follows.

For the converse, assume $\mathfrak{o}$ is not alternating. Then, there exists
a vertex which is both a source and a target – otherwise, the partition of
$V({\tt G}_{\mathfrak{o}})$ into indegree $0$ and outdegree $0$ vertices would
imply that $\mathfrak{o}$ is alternating. Therefore, there is at least a
multipath of length two which is not made of disjoint edges. Thus, $|P({\tt
G}_{\mathfrak{o}})|>|F(\overline{\mathrm{M}}({\tt G}))|$, concluding the
proof. ∎

######  Corollary 4.2.

All alternating orientations on a graph have isomorphic path posets.

Note however that is it possible to find two orientations (not both
alternating) yielding isomorphic path posets. As an example, consider the two
orientations on the “Y”-shaped graph shown in Figure 7.

$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$ Figure 7. Two non-
isomorphic Y-shaped oriented graphs with isomorphic path posets.

######  Definition 4.3.

For an oriented graph ${\tt G}$, the _multipath complex_ $X({\tt G})$ is the
simplicial complex whose face poset is the path poset $P({\tt G})$.

The existence of $X({\tt G})$ is guaranteed by Remark 2.10 – see also
[CCDT21a, Definition 6.4].

$v_{0}$$v_{1}$…$v_{n-1}$$v_{n}$(A) ${\tt
L}_{n}$$v_{0}$$v_{1}$…$v_{n-1}$$v_{n}$(B) ${\tt A}_{n}$ (for $n$ even) Figure
8. (A) The coherently oriented linear graph ${\tt L}_{n}$, and (B) the
alternating linear graph ${\tt A}_{n}$.

######  Example 4.4.

The multipath complex $X({\tt A}_{n})$ of the alternating graph on $n$ edges –
see Figure 8 – is isomorphic to the graph matching complex
$\overline{\mathrm{M}}({\tt A}_{n})$, associated to the unoriented linear
graph underlying ${\tt A}_{n}$. This is coherent with the computations in
[Koz99, Proposition 4.6] and in [CCDT21a, Corollary 5.7].

The study of the topology of multipath complexes was initiated in [CCDT21a,
Section 6]. All examples provided therein are wedges of spheres. A consequence
of Theorem 4.1 implies that this is not always the case. The following
proposition provides an affirmative answer to [CCDT21a, Question 6.18].

######  Proposition 4.5.

Multipath cohomology with coefficients in $R=\mathbb{Z}$, for $A=R$, can have
torsion. In particular, the multipath complex is not always homotopic to a
wedge of spheres.

###### Proof.

Denote by ${\tt K}_{n,m}$ the complete bipartite graph on $m,n$ vertices. By
[SW04, Theorem 1.7], the integral homology of the graph matching complex of
$\overline{\mathrm{M}}({\tt K}_{n,m})$ can have torsion for certain values of
$m$ and $n$ (the minimal example being $m=n=5$, containing $3$-torsion).
Therefore, $\overline{\mathrm{M}}({\tt K}_{n,m})$ is not necessarily homotopic
to a wedge of spheres. Theorem 4.1 implies that, for any alternating
orientation $\mathfrak{o}$ on ${\tt K}_{n,m}$, we have

$X(({\tt K}_{m,n})_{\mathfrak{o}})\cong\overline{\mathrm{M}}({\tt K}_{n,m})\
.$

The multipath homology for $A=R$ is (up to a shift by $1$ in homological
degree) the reduced simplicial cohomology of the multipath complex – [CCDT21a,
Theorem 6.8]. The statement follows from the above isomorphism. ∎

We now want to interpret the poset of oriented matchings on a oriented graph
as the path poset of a suitable oriented graph.

######  Definition 4.6.

Let ${\tt G}$ be an oriented graph, its _source resolution_ is the oriented
graph ${\tt G}_{sr}$ whose vertices are:

$V({\tt G}_{sr})\coloneqq\\{v\in V({\tt G})\mid v={\tt t}(e)\text{ for some
}e\in E({\tt G})\\}\cup\\{(v,e)\in V({\tt G})\times E({\tt G})\mid{\tt
s}(e)=v\\},$

and whose edges are

$E({\tt G}_{sr})\coloneqq\\{\left((v,e),w\right)\mid v,w\in V({\tt G}),\,e\in
E({\tt G})\text{ and }{\tt s}(e)=v,{\tt t}(e)=w\\}.$

Intuitively, we are splitting the sources of the edges in ${\tt G}$ (see the
top-right part of Figure 9).

$v_{1}$$v_{2}$$v_{3}$ (a) ${\tt G}$

$(v_{1},(v_{1},v_{3}))$$(v_{2},(v_{2},v_{3}))$$v_{3}$$(v_{1},(v_{1},v_{2}))$$v_{2}$
(b) ${\tt G}_{sr}$

(c) The path poset of ${\tt G}_{sr}$.

Figure 9. (A) A graph, (B) its source resolution and (C) the path poset of its
source resolution.

######  Remark 4.7.

Given a graph ${\tt G}$, each connected component of ${\tt G}_{sr}$ is a sink.
Therefore, multipaths in ${\tt G}_{sr}$ are collections of edges whose targets
are pairwise distinct.

It is not hard to see that the association ${\tt G}\mapsto P({\tt G})$ is
functorial (_cf_. [CCDT21b, Remark 2.33]). This association can be promoted to
a functor ${\tt G}\mapsto P({\tt G}_{sr})$, with respect to regular graph
morphisms. To this end, we have to prove that a regular morphism of oriented
graphs $f\colon{\tt G}\to{\tt G}^{\prime}$ induces a regular morphism
$f_{sr}\colon{\tt G}_{sr}\to{\tt G}^{\prime}_{sr}$. Recall that a vertex
$v_{sr}$ in ${\tt G}_{sr}$ is either of the form $v\in V({\tt G})$ or of the
form $(v,e)$, where $v$ is the source of the edge $e\in E({\tt G})$. Define

$f_{sr}:V({\tt G}_{sr})\to V({\tt
G}^{\prime}_{sr})\;\;\;\;v_{sr}\mapsto\begin{cases}f(v)&\text{if }v_{sr}=v\in
V({\tt G});\\\ (f(v),f(e))&\text{if }v_{sr}=(v,e)\in V({\tt G})\times E({\tt
G}).\end{cases}$

The definition of $f_{sr}$ is well-posed; if $v\in V({\tt G}_{sr})$, then $v$
is the target of at least one edge in ${\tt G}$. Since $f$ is regular, then
$f(v)$ is the target of at least one edge in ${\tt G}^{\prime}$. As a
consequence, $f(v)$ is a vertex of ${\tt G}^{\prime}_{sr}$. Similarly, one can
show that if $(v,e)\in V({\tt G}_{sr})$, then $(f(v),f(e))\in V({\tt
G}^{\prime}_{sr})$. It follows directly from the definitions that $f_{sr}$ is
a regular morphism of oriented graphs, and that $f\mapsto f_{sr}$ preserves
compositions, proving the desired functoriality.

######  Proposition 4.8.

Let ${\tt G}$ be an oriented graph. There is an isomorphism
$F(\mathrm{M}^{o}({\tt G}))\cong P({\tt G}_{sr})$ between the face poset of
$\mathrm{M}^{o}({\tt G})$ and the path poset of ${\tt G}_{sr}$.

###### Proof.

The map

$\displaystyle\Phi\colon\,$ $\displaystyle E({\tt G})\longrightarrow E({\tt
G}_{sr})$ $\displaystyle e\longmapsto(({\tt s}(e),e),{\tt t}(e)),$

is a bijection, with inverse $((v,e),w)\mapsto e$. Therefore, we have a
bijection between the (coherently oriented) edges in the barycentric
subdivision of ${\tt G}$ and the edges of ${\tt G}_{sr}$.

We claim that this bijection sends an oriented matching in ${\tt G}$ to the
set of edges in a multipath in ${\tt G}_{sr}$. An oriented matching is a
collection of coherently oriented edges in the barycentric subdivision of
${\tt G}$ not sharing the same target. These can be seen as a collection of
edges in ${\tt G}$ not sharing the same target. Since ${\tt t}(e)\neq{\tt
t}(e^{\prime})$ if and only if ${\tt t}(\Phi(e))\neq{\tt
t}(\Phi(e^{\prime}))$, our claim is a consequence of Remark 4.7. A similar
reasoning, replacing $\Phi$ with its inverse, can be used to prove that all
oriented matchings arise in this way.

Finally, the bijection between $\mathrm{M}^{o}({\tt G})$ and $P({\tt G}_{sr})$
just described clearly respects inclusions, and the statement follows. ∎

##  5\. Oriented matchings and monotone cohomology

The aim of this section is to study the homotopy type of the oriented matching
complexes. We show that the oriented matching complexes are homotopy
equivalent to wedges of spheres; futher, the sphere’s dimensions are related
to simple combinatorial information of graphs. By providing an isomorphism
with multipath cohomology, we show that computations can be carried on also in
the case of non-constant functor coefficients.

###  5.1. Homotopy type of oriented matchings

Recall that if $X$ is a finite simplicial complex, then its _$n$ -point
suspension_333This operation is also sometimes referred to as _iterated
suspension_. $\Sigma_{n}(X)$ is the join $\\{n\text{ points}\\}*X$. So, for
example, $\Sigma_{0}(X)=X$, $\Sigma_{1}(X)=\text{Cone}(X)$ and $\Sigma_{2}(X)$
is the usual suspension of $X$. If $\alpha=(n_{1},n_{2},\ldots,n_{k})$ is a
sequence of non-negative integers, we denote by $\Sigma_{\alpha}(X)$ the
$\alpha$-suspension of $X$, _i.e._ the composition
$\Sigma_{n_{1}}\circ\dots\circ\Sigma_{n_{k}}$ applied to $X$. We also set
$\Sigma(\alpha)\coloneqq\Sigma_{\alpha}(\emptyset)$.

Note that

$\Sigma(\underbrace{1,\ldots,1}_{n})=\Delta^{n-1}$

and

(8)
$\Sigma_{(\ldots,n_{i},0,n_{i+2},\ldots)}(X)=\Sigma_{(\ldots,n_{i},n_{i+2},\ldots)}(X)\
.$

As a consequence of Equation (8), from now on we assume that all strings
$\alpha$ do not contain any zero entry. Since the join of complexes is
commutative and associative [Mun84, Lemma 62.4], we see that
$\Sigma_{(a,b)}(X)\simeq\Sigma_{(b,a)}(X)$. In particular, if
$a_{1},\ldots,a_{k}$ are positive integers, then $\Sigma(1,a_{1},...,a_{k})$
is always a cone.

######  Lemma 5.1.

Let $m$ be an integer greater than $1$. The $m$-point suspension
$\Sigma_{m}(S)$ of a wedge of spheres
$S=\mathbb{S}^{n_{1}}\vee\dots\vee\mathbb{S}^{n_{k}}$ is homotopy equivalent
to

$\bigvee_{i=1}^{m-1}\left(\mathbb{S}^{n_{1}+1}\vee\dots\vee\mathbb{S}^{n_{k}+1}\right)\
.$

###### Proof.

Recall that for a sphere $\mathbb{S}^{n}$, the join with $m\geq 1$ points is a
wedge of $m-1$ spheres:

$\mathbb{S}^{n}*\\{m\text{
points}\\}\simeq\bigvee_{i=1}^{m-1}\mathbb{S}^{n+1}\ .$

Since the join and the wedge commute up to homotopy, the assertion follows. ∎

We obtain the following observation as a straightforward consequence of Lemma
5.1:

######  Remark 5.2.

Let $\alpha=(n_{1},n_{2},\ldots,n_{k})$ be a sequence of positive integers,
and assume $k>0$. Then, if $n_{i}=1$ for some $i$, $\Sigma(\alpha)$ is
contractible. Otherwise, we have a homotopy equivalence:

$\Sigma(\alpha)\simeq\bigvee^{q(\alpha)}\mathbb{S}^{|\alpha|-1}\ ,$

where $q(\alpha)=\prod_{i}(n_{i}-1)$, and $|\alpha|=\sum_{i=1}^{k}n_{i}$. As a
consequence, $\Sigma(\alpha)$ is simply connected if and only if $k\geq 3$.

This next result implies that the whole homotopy type of the complexes
$\mathrm{M}^{o}({\tt G})$ is completely determined by the collection of
indegrees of the vertices in ${\tt G}$.

######  Proposition 5.3.

Let ${\tt G}$ be an oriented graph on $n$ vertices. Then, the oriented
matching complex of ${\tt G}$ is either contractible or a wedge of spheres.
More precisely, if there is $v$ such that $\mathrm{indeg}(v)=1$, then
$\mathrm{M}^{o}({\tt G})$ is contractible. Otherwise,

$\mathrm{M}^{o}({\tt G})\simeq\bigvee^{q}\mathbb{S}^{N-1}$

where

$q\coloneqq\prod_{\begin{subarray}{c}v\in V({\tt G})\\\
\mathrm{indeg}(v)>1\end{subarray}}(\mathrm{indeg}(v)-1),$

and $N$ is the number of vertices whose indegree is positive.

###### Proof.

By Proposition 4.8 we have the following isomorphism of simplicial complexes

$\mathrm{M}^{o}({\tt G})\cong X({\tt G}_{sr})\ .$

Observe that ${\tt G}_{sr}$ is the disjoint union of graphs ${\tt H}_{v}$, for
$v\in V({\tt G})$ with $\mathrm{indeg}(v)\geq 1$, each one isomorphic to a
sink with $\mathrm{indeg}(v)+1$ vertices. The multipath complex of a disjoint
union is the join of the corresponding multipath complexes, and the multipath
complex of a sink with $m$ vertices is the disjoint union of $m$ points – see
[CCDT21a, Subsection 6.4]. The statement now follows from Remark 5.2. ∎

A simplicial complex $X$ is _shellable_ if there exists an ordering
$\Delta_{1}$, …, $\Delta_{k}$ of the maximal simplices in $X$ such that the
complex

$Y_{h}=\Delta_{h}\cap\bigcup_{i=1}^{h-1}\Delta_{i}$

is pure (_i.e._ all its maximal simplices have the same dimension) of
dimension $\mathrm{dim}(\Delta_{h})-1$, for each $h$. There is also a related
notion of _strong shellability_ ; we will not recall it here and refer the
reader to [GSW19, Definition 2.2]. For a simplicial complex, strong
shellability implies shellability.

It is well-known that shellable (and strongly shellable) complexes are
homotopy equivalent to wedges of spheres, one for each maximal simplex (see,
for instance, [Koz08, Theorem 12.3]). Therefore, it is natural to ask whether
non-contractible oriented matching complexes are shellable. We affermatively
answer this question;

######  Corollary 5.4.

Oriented matching complexes are either strongly shellable or contractible.
In particular, all non-contractible matching complexes are Cohen-Macaulay.

###### Proof.

The join of strongly shellable complexes is strongly shellable by [GSW19,
Proposition 2.16] (see also [BW97, Remark 10.22] for “non-strong”
shellability). Wedges of $0$-dimensional spheres, that is finite unions of
more than one points, are strongly shellable. The oriented matching complex
can be obtained as the iterated join of wedges of $0$-dimensional spheres if
and only if there are no vertices with indegree $1$ (_cf._ the proof of
Proposition 5.3). Since, by Proposition 5.3, the presence of vertices of
indegree $1$ is equivalent to the oriented matching complex being
contractible, we obtain the first part of the statement.

Oriented matching complexes are pure; the dimension of each maximal simplex is
the number of vertices with positive indegree. It was observed in [GSW19] that
pure strongly shellable complexes are Cohen-Macaulay, giving the second part
of the statement. ∎

Following this last proposition, it is easy to give a complete
characterisation of the graphs whose oriented matching complex is
homotopically equivalent to a sphere.

######  Corollary 5.5.

The oriented matching complex $\mathrm{M}^{o}({\tt G})$ of an oriented graph
${\tt G}$ is homotopically equivalent to a sphere $\mathbb{S}^{n}$ if and only
if all vertices have indegree $0$ or $2$.

###### Proof.

Apply Proposition 5.3. The number of vertices of indegree $2$ gives the
dimension of the sphere. ∎

###  5.2. Oriented matching homology with non-constant coefficients

We proved in Proposition 4.8 that the isomorphism between the face poset of
$\mathrm{M}^{o}({\tt G})$ and the path poset of ${\tt G}_{sr}$ induces an
isomorphism of the associated monotone cohomologies with constant
coefficients. It is however possible to show that this isomorphism cannot be
straightforwardly extended to an isomorphism between the respective monotone
cohomologies groups with arbitrary functor-valued coefficients.

One easy example is given by the graph ${\tt G}$ shown in Figure 9(A); let $A$
be a finitely generated $\mathbb{F}$-algebra of dimension $\alpha$. Consider
$C_{\mu}^{*}({\tt G};A)$, the multipath cochain complex of ${\tt G}$ with
coefficients in the functor $\mathcal{F}_{A}\colon\mathbf{P}({\tt
G})\to\mathbf{Vect}_{\mathbb{F}}$ and denote by $C_{o}^{*}({\tt G};A)$ the
monotone cochain complex associated to oriented matchings (_cf._ Definition
2.13). The chain complex $C_{o}^{*}({\tt G};A)$ admits the following
description:

${\cdots}$${0}$${A^{\otimes 6}}$${A^{\otimes 5}\oplus A^{\otimes 5}\oplus
A^{\otimes 5}}$${A^{\otimes 4}\oplus A^{\otimes 4}}$${0}$${\cdots}$

where $A^{\otimes 6}$ sits in degree $0$. The Euler characteristic of
$C_{o}^{*}({\tt G};A)$ is easily computed to be
$\alpha^{4}(\alpha-2)(\alpha-1)$. On the other hand, the cochain complex
associated to ${\tt G}_{sr}$, using the poset in Figure 9(C) is

${\cdots}$${0}$${A^{\otimes 5}}$${A^{\otimes 4}\oplus A^{\otimes 4}\oplus
A^{\otimes 4}}$${A^{\otimes 3}\oplus A^{\otimes 3}}$${0}$${\cdots}$

where $A^{\otimes 5}$ sits in cohomological degree $0$. This time, the Euler
characteristic of $C_{o}^{*}({\tt G};A)$ is given by
$\alpha^{3}(\alpha-2)(\alpha-1)$. Therefore, these complexes must have
distinct homologies for any choice of $A$ of dimension $\alpha\geq 3$.

Despite being distinct, the above example suggests that the multipath
cohomology groups of ${\tt G}_{sr}$ and the oriented matching cohomology
groups of ${\tt G}$ might still be closely related. We will make use of the
following general observation.

######  Remark 5.6.

Let ${\tt G}$ and ${\tt G}^{\prime}$ two graphs, and let
$P_{\mathscr{M}}\subseteq SS({\tt G})$ and $P_{\mathscr{M}^{\prime}}\subseteq
SS({\tt G}^{\prime})$ be posets associated to the monotone properties
$\mathscr{M}$ and $\mathscr{M}^{\prime}$, respectively. If there exists an
isomorphism of posets

$\varphi:P_{\mathscr{M}}\longrightarrow P_{\mathscr{M}^{\prime}}$

such that there is a graph isomorphism $\varphi({\tt H})\cong{\tt H}$ for each
${\tt H}\in P_{\mathscr{M}}$, then we have an isomorphism

$(C^{*}_{\mathcal{F}_{A}}(P_{\mathscr{M}}({\tt
G})),d^{*})\cong(C^{*}_{\mathcal{F}_{A}}(P_{\mathscr{M}^{\prime}}({\tt
G}^{\prime})),d^{*})$

of cochain complexes.

######  Theorem 5.7.

Let ${\tt G}$ be an oriented graph. Then, there is an isomorphism of cochain
complexes

$C_{o}^{*}({\tt G};A)\cong C_{\mu}^{*}({\tt G}_{sr};A)\otimes A^{\otimes s}$

where $s$ is the number of vertices of indegree $0$ in ${\tt G}$.

###### Proof.

Consider the graph ${\tt G}^{\prime}_{sr}$ obtained from $\mathtt{F}({\tt G})$
by removing all “non-coherent” edges, _i.e._ the edges of the form
$((v_{i},v_{j}),v_{i})\in E(\mathtt{F}({\tt G}))$ – _cf._ Definition 3.3.
Then, ${\tt G}_{sr}^{\prime}$ is isomorphic to ${\tt G}_{sr}$ together with
the disjoint union of $s$ isolated vertices, $s$ being is the number of
vertices of indegree $0$ in ${\tt G}$.

The graph ${\tt G}^{\prime}_{sr}$ is bipartite, and its orientation is
alternating. This yields a bijection $\Phi$ between the set of multipaths in
${\tt G}^{\prime}_{sr}$ and the set of oriented matching in ${\tt G}$. The
bijection is such that a multipath is isomorphic, as a graph, to the
corresponding oriented matching on ${\tt G}$, thus providing an identification
of the two. Moreover, inclusions of multipaths correspond to inclusions of
matchings. Therefore, as an application of Remark 5.6, the associated cochain
complexes are isomorphic. We conclude by noting that the multipath cochain
complex of ${\tt G}^{\prime}_{sr}$ is isomorphic to the tensor product
$C_{\mu}^{*}({\tt G}_{sr};A)\otimes A^{\otimes s}$, since the multipath
cochain complex of a disjoint union is the tensor product of the multipath
cochain complexes (_cf._ [CCDT21a, Remark 3.2]). ∎

Theorem 5.7 shows how to carry out computations with general functor
coefficients; for sake of completeness, we provide an example:

######  Example 5.8.

Consider the $3$-clique graph in Figure 9(A), and set
$A=\mathbb{F}[X]/(X^{2})$, for some field $\mathbb{F}$. By Theorem 5.7, we
know that $C_{o}^{*}({\tt G};A)\cong C_{\mu}^{*}({\tt G}_{sr};A)\otimes
A^{\otimes s}$, where $s=1$ and ${\tt G}_{sr}={\tt A}_{2}\sqcup{\tt L}_{1}$ is
the graph in in Figure 9(B). As mentioned in the proof above, the multipath
cochain complex of a disjoint union is the tensor product of the multipath
cochain complexes. It follows that

$C_{\mu}^{*}({\tt G}_{sr};A)\cong C_{\mu}^{*}({\tt A}_{2};A)\otimes
C_{\mu}^{*}({\tt L}_{1};A)\ ,$

where the tensor product is a graded tensor product. Using [CCDT21b, Example
4.16] we can deduce that the multipath cohomology $\mathrm{H}_{\mu}^{*}({\tt
A}_{2};A)$ is isomorphic to $\mathbb{F}^{2}$ in degree $0$, and
$\mathbb{F}^{2}$ in degree $1$. It is then easy to see that
$\mathrm{H}_{\mu}^{*}({\tt L}_{1};A)$ is isomorphic to $\mathbb{F}^{2}$ in
degree $0$. Putting everything together, we see that the cohomology of the
cochain complex $C_{o}^{*}({\tt G};A)$ is

${\rm H}_{o}^{i}({\tt G};A)\cong\begin{cases}\mathbb{F}^{8}&i=0,1,\\\
(0)&\text{otherwise},\end{cases}$

concluding the computation.

##  6\. Oriented homology and free-flow orientations

In this final section, we introduce a special instance of the general
constructions outlined in Section 2. This is a novel homology theory for
oriented graphs; its construction is formally similar to the homology theories
introduced in a different context in [CCC22].

Let ${\tt G}$ be an unoriented graph. Recall that $O({\tt G})$ denotes the set
of all possible orientations on ${\tt G}$. For a fixed orientation
$\mathfrak{o}$, we can define a poset $\mathcal{O}({\tt
G},\mathfrak{o})\coloneqq(O({\tt G}),<_{\mathfrak{o}})$ where
$<_{\mathfrak{o}}$ is the order relation associated to the covering relation
$\prec_{\mathfrak{o}}$ defined as follows: given
$\mathfrak{o}_{1},\mathfrak{o}_{2}\in O({\tt G})$, we say
$\mathfrak{o}_{1}\prec_{\mathfrak{o}}\mathfrak{o}_{2}$ if and only if the
oriented graphs ${\tt G}_{\mathfrak{o}_{1}}$ and ${\tt G}_{\mathfrak{o}_{2}}$
differ only on a single edge $e$, and the orientation of $e$ in ${\tt
G}_{\mathfrak{o}_{1}}$ is the same as in ${\tt G}_{\mathfrak{o}}$. Clearly, we
have a poset isomorphism between $\mathcal{O}({\tt G},\mathfrak{o})$ and the
Boolean poset $(\wp(|E({\tt G})|),\subseteq)$.

Consider the functor $\mathcal{F}\colon\mathcal{O}({\tt
G},\mathfrak{o})\to\mathbf{grVect}_{\mathbb{F}}$ to graded vector spaces over
the field $\mathbb{F}$, defined on each object $\mathfrak{o}^{\prime}$ of (the
category associated to) $\mathcal{O}({\tt G},\mathfrak{o})$ as

(9)
$\mathcal{F}(\mathfrak{o}^{\prime})\coloneqq\mathbb{F}\langle\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}^{\prime}})\rangle$

where $\mathbb{F}\langle\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}^{\prime}})\rangle$ is the $\mathbb{F}$-vector space spanned
by the simplices in the oriented matching complex $\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}^{\prime}})$, graded by the simplices’ dimension.

For each covering relation
$\mathfrak{o}_{1}\prec_{\mathfrak{o}}\mathfrak{o}_{2}$, we set

(10)
$\mathcal{F}(\mathfrak{o}_{1}\prec_{\mathfrak{o}}\mathfrak{o}_{2})\colon\mathcal{F}(\mathfrak{o}_{1})\to\mathcal{F}(\mathfrak{o}_{2})$

to be the map induced by the inclusion $\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}_{1}})\cap\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}_{2}})\hookrightarrow\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}_{2}})$. Arguing as in Proposition 2.11, this data describes a
functor on the category associated to $\mathcal{O}({\tt G},\mathfrak{o})$.

Now, for any sign assignment $\varepsilon$ on $\mathcal{O}({\tt
G},\mathfrak{o})$, we consider the associated poset homology as the homology
of the associated cochain complex $\mathrm{OC}({\tt
G},\mathfrak{o})\coloneqq(C_{\mathcal{F}}^{*}(\mathcal{O}({\tt
G},\mathfrak{o})),d^{*})$ – _cf._ Theorem 2.4. Note that this homology is
bigraded; the first degree increases by one under the action of the
differential. The second degree, called _simplicial degree_ , is just given by
the simplices’ dimensions. This latter degree is preserved under the
differential, making the homology bigraded.

######  Definition 6.1.

We define the oriented homology of ${\tt G}$ with respect to $\mathfrak{o}$,
$\mathrm{OH}({\tt G}_{\mathfrak{o}})$, as the homology of the bigraded cochain
complex $(\mathrm{OC}({\tt G},\mathfrak{o}),d^{*})$.

As an example, consider the linear graph ${\tt L}_{2}$ and the alternating
graph ${\tt A}_{2}$ – _cf._ Figure 8. First, note that the oriented matching
complex of ${\tt L}_{2}$ is an interval, whereas the oriented matching complex
of ${\tt A}_{2}$ is given by the disjoint union of two points. The posets
$\mathcal{O}({\tt L}_{2},\mathfrak{o})$ and $\mathcal{O}({\tt
A}_{2},\mathfrak{o})$ are two Boolean posets of the same dimension. The
functor $\mathcal{F}$ defined above, applied to these posets, yields the
graded vector spaces shown in Figure 10, where the homological degrees go from
$0$ to $2$ and the simplicial degrees are supported in degree $0$ and $1$.

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} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{76.82388pt}{0.0pt}\pgfsys@moveto{77.25069pt}{0.0pt}\pgfsys@curveto{77.25069pt}{0.23572pt}{77.0596pt}{0.4268pt}{76.82388pt}{0.4268pt}\pgfsys@curveto{76.58817pt}{0.4268pt}{76.39708pt}{0.23572pt}{76.39708pt}{0.0pt}\pgfsys@curveto{76.39708pt}{-0.23572pt}{76.58817pt}{-0.4268pt}{76.82388pt}{-0.4268pt}\pgfsys@curveto{77.0596pt}{-0.4268pt}{77.25069pt}{-0.23572pt}{77.25069pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{76.82388pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.8}{0.0}{0.0}\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.8,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.0,0.0}\pgfsys@color@rgb@stroke{0.8}{0.0}{0.0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.8}{0.0}{0.0}\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{60.90895pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}})\ \
$$\mathcal{F}(\text{\raisebox{0.0pt}{ \leavevmode\hbox to57.88pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-22.27473pt\lower-3.33301pt\hbox
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{25.60796pt}{0.0pt}\pgfsys@moveto{26.03476pt}{0.0pt}\pgfsys@curveto{26.03476pt}{0.23572pt}{25.84367pt}{0.4268pt}{25.60796pt}{0.4268pt}\pgfsys@curveto{25.37224pt}{0.4268pt}{25.18115pt}{0.23572pt}{25.18115pt}{0.0pt}\pgfsys@curveto{25.18115pt}{-0.23572pt}{25.37224pt}{-0.4268pt}{25.60796pt}{-0.4268pt}\pgfsys@curveto{25.84367pt}{-0.4268pt}{26.03476pt}{-0.23572pt}{26.03476pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.60796pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{51.21591pt}{0.0pt}\pgfsys@moveto{51.64272pt}{0.0pt}\pgfsys@curveto{51.64272pt}{0.23572pt}{51.45163pt}{0.4268pt}{51.21591pt}{0.4268pt}\pgfsys@curveto{50.9802pt}{0.4268pt}{50.78911pt}{0.23572pt}{50.78911pt}{0.0pt}\pgfsys@curveto{50.78911pt}{-0.23572pt}{50.9802pt}{-0.4268pt}{51.21591pt}{-0.4268pt}\pgfsys@curveto{51.45163pt}{-0.4268pt}{51.64272pt}{-0.23572pt}{51.64272pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{51.21591pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{76.82388pt}{0.0pt}\pgfsys@moveto{77.25069pt}{0.0pt}\pgfsys@curveto{77.25069pt}{0.23572pt}{77.0596pt}{0.4268pt}{76.82388pt}{0.4268pt}\pgfsys@curveto{76.58817pt}{0.4268pt}{76.39708pt}{0.23572pt}{76.39708pt}{0.0pt}\pgfsys@curveto{76.39708pt}{-0.23572pt}{76.58817pt}{-0.4268pt}{76.82388pt}{-0.4268pt}\pgfsys@curveto{77.0596pt}{-0.4268pt}{77.25069pt}{-0.23572pt}{77.25069pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{76.82388pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.8}{0.0}{0.0}\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}})\ \
$$\mathcal{F}(\text{\raisebox{0.0pt}{ \leavevmode\hbox to57.88pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-22.27473pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{25.60796pt}{0.0pt}\pgfsys@moveto{26.03476pt}{0.0pt}\pgfsys@curveto{26.03476pt}{0.23572pt}{25.84367pt}{0.4268pt}{25.60796pt}{0.4268pt}\pgfsys@curveto{25.37224pt}{0.4268pt}{25.18115pt}{0.23572pt}{25.18115pt}{0.0pt}\pgfsys@curveto{25.18115pt}{-0.23572pt}{25.37224pt}{-0.4268pt}{25.60796pt}{-0.4268pt}\pgfsys@curveto{25.84367pt}{-0.4268pt}{26.03476pt}{-0.23572pt}{26.03476pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.60796pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{51.21591pt}{0.0pt}\pgfsys@moveto{51.64272pt}{0.0pt}\pgfsys@curveto{51.64272pt}{0.23572pt}{51.45163pt}{0.4268pt}{51.21591pt}{0.4268pt}\pgfsys@curveto{50.9802pt}{0.4268pt}{50.78911pt}{0.23572pt}{50.78911pt}{0.0pt}\pgfsys@curveto{50.78911pt}{-0.23572pt}{50.9802pt}{-0.4268pt}{51.21591pt}{-0.4268pt}\pgfsys@curveto{51.45163pt}{-0.4268pt}{51.64272pt}{-0.23572pt}{51.64272pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{51.21591pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{76.82388pt}{0.0pt}\pgfsys@moveto{77.25069pt}{0.0pt}\pgfsys@curveto{77.25069pt}{0.23572pt}{77.0596pt}{0.4268pt}{76.82388pt}{0.4268pt}\pgfsys@curveto{76.58817pt}{0.4268pt}{76.39708pt}{0.23572pt}{76.39708pt}{0.0pt}\pgfsys@curveto{76.39708pt}{-0.23572pt}{76.58817pt}{-0.4268pt}{76.82388pt}{-0.4268pt}\pgfsys@curveto{77.0596pt}{-0.4268pt}{77.25069pt}{-0.23572pt}{77.25069pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{76.82388pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.8,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.0,0.0}\pgfsys@color@rgb@stroke{0.8}{0.0}{0.0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.8}{0.0}{0.0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.0,0.0}{}{}{}{}{{}}\pgfsys@moveto{35.301pt}{0.0pt}\pgfsys@lineto{47.28287pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{35.301pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.8,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.0,0.0}\pgfsys@color@rgb@stroke{0.8}{0.0}{0.0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.8}{0.0}{0.0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.0,0.0}{}{}{}{}{{}}\pgfsys@moveto{60.90895pt}{0.0pt}\pgfsys@lineto{72.89084pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{60.90895pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}})$ (a) The Boolean
poset $\mathcal{O}({\tt L}_{2})$ and ${\rm OC}({\tt L}_{2})$.

$\mathbb{F}_{0}^{2}$$\begin{array}[]{c}\mathbb{F}_{0}^{2}\oplus\mathbb{F}_{1}\\\
\oplus\\\
\mathbb{F}_{0}^{2}\oplus\mathbb{F}_{1}\end{array}$$\mathbb{F}_{0}^{2}$$C^{0}(X)$$C^{0}(X)$$C^{0}(X)$$d^{0}$$d^{1}$$\oplus$$\oplus$$\mathcal{F}(\text{\raisebox{0.0pt}{
\leavevmode\hbox to57.88pt{\vbox
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{25.60796pt}{0.0pt}\pgfsys@moveto{26.03476pt}{0.0pt}\pgfsys@curveto{26.03476pt}{0.23572pt}{25.84367pt}{0.4268pt}{25.60796pt}{0.4268pt}\pgfsys@curveto{25.37224pt}{0.4268pt}{25.18115pt}{0.23572pt}{25.18115pt}{0.0pt}\pgfsys@curveto{25.18115pt}{-0.23572pt}{25.37224pt}{-0.4268pt}{25.60796pt}{-0.4268pt}\pgfsys@curveto{25.84367pt}{-0.4268pt}{26.03476pt}{-0.23572pt}{26.03476pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.60796pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{51.21591pt}{0.0pt}\pgfsys@moveto{51.64272pt}{0.0pt}\pgfsys@curveto{51.64272pt}{0.23572pt}{51.45163pt}{0.4268pt}{51.21591pt}{0.4268pt}\pgfsys@curveto{50.9802pt}{0.4268pt}{50.78911pt}{0.23572pt}{50.78911pt}{0.0pt}\pgfsys@curveto{50.78911pt}{-0.23572pt}{50.9802pt}{-0.4268pt}{51.21591pt}{-0.4268pt}\pgfsys@curveto{51.45163pt}{-0.4268pt}{51.64272pt}{-0.23572pt}{51.64272pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{51.21591pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
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