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$$\mathcal{F}(\text{\raisebox{0.0pt}{ \leavevmode\hbox to57.88pt{\vbox
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{}{}{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\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{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.0,0.0}{}{}{}{}{{}}\pgfsys@moveto{29.541pt}{0.0pt}\pgfsys@lineto{41.52287pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{41.52287pt}{0.0pt}\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{
}\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{
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}\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}}}})$ (b) The Boolean
poset $\mathcal{O}({\tt A}_{2})$ and ${\rm OC}({\tt A}_{2})$.

Figure 10. The complexes ${\rm OC}({\tt L}_{2})$ (above) and ${\rm OC}({\tt
A}_{2})$ (below) associated to the linear graph ${\tt L}_{2}$ and the
alternating graph ${\tt A}_{2}$, respectively. The symbol $\mathbb{F}^{n}_{b}$
denotes a copy of $\mathbb{F}^{n}$ in simplicial degree $b$.

Crucially, the cohomology $\mathrm{OH}$ does depend on the fixed orientation
on ${\tt G}$. Indeed, the two graphs just considered correspond to two
different orientations on the linear unoriented graph with two edges. A simple
computation based on the graded vector spaces in Figure 10 shows that the
Euler characteristic of $\mathrm{OH}({\tt L}_{2})$ and $\mathrm{OH}({\tt
A}_{2})$ are equal in absolute value, but have different sign; hence, their
poset cohomologies are not isomorphic. Furthermore, if we denote by
$\mathbb{F}^{n}_{a,b}$ a copy of $\mathbb{F}^{n}$ in homological degree $a$
and simplicial degree $b$, explicitly computing the homology for both graphs
we obtain that ${\rm OH}^{*,*}({\tt
L}_{2})\cong\mathbb{F}_{0,1}\oplus\mathbb{F}_{2,1}$, and ${\rm OH}^{*,*}({\tt
L}_{2})\cong\mathbb{F}^{2}_{1,1}$.

We conclude by proving the following unexpected result; namely, the oriented
homology $\mathrm{OH}({\tt G},\mathfrak{o})$ admits an extremely simple and
explicit description, regardless of the initial orientation $\mathfrak{o}$
chosen. In particular, we show that $\mathrm{OH}({\tt G},\mathfrak{o})$ is
simply a count of the free-flow orientations on ${\tt G}$.

######  Theorem 6.2.

Let ${\tt G}$ be a connected unoriented graph, and let $\mathfrak{o}$ be an
orientation on ${\tt G}$. Then, the generators of $\mathrm{OH}^{*}({\tt
G}_{\mathfrak{o}})$ are in bijection with free-flow orientations on ${\tt G}$.
More precisely, $\dim({\rm OH}^{i}({\tt G}_{\mathfrak{o}}))$ is the number of
free-flow orientations on ${\tt G}$ obtained from $\mathfrak{o}$ by changing
the orientation of exactly $i$ edges. In particular, ${\rm OH}^{*}({\tt
G}_{\mathfrak{o}})$ is non-trivial if and only if ${\tt G}$ is a pseudotree.

###### Proof.

By definition, ${\rm OC}^{i}({\tt G},\mathfrak{o})$ is the $\mathbb{F}$-vector
space spanned by oriented matchings on ${\tt G}_{\mathfrak{o}^{\prime}}$, for
all orientations $\mathfrak{o}^{\prime}$ differing from $\mathfrak{o}$ on
precisely $i$ edges.

We call a chain complex _Boolean of dimension $n+1$_ if it is isomorphic to
the reduced simplicial chain complex of the standard $n$-simplex (the
$-1$-simplex being the empty set). We claim that for each matching $m$ in
$\mathrm{M}({\tt G})=\bigcup_{\mathfrak{o}}\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}})$ (_cf._ Equation (7)), we have a Boolean subcomplex $C(m)$
of ${\rm OC}({\tt G},\mathfrak{o})$, and that moreover ${\rm OC}({\tt
G},\mathfrak{o})$ splits as the direct sum of these complexes.

Fix $m\in\mathrm{M}({\tt G})$, and define

$C(m)=\mathbb{F}\langle m^{\prime}\in\mathrm{M}^{o}({\tt
G},\mathfrak{o}^{\prime})\mid\mathfrak{o}^{\prime}\in\mathcal{O}({\tt G}),\
m^{\prime}=m\rangle\subseteq{\rm OC}({\tt G},\mathfrak{o})\ .$

Since the differential $d$ of ${\rm OC}^{*}({\tt G},\mathfrak{o})$ is induced
by the inclusions $\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}_{1}})\cap\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}_{2}})\hookrightarrow\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}_{2}})$ (_cf._ Equations (2) and (10)), it follows that
$C(m)$, endowed with its induced differential, is a subcomplex. Furthermore,
if $d(x)\in C(m)$ then $x\in C(m)$. This last fact implies that ${\rm OC}({\tt
G},\mathfrak{o})$ splits as the direct sum of these complexes. It is not hard
to see that $C(m)$ is Boolean; first, consider the set $F$ of edges of ${\tt
G}$ identified by the matching $m\subset E(\mathtt{F}({\tt G}))$. For an
orientation $\mathfrak{o}^{\prime}$ on ${\tt G}$, $m$ will belong to
$\mathrm{M}^{o}({\tt G}_{\mathfrak{o}^{\prime}})$ if and only if the target of
each edge in $F$ is the same as the corresponding edge in $m$ (_cf._
Definition 3.3). Let $\widehat{\mathfrak{o}}$ be the unique orientation on
${\tt G}$ such that the edges in $F$ are coherent with $m$, and the remaining
edges are oriented as in $\mathfrak{o}$. Starting from
$\widehat{\mathfrak{o}}$ and changing the orientations of the edges in $E({\tt
G})\setminus F$ we get all orientations $\mathfrak{o}^{\prime}$ such that

$C(m)\cap\mathbb{F}\langle\mathrm{M}^{o}({\tt
G}_{\mathfrak{o}^{\prime}})\rangle\neq(0)\ .$

In particular, $C(m)$ is Boolean of dimension $|E({\tt G})\setminus F|$.

Boolean complexes have trivial homology, unless they are of dimension $0$ – in
which case the homology is free of rank $1$. Therefore, the generators of
${\rm OH}({\tt G}_{\mathfrak{o}})$ correspond precisely to the oriented
matchings on ${\tt G}$ with the same number of edges as ${\tt G}$. In turn,
each of these matchings induce an orientation on ${\tt G}$ where all vertices
have indegree $1$ or $0$. Thus, by Lemma 3.7 these orientations are exactly
the free-flow orientations on ${\tt G}$. ∎

This last result implies in particular that the rank of the oriented homology
is independent of the initial choice of orientation. However, as showed in the
previous example and in Figure 10, the oriented homology of a given graph with
respect to two distinct orientations can be different. Namely, the generators
might be supported in different degrees.

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