is equivalent to the definition of dual cell used in [29] (which explains our
choice for the letter $B$ in the notation $\overline{x}^{B}$). The dual cell
$\overline{x}^{B}$ is therefore seen as a sub-simplicial complex of
$\widecheck{K}.$

Figure 7: This illustration represents certain cells of the barycentric
subdivision $\widecheck{K}$ of a 2-cc $K$ and gives three examples of cells in
$\overline{K}^{B}.$ Being a simplicial complex, each 2-cell of the barycentric
subdivision is a triangle and the vertices of the barycentric subdivision are
represented using three different symbols depending of the rank of the cells
it represent: a vertex of $\widecheck{K}$ of the form $\\{C\\}$ where $C\in
K^{[2]}$ is represented by a cross, a vertex of the form $\\{e^{\prime}\\}$
where $e^{\prime}\in K^{[1]}$ is represented by a dash and a vertex of the
form $\\{v\\}$ simply corresponds to a vertex of $K.$ There are two 2-cells of
$\overline{K}^{B},$ the first $\overline{v}^{B}$ is filled the black wavy
pattern, the other $\overline{w}^{B}$ is filled with the white wavy pattern.
We also have the 1-cell $\overline{e}^{B},$ the dual of $e=\\{v,w\\},$
represented by the two thick dashed lines. As a comparison, the dual cell
$\overline{v}$ used in our framework simply corresponds to the triangle whose
vertices are the three 2-cells containing $v.$

We can then define the poset
$\overline{K}^{B}:=\\{\overline{x}^{B}\leavevmode\nobreak\
|\leavevmode\nobreak\ x\in K\\}$ with rank function

$\operatorname{rk}_{\overline{K}^{B}}(\overline{x}^{B}):=\operatorname{Rk}(\widecheck{\overline{K}\cap\overline{x}})=\overline{\operatorname{rk}}(\overline{x}).$

It is possible to show that the map $\phi(\overline{x}^{B}):=\overline{x}$
defines an isomorphism of posets from $\overline{K}^{B}$ to $\overline{K}$
such that
$\overline{\operatorname{rk}}(\phi(\overline{x}^{B}))=\operatorname{rk}_{\overline{K}^{B}}(\overline{x}^{B}).$
Nevertheless $\overline{K}^{B}$ is not a cc since the cells in
$\overline{K}^{B}$ are not subsets of $(\overline{K}^{B})^{[0]}$ as these
cells also contain elements in
$\widecheck{K}^{[0]}\setminus(\overline{K}^{B})^{[0]}.$

The remainder of this section aims to present an adaptation of the Theorem 3
given by Bayer in [21], stating that the barycentric subdivision of a
polyhedral complex determines the underlying complex up to duality. The notion
of polyhedral complex from [21] is identical to the notion of complex of
polytopes defined for example in [77], which will be discussed in more detail
in Section 2.3. We sometimes designate such convex polytopes in
$\mathbb{R}^{n}$ as ”embedded polytopes”. Bayer’s proof can nevertheless be
easily adapted to our purely combinatorial framework. Theorem 1.28 stated
below can then be seen as a relatively straightforward generalization of
Bayer’s result to local cell complexes. The main point from Bayer’s proof that
needs an additional justification is an observation about the existence of a
type of ordering of the faces of an embedded polytope. This observation is
also valid for the sub-cells of a cell in a cell complex and this constitutes
our Lemma 1.26. Note that by ”face of a polytope”, we mean any sub-polytope
and not necessarily of dimension one lower than the original polytope (as is
defined for the cells of a cc).

We start by summarizing the notions needed to state the proof of Theorem 1.28,
closely following the section 3 of [21]. First, we define what is an
orientation in a graph-based cc. For a set of edges
$\mathcal{E}\subset\binom{\mathcal{V}}{2}$ we use the notation
$\mathcal{E}_{\rightarrow}$ to denote the set of ordered two-sets of
$\mathcal{E}.$ An element $e=(x,y)\in\mathcal{E}_{\rightarrow}$ is called an
directed edge, $e_{s}:=x$ is called its source and $e_{t}:=y$ its target.

######  Definition 1.25 (Orientation of a cc).

An orientation of a graph-based cc $K$ is a map $g:K^{[1]}\mapsto
K^{[1]}_{\rightarrow}$ such that $g(e)^{[0]}=e.$

We say that an orientation $g$ is transitive if, whenever

$g(\\{x,y\\})=(x,y),\quad g(\\{y,z\\})=(y,z)$

for $x,y,z\in K^{[0]}$ such that $\\{x,y\\},\\{y,z\\}\in K^{[1]}$ then
$\\{x,z\\}\in K^{[1]}$ and

$g(\\{x,z\\})=(x,z).$

In particular an oriented graph $(\mathcal{G},g)$ is a graph
$\mathcal{G}=(\mathcal{V},\mathcal{E}),$ for example the 1-skeleton of a
graph-based cc, with an orientation
$g:\mathcal{E}\longrightarrow\mathcal{E}_{\rightarrow}.$ An isomorphism of
oriented graphs is a cc-isomorphism
$\phi:\mathcal{G}\longrightarrow\mathcal{G}^{\prime}$ between two oriented
graphs $(\mathcal{G},g)$ and $(\mathcal{G}^{\prime},g^{\prime})$ such that
$g^{\prime}$ is the image of $g$ through $\phi$ in the sense that

$g^{\prime}(\phi(e))=(\phi(g(e)_{s}),\phi(g(e)_{t})).$

We now turn to the notion of uniquely partially orderable cc, a key notion in
the proof of the Theorem 1.28, in which we essentially prove that the
barycentric subdivision of a local cell complex is uniquely partially
orderable. Let $g$ be a transitive orientation of a graph-based cc $K$ and let
$(x,y),(w,z)\in K^{[1]}_{\rightarrow}.$ We say that $(x,y)$ directly forces
$(w,z)$ if either $x=w$ and $\\{y,z\\}\notin K^{[1]}$ or $y=z$ and
$\\{x,y\\}\notin K^{[1]}.$ More generally, we say that $(x,y)$ forces $(y,z)$
if there is a finite sequence of directed edges starting from $(x,y)$ and
ending at $(y,z)$ such that each directed edge in the sequence directly forces
the next one. This relation is an equivalence relation, which allows to
decompose $K^{[1]}_{\rightarrow}$ into equivalence classes called implication
classes. $K$ is called uniquely partially orderable if and only if $K$ admits
exactly two transitive orientations, each being the reverse of the other, or
equivalently, if and only if there exists only two possible implication
classes in $K^{[1]}_{\rightarrow}.$ For example, a transitive orientation on
$\widecheck{K}$ is simply given by
$g_{K}:\widecheck{K}^{[1]}\longrightarrow\widecheck{K^{[1]}}_{\rightarrow},$
the orientation induced by the inclusion relation in $K,$ i.e. if $x,y\in K$
satisfy $x\subsetneq y,$ or equivalently $\\{x\subsetneq
y\\}\in(\widecheck{K})^{[1]},$ then $g_{K}(\\{x\subsetneq y\\})=(x,y).$
(Recall that our notation $\\{x\subsetneq y\\}$ is a shorthand for an -
unordered - set $\\{x,y\\}$ whose elements are totally ordered by the
inclusion relation in $K,$ so that we also specify this total order by writing
$\\{x\subsetneq y\\}$). A second transitive orientation of $\widecheck{K}$ is
the reverse of $g_{K},$ denoted $g^{-1}_{K},$ and defined by
$g^{-1}_{K}(\\{x\subsetneq y\\}):=(y,x).$ In case where $K$ is a closed cc
then Proposition 1.24 implies

$\widecheck{K}\cong\widecheck{\overline{K}}.$

Using the later cc-isomorphism, we can identify $\\{x\subsetneq
y\\}\in\widecheck{K}^{[1]}$ with
$\\{\overline{y}\subsetneq\overline{x}\\}\in\widecheck{\overline{K}}^{[1]}$
and using this identification we have that $g^{-1}$ corresponds to the
transitive orientation $g_{\overline{K}}$ induced by the inclusion relation in
$\overline{K}.$

We prove the following lemma, which, in the proof from [21], corresponds to an
observation about the faces of convex polyhedra, where by faces we mean any
sub-polyhedra. We will see that it also holds in the case of cc of the form
$K\cap z,$ for $z\in K^{[R]},$ as a consequence of Axioms iv) and ii).

######  Lemma 1.26.

Let $K$ be a $R$-lcc and let $z\in K^{[R]}.$ For any $0\leq r\leq R-2$ and any
$r$-cell $y\in K\cap z,$ there is a $(r+1)$-cell $w\in K\cap z$ such that
$y\not\subset w.$

###### Proof.

It is sufficient to prove the claim for the case $r=R-2.$ The case $r=0,$ i.e.
$R=2$ and $y$ is a vertex of a 2-cell $z,$ is direct since by Lemma 1.12 $z$
is a cycle of at least 3 edges. Suppose that $r>0$ and take
$x\in\triangle\left(y\right),$ i.e. such that $\operatorname{rk}(x)=r-1.$ By
the diamond property iv) there exist $\\{w_{1},w_{2}\\}\subset K^{[r+1]}$ and
$y^{\prime}\in K^{[r]}$ such that

$\bigtriangledown\left(y\right)\cap\triangle\left(z\right)=\\{w_{1},w_{2}\\},\quad\bigtriangledown\left(x\right)\cap\triangle\left(w_{1}\right)=\\{y,y^{\prime}\\}.$

Moreover there exists $w\in K^{[r+1]}$ such that

$\bigtriangledown\left(y^{\prime}\right)\cap\triangle\left(z\right)=\\{w_{2},w\\}$

and $w\in K^{[r+1]}\setminus\\{w_{1},w_{2}\\}$ since otherwise
$\\{y,y^{\prime}\\}\subset w_{1}\cap w_{2}$ which contradicts ii). Also
$y\not\subset w$ since
$\bigtriangledown\left(y\right)\cap\triangle\left(z\right)=\\{w_{1},w_{2}\\}$
and this concludes the proof. ∎

We also add the following lemma, which is considered as an obvious statement
in [21] for the case of face lattices of polytopes.

######  Lemma 1.27.

A cell complex $K$ is determined up to cc-isomorphism by the oriented
1-skeleton of its barycentric subdivision $(\widecheck{K}^{(1)},g_{K}).$

###### Proof.

Let $\\{x\\}$ be a vertex of $\widecheck{K}.$ Lets us prove this result by
first indicating how to associated to $\\{x\\}$ a set of vertices of
$\widecheck{K}$ of the form $\\{v\\}$ where $v\in K^{[0]}$ corresponding to
the vertices of $K$ contained in $x.$ The elements of the form
$\\{v\\}\in\widecheck{K}^{[0]}$ such that $v\in x$ are the starting points of
all directed paths (i.e. paths using directed edges which target is equal to
the source of the next directed edges or the endpoint of the directed path) in
$(\widecheck{K}^{(1)},g_{K})$ having $\\{x\\}$ as an endpoint and which length
is maximal with this property. The rank of $\\{x\\}\in\widecheck{K}^{[0]}$ is
the length of any of such directed path. This is well defined by Axiom iii). ∎

We now have the required notions and preliminary results to state and prove
the following theorem. This result seems important to us as a general fact
about lcc and will be interesting for us in relation to the reconstruction
results of Chapter 2, as explained in Remark 2.47. It also lead us to
introduce a notion of orientation in a cc, but this notion as defined here
will not be used again in this work. The work of Basak [20] introduces another
notion of orientation which is central to his treatment of the Poincaré
duality for his definition of combinatorial cell complexes.

######  Theorem 1.28.

Let $K$ and $J$ be two lcc having at least two maximal cells and such that
$\widecheck{K}\cong\widecheck{J}.$ Then we have the following possibilities:

  * •

if $\partial K$ is non-empty then $\partial J$ is also non-empty and $K\cong
J;$

  * •

if $K$ and $J$ are closed then either $K\cong J$ or $\overline{K}\cong J.$

###### Proof.

Let $(\widecheck{K}^{(1)},g_{K})$ and $(\widecheck{J}^{(1)},g_{J})$ be the
oriented 1-skeletons of the barycentric subdivision of $K$ and $J,$ where both
$g$ and $h$ are the orientation induced by the inclusion relation in $K$ and
$J$ respectively. Our assumption implies in particular that
$\widecheck{K}^{(1)}$ and $\widecheck{J}^{(1)}$ are cc-isomorphic, we
therefore consider these two graphs as identical for simplicity. The strategy
of the proof is to show that $\widecheck{K}$ is uniquely partially orderable,
which would directly imply that $g_{J}=g_{K}$ or $g_{J}=g_{K}^{-1}$ as defined
above.

In case $K$ and $J$ are closed, the two possibilities $g_{J}=g_{K}$ and
$g_{J}=g_{K}^{-1}$ lead consistent cases since $g_{K}^{-1}=g_{\overline{K}}$
and Lemma 1.27 lead $K\cong J$ or $\overline{K}\cong J.$

If $\partial K$ is non-empty then $g^{-1}_{K}$ cannot be a transitive
orientation induced by the inclusion relation of a local cell complex. To
justify this statement, let us assume by contradiction that

$g_{K}^{-1}=g_{L}:\widecheck{L}^{[1]}\longrightarrow\widecheck{L}_{\rightarrow}^{[1]}$

is a transitive orientation induced by the inclusion relation of a lcc $L.$
Consider a maximal cell $y\in\partial K,$ which is therefore included in a
unique maximal cell $z$ of $K.$ Then, as shown in Lemma 1.27, $\\{z\\}$
corresponds to a vertex of $L$ and $\\{y\\}$ corresponds to an edge of $L$ and
this is a contradiction with the fact that every edge of $L$ has two vertices.
As a consequence, it only leaves $g_{K}=g_{J}$ as a possibility and Lemma 1.27
implies $K\cong J.$

We therefore turn to the proof that $\widecheck{K}$ is uniquely partially
orderable by induction on the rank of $K.$

For $\operatorname{Rk}(K)=1$ we have that $K$ is a connected graph and so is
$\widecheck{K}.$ Every edge $e=\\{v,w\\}$ of $K$ is subdivided into two ”half-
edges” $\\{\\{v\\},\\{e\\}\\}$ and $\\{\\{w\\},\\{e\\}\\}$ in
$\widecheck{K}^{[1]}.$ Moreover, any two edges
$\varepsilon,\varepsilon^{\prime}$ of $\widecheck{K}$ sharing a vertex in
$\widecheck{K}^{[0]}$ correspond either to a subdivided edge of $K$ or to two
”half-edges of $K$” i.e. $\varepsilon=\\{\\{v\\},\\{e\\}\\},$
$\varepsilon^{\prime}=\\{\\{v\\},\\{e^{\prime}\\}\\}$ where $e,e^{\prime}\in
K^{[1]}$ and $v=e\cap e^{\prime}\in K^{[0]}.$ In both cases there is no edge
between the other ends of $\varepsilon$ and $\varepsilon^{\prime}$ and this
means that an orientation of $\varepsilon$ directly forces an orientation of
$\varepsilon^{\prime}.$ The connectedness of $\widecheck{K}$ then implies that
an orientation of any edge in $\widecheck{K}$ force an orientation of any
other edge in $\widecheck{K},$ which shows that $K$ is uniquely partially
orderable.

Let $K$ be a $R$-lcc with $\left|K^{[R]}\right|\geq 2$ and assume that
$\widecheck{(K^{\prime})}^{(1)}$ is uniquely partially orderable for any lcc
$K^{\prime}$ of rank smaller or equal to $R-1$ with more than one maximal
cell. Let $K^{\prime}:=K^{(R-1)}$ and
$G^{\prime}:=(\widecheck{K^{\prime}})^{(1)}.$ Then $K^{\prime}$ is a
$(R-1)$-lcc with more than one maximal cell and $G^{\prime}$ is the sub-graph
of $G$ restricted to the vertices with corresponding cell in $K$ having a rank
smaller or equal to $R-1.$ By induction $G^{\prime}$ is uniquely partially
orderable and we show that an orientation of $G^{\prime}$ forces an
orientation of $G.$

Let $z\in K^{[R]}$ and let $\\{z\\}$ be the corresponding vertex in $G.$ Since
there are at least two maximal cells in $K,$ there is one vertex $v$ of $z$
contained in an edge $e\in K^{[1]}$such that $e\not\subset z.$ Therefore we
obtain

$\\{\\{v\\},\\{z\\}\\}\in
G^{[1]},\quad\\{\\{v\\},\\{e\\}\\}\in(G^{\prime})^{[1]}\text{ and
}\\{\\{z\\},\\{e\\}\\}\notin G^{[1]}$

and this implies that $(\\{v\\},\\{e\\})$ directly forces $(\\{v\\},\\{z\\}).$
We conclude the proof by showing that $(\\{v\\},\\{z\\})$ directly forces all
edges of $G$ containing $\\{z\\}.$

Indeed, all edges of $G$ containing $\\{z\\}$ are of the form
$\\{\\{z\\},\\{w\\}\\}$ where $w$ is a sub-cell of $z.$ By Lemma 1.26 we can
find an ordering of the cells in $K\cap z,$ starting with any vertex in $z,$
so that no two consecutive sub-cells contain one another. This corresponds to
an ordering of the vertices in $G$ neighbouring $\\{z\\},$ so that no two
consecutive vertices are linked by an edge. We can suppose that this vertex
ordering starts with $v$ and this means that $(\\{v\\},\\{z\\})$ forces each
edge containing $\\{z\\}$ in G and this concludes the proof. ∎

###  1.6 Topological considerations

In this section, we mention two different ways to assign a topology to a cc:
the first one is from [20] and the second, using a notion of a geometrical
realization, is from [75]. It seems clear that these two topologies are
equivalent, but we have not proven this here. We then give the definitions of
star, link and homogeneity in a cc and define the notions of cellular and
simplicial manifold. All homeomorphisms here are piecewise-linear
homeomorphisms (as defined e.g. in [29]).

In [20] (setion 2.4) the topology of a cc $K$ is defined by setting all closed
subsets $C$ of $K$ to be the subsets of $K$ such that if $x\in
K,\leavevmode\nobreak\ y\in C,\leavevmode\nobreak\ x\subset y$ then $x\in C.$
If we define the closure of a subset $S\subset K$ to be the set of cells of
$K$ that are contained in some cell in $S$ then the closed subsets of $K$ are
given by the closures of subsets of $K.$ As a consequence, $K$ is not
Hausdorff if it has rank one or above, since each set of the form $\\{x\\}$
where $x\in K$ has a rank higher or equal to 1 is a non-closed ”point” in this
topology. For example if $x=\\{v,w\\}\in K^{[1]},$ this topology implies that
$\\{\\{v,w\\}\\}$ is an ”open edge” whereas the closure of $\\{x\\}$
corresponds to a ”closed edge” $\\{\\{v,w\\},\\{v\\},\\{w\\}\\}$ and this is
coherent with the following definitions obtained using a geometrical
realization.

Another way to define a topology on a cc $K,$ described for example in [75],
is to consider $K$ simply as a poset to which one associates the topology
obtained from the geometrical realization of its barycentric subdivision
$\widecheck{K}.$ The geometrical realization of a simplicial complex is
defined in the following way. A $d$-dimensional geometric simplex $\sigma$ in
$\mathbb{R}^{n}$ is defined as the convex hull of $d+1$ points in
$\mathbb{R}^{n},$ each such point being a vertex of $\sigma.$ The set of sub-
simplicies of $\sigma,$ $B(\sigma),$ contains all the convex hulls of subsets
of vertices of $\sigma.$ A geometric simplicial complex $\Lambda$ in
$\mathbb{R}^{n}$ is defined as a non-empty collection of geometric simplicies
satisfying that if $\sigma\in\Lambda$ and $\tau\in B(\sigma)$ then
$\tau\in\Lambda$ and if $\tau,\sigma\in K$ then $\tau\cap\sigma\in
B(\sigma)\cap B(\sigma)$ (the roles of $K$ and $\Lambda$ are exchanged
compared to the notations used in [75]). From a geometric simplicial complex
$\Lambda,$ one can define a (abstract) simplicial complex $K(\Lambda)$
corresponding to the poset of non-empty faces of $\Lambda$ and say that
$\Lambda$ realizes $K.$ It is an exercise in [67] (2.27(1)) to show that every
simplicial complex can be realized as a geometric simplicial complex and such
a realization is unique up to homeomorphism. We can therefore define the
topology of $K$ to be the topology of $\Lambda$ as a subspace of
$\mathbb{R}^{n},$ with usual topology. As a direct consequence of Proposition
1.24, $K$ and $\overline{K}$ are homeomorphic as topological spaces.

The notions of star and link are usually defined in the context of simplicial
complexes. These definitions are easily adapted to the case of cc in the
following way.

######  Definition 1.29 (Star and link of a cell).

The star of a cell $x$ of a cell complex $K$ is defined by

$\operatorname{St}_{K}(x)=\\{y\in K\leavevmode\nobreak\ |\leavevmode\nobreak\
\exists w\in K\text{ such that }x\vee y=w\\}.$

The link of $x$ is defined as

$\operatorname{Lk}_{K}(x)=\\{y\in\operatorname{St}_{K}(x)\leavevmode\nobreak\
|\leavevmode\nobreak\ x\cap y=\emptyset\\}.$

The previous definitions of star and link are then equivalent to the
definitions given for example in [29] for the case of simplicial complex,
since if $\sigma$ and $\eta$ are sub-simplicies of a simplex in a simplicial
complex $K$ then $\sigma\vee\eta=\sigma\cup\eta\in K.$

Homogeneity in a cc is then defined using the star of each vertex and the
topology associated to a cc introduced before.

######  Definition 1.30 (Homogeneous vertex in a cc).

A vertex $v$ in a $R$-cc is homogeneous if any geometrical realization of the
cc $K\cap\operatorname{St}_{K}(v)$ is homeomorphic to the closed
$R$-dimensional Euclidean ball. If every vertex in $K$ is homogeneous, we say
that $K$ is homogeneous.

An example of non-homogeneous vertex is given in Figure 17 of Chapter 3. It is
then natural to introduce the following definitions of cellular manifold and
simplicial manifold.

######  Definition 1.31 (Cellular and simplicial manifold).

A cellular $R$-manifold is a homogeneous $R$-cc. A simplicial (cellular)
manifold is a cellular manifold such that the underlying cc is simplicial.

A discretization of a piecewise-linear manifold $M$ is a cellular manifold $K$
which geometrical realization is homeomorphic to $M$ and a triangulation of
$M$ is a simplicial discretization of $M.$

## 2 Structures in a simple cell complex and reconstruction

The aim of this chapter is to prove Theorem 2.42 as well as Corollaries 2.45
and 2.46 of Section 2.6. These results can be interpreted as a method for
reconstructing a local even simply connected simple cell complex from its
2-skeleton. The core of this chapter is thus devoted to the introduction of
the notions of fullness, evenness, monodromy-freedom and simple connectedness
and prove some related basic facts, with the exception of Section 2.3 where we
make links with some well-known notions of discrete geometry. A key technical
result that allows to prove the results on reconstruction is the ”Extension
Lemma” 2.40. This lemma will be the first result involving all notions
required for the main results of Section 2.6 as well as two newly introduced
notions of this chapter defined in the context of a cc of rank 2 or higher: a
connection and covariant edge fields. We start with a short section
introducing these two new notions, followed by Section 2.2 where we introduce
the notions of full and simple cell complexes that characterize the duals of
simplicial cell complexes. In Section 2.3, we make a digression from the
reconstruction problem about an abstract definition of polytopes and
shellability, using the notion of simplicity. We first show the Euler-Poincaré
formula for shellable local cell complexes. We then introduce a notion of
2-shellability, only involving the cells of the 2-skeleton, and show in
Theorem 2.25 that the 2-shellability of a simple cell complex is equivalent to
the shellability of its dual. We then come back to the discussion of
reconstruction-related notions in Section 2.4, a mostly technical section
whose principal goal is to define simply connected cc. Simple connectedness in
a cc is defined using a combinatorial notion of homotopy that naturally arise
from the structure of 2-cc. In Section 2.5, we introduce the concept of
monodromy-free and even cc. We then have all the ingredients needed to give a
relatively quick proof of the Extension Lemma at the beginning of Section 2.6,
which allows us to prove the main results on reconstruction in the remainder
of this last section.

In this chapter 2-cc play an important role. We often use the letter $C$ to
denote a $2$-cell of a cc and we also recall that if $K$ is a pure 2-cc then
the dual set of $A\subset K^{[0]}$ corresponds to

$\overline{A}:=\\{C\in K^{[2]}\leavevmode\nobreak\ |\leavevmode\nobreak\
A\subset C\\}.$

In what follows, we will choose to state certain results using a 2-cc $K$ in
order to formulate the proof in terms of dual sets as above. In these cases
$K$ often play the role of the 2-skeleton of a cc of higher rank for which the
corresponding result is also true.

It will also be convenient to use the following notation to denote the set of
edges included in a cell $x\in K$ and containing a vertex $v\in x,$

$\mathcal{E}_{v}^{x}:=\\{e\in\mathcal{E}_{v}\leavevmode\nobreak\
|\leavevmode\nobreak\ e\subset x\\}.$

More generally, if $\mathcal{A}\subset K,$ we define

$\mathcal{E}_{v}^{\mathcal{A}}:=\bigcup_{x\in\mathcal{A}}\mathcal{E}_{v}^{x}.$

As an example used in Section 2.1, if $v,w$ are vertices of a pure graph-based
$2$-cc $K$ such that $\\{v,w\\}\in K^{[1]},$ we consider the following subset
of $\mathcal{E}_{v}:$

$\mathcal{E}_{v}^{\overline{w}}:=\\{e\in\mathcal{E}_{v}\leavevmode\nobreak\
|\leavevmode\nobreak\ \exists C\in\overline{w}\text{ s.t. }e\subset
C\\}=\\{e\in\mathcal{E}_{v}\leavevmode\nobreak\ |\leavevmode\nobreak\
\overline{\\{v,w\\}}\cap\overline{e}\neq\emptyset\\}.$

In Section 2.2 we introduce the notion of full cc and this notion in
particular implies that $\mathcal{E}_{v}^{\overline{w}}=\mathcal{E}_{v}$ for
all $w\in\mathcal{N}_{v}.$ Since these edge sets will for the most part be
used in the case of full cc, the set $\mathcal{E}_{v}^{\overline{w}}$ is
almost exclusively used to introduce a general definition of connection in the
next section.

###  2.1 Connection and covariant edge fields

In this first section, we define the notion of a connection as a mapping
between edges of neighbouring vertices canonically associated to a pure graph-
based cc of rank 2 or higher. This connection is an important tool for this
chapter, as well as the notion of covariant edge fields introduced next which
is the main object of the Extension Lemma 2.40.

The next Lemma is used to introduce the connection $\nabla$ for the case of a
pure graph-based cc $K$ of rank 2. The connection associated to a pure graph-
based cc $L$ of rank higher or equal to 2 is simply defined as the connection
associated to its 2-skeleton $K=L^{(2)}.$

######  Lemma 2.1.

Let $K$ be a pure graph-based 2-cc and $\\{v,w\\}\in K^{[1]}.$ There is a
bijection

$\displaystyle\nabla_{w}^{v}:\mathcal{E}_{v}^{\overline{w}}$
$\displaystyle\longrightarrow\mathcal{E}_{w}^{\overline{v}}$ $\displaystyle
e_{v}$ $\displaystyle\longmapsto\nabla_{w}^{v}e_{v}=:e_{w},$

as illustrated in Figure 8, such that

  1. 1)

$\nabla_{w}^{v}\\{v,w\\}=\\{v,w\\},$

  2. 2)

If $C\in\overline{\\{v,w\\}}$ then $e_{v}\subset C$ if and only if
$e_{w}\subset C,$

  3. 3)

$(\nabla_{w}^{v})^{-1}=\nabla_{v}^{w}.$

Figure 8: This figure illustrates how the connection $\nabla_{w}^{v}$ maps an
edge $e_{v}\in\mathcal{E}_{v}\setminus\\{v,w\\}$ to an edge
$e_{w}\in\mathcal{E}_{w}\setminus\\{v,w\\}$ using a $2$-cell $C_{e_{v}}$
defined as the unique $2$-cell in $\overline{v}$ containing $e_{v}$ and
$\\{v,w\\}.$

###### Proof.

We set $\nabla_{w}^{v}\\{v,w\\}=\\{v,w\\}.$ Let
$e_{v}\in\mathcal{E}_{v}^{\overline{w}}\setminus\\{v,w\\}$ and let
$C_{e_{v}}\in\overline{e_{v}}\cap\overline{\\{v,w\\}}.$ Since
$e_{v}\neq\\{v,w\\}$ we have
$\left|\overline{e_{v}}\cap\overline{\\{v,w\\}}\right|=1$ as two 2-cells
cannot intersect on more than one edge by Axiom ii). $C_{e_{v}}$ is then the
unique 2-cell containing $e_{v}$ and $\\{v,w\\}.$ In particular
$C_{e_{v}}\in\overline{w},$ so there exists a unique edge
$e_{w}\in\mathcal{E}_{w}\setminus\\{v,w\\}$ such that
$\\{\\{v,w\\},e_{w}\\}\subset C_{e_{v}},$ which implies
$e_{w}\in\mathcal{E}_{w}^{\overline{v}},$ so that we can set
$\nabla_{w}^{v}e_{v}=e_{w}.$ The uniqueness of $C_{e_{v}}$ and $e_{w}$ ensures
2) and $\nabla_{v}^{w}e_{w}=e_{v}.$ ∎

The connection is also a convenient tool in the formulation of Lemma 2.7 and
in Definition 2.33 of monodromy-free cc. From this mapping, one is also
naturally lead to consider the notion of covariant edge fields, introduced
next.

######  Definition 2.2 (Covariant Edge field).

Let $K$ be a cc a let $\mathcal{D}\subset K^{[0]}$ be non-empty. An edge field
on $\mathcal{D}$ is a map

$\phi:\mathcal{D}\longrightarrow K^{[1]}$

such that $\phi(v)\in\mathcal{E}_{v}$ for all $v\in\mathcal{D}.$ We will
sometimes denote the domain of $\phi$ $\mathcal{D}$ by $\mathcal{D}_{\phi}.$

An edge field $\phi$ in a full graph-based cc $K$ is said to be covariant if,
whenever an edge $\\{v,w\\}\in K^{[1]}$ is included in $\mathcal{D}_{\phi}$
then $\nabla_{w}^{v}\phi(v)=\phi(w).$

Covariant edge fields, encoding the edge mapping induced by the connection,
are a key tool in the proof of the Theorem 2.42 where they allow to construct
”induced cells” by identifying edge fields that can be interpreted as
”tangent” and ”normal” to a given cell. In this sense the connection can in
fact be interpreted as defining a notion of ”parallel transport” in a cc.

###  2.2 Full and simple cc

In this section we introduce the notions of regularity and edge-regularity in
a cc and we explain how these notions are implied by the notion of fullness in
a cc that is defined next. Fullness is the object of two lemmas and is used as
a characteristic property in our definition of simple cc. We finally show that
simple cc are precisely the duals of simplicial complexes.

The following notion of $n$-regularity is common to many contexts involving
graphs.

######  Definition 2.3 ($n$-regular cc).

A graph-based cc $K$ is said to be $n$-regular if
$\left|\mathcal{E}_{v}\right|=n$ for all $v\in K^{[0]}$ or equivalently if
$K^{(1)}$ is a $n$-regular graph.

When considering cc of rank 2, it is natural to also define the notion of
edge-regularity as follows.

######  Definition 2.4 ($n$-edge-regular 2-cc).

A 2-cc $K$ is $n$-edge-regular if $\left|\overline{e}\right|=n$ for all $e\in
K^{[1]}.$

As a consequence of the diamond property iv), the notions of regularity and
edge-regularity in a cc are related. The precise relation is given in the next
remark mentioned in Lemma 2.7.

######  Remark 2.5.

Let $n\geq 1$ and $n-1\geq m\geq 0.$ An $n$-regular 2-cc is $m$-edge regular
if and only if $\left|\overline{v}\right|=nm/2$ for all $v\in K^{[0]},$ as
showed by the following counting argument:

$\displaystyle 2\left|\overline{v}\right|$
$\displaystyle=\sum_{C\in\overline{v}}2=\sum_{C\in\overline{v}}\left|\mathcal{E}_{v}^{C}\right|$
$\displaystyle=\left|\\{(e,C)\in\mathcal{E}_{v}\times\overline{v}\leavevmode\nobreak\
|\leavevmode\nobreak\ e\subset C\\}\right|$
$\displaystyle=\sum_{e\in\mathcal{E}_{v}}\left|\overline{e}\right|=m\left|\mathcal{E}_{v}\right|=m\deg(v)=mn,$

where the second equality uses that $v$ as two neighbors in $C$ by the diamond
property iv) and the penultimate equality uses that $K$ is $m$-edge-regular.
In particular, in a 2-edge-regular 2-cc we have
$\deg(v)=\left|\overline{v}\right|.$

Fullness in a cc, introduced next, is a key characteristic of cc dual to
simplicial complexes and formalizes the idea that one cannot add more cell to
the set of cells containing a given vertex while complying with Axiom ii).
This dualizes the property of (maximal) simplices in a simplicial complex for
which one cannot add more faces, as their faces are already given by all
subsets of their set of vertices. Since in what follows we will particularly
need to use this assumption for the case of 2-cc, or 2-skeletons of higher
dimensional cc, we chose to reserve the terminology ”full” to denote the
fullness property for the case of 2-cc and only specify a rank $r$ when using
the more general notion of $r$-fullness.

######  Definition 2.6 (Full cc).

Let $K$ be a cc of rank $R\geq 2$ and let $2\leq r\leq R.$ We say that $K$ is
$r$-full if for all $v\in K^{[0]},$ for all $2\leq k\leq r$ and for all
$S\subset\mathcal{E}_{v}$ such that $\left|S\right|=k,$ there is a (unique, by
Lemma 2.8) $r$-cell $x\in K$ such that $\mathcal{E}_{v}^{x}=S.$ We simply say
that $K$ is full if $K$ is 2-full.

In particular a $R$-full $R$-cc is pure. The following lemma constitutes a
first useful observation one can make about full connected cc.

######  Lemma 2.7.

A connected cc $K$ is full if and only if there exists $n\in\mathbb{N}^{*}$
such that $K$ is a $(n-1)$-edge-regular $n$-regular 2-cc.

###### Proof.

First, as a consequence of Remark 2.5, a $n$-regular 2-cc is $(n-1)$-edge
regular if and only if $\left|\overline{v}\right|=n(n-1)/2.$ Therefore if $K$
is full and $\\{v,w\\}\in K^{[1]}$ then $\nabla_{w}^{v}$ is a bijection from
$\mathcal{E}_{v}$ to $\mathcal{E}_{w}.$ Also, since $K^{(1)}$ is connected,
$K^{(1)}$ is $n$-regular where $n=\left|\mathcal{E}_{v}\right|$ and

$\left|\overline{v}\right|=\left|\binom{\mathcal{E}_{v}}{2}\right|=\binom{\left|\mathcal{E}_{v}\right|}{2}=\frac{n(n-1)}{2}.$

Conversely, if $K$ is a $(n-1)$-edge-regular $n$-regular 2-cc and $v\in
K^{[0]}$ then

$\frac{n(n-1)}{2}=\left|\overline{v}\right|\leq\left|\binom{\mathcal{E}_{v}}{2}\right|=\frac{n(n-1)}{2},$

so
$\left|\overline{v}\right|=\left|\\{\\{e_{1},e_{2}\\}\subset\mathcal{E}_{v}\\}\right|$
and $K$ is full. ∎

An important consequence of $k$-fullness is the following useful result, which
will be used several times in the remainder of this chapter.

######  Lemma 2.8.

Let $K$ be an $R$-cc and suppose that $K$ is $k$-full for a given $k$ such
that $2\leq k\leq R.$ Then for all $x\in K$ such that
$1\leq\operatorname{rk}(x)\leq k$ and all $v\in x,$ we have

$\left|\mathcal{E}_{v}^{x}\right|=\operatorname{rk}(x)$

and $x$ is the unique $\operatorname{rk}(x)$-cell $y$ such that
$\mathcal{E}_{v}^{y}=\mathcal{E}_{v}^{x}.$

###### Proof.

We proceed by induction on $1\leq\operatorname{rk}(x)\leq k.$

The claim for $\operatorname{rk}(x)=1$ is direct. For $x\in K^{[2]}$ and $v\in
x,$ we have $\left|\mathcal{E}_{v}^{x}\right|=2$ by the diamond property iv)
and if $x,y\in K^{[2]}$ satisfy $\mathcal{E}_{v}^{x}=\mathcal{E}_{v}^{y}$ then
$e\subset x\cap y$ for all $e\in\mathcal{E}_{v}^{x}.$ This implies
$\operatorname{rk}(x\cap y)\geq 2$ and therefore $x\cap y=x=y.$

Suppose that the claim holds for all cells in $K^{(r)}$ for some $2\leq r\leq
k-1$ and let $x\in K^{[r+1]}$ be such that $v\in x.$ By the property iii),
there exists $x_{s}\in K^{[s]},$ $s=r-1,r$ such that

$v\in x_{r-1}\subset x_{r}\subset x.$

By the diamond property there exists $x_{r}^{\prime}\in
K^{[r]}\setminus\\{x_{r}\\}$ such that $x_{r-1}\subset x_{r}^{\prime}\subset
x$ and by the induction hypothesis there is an edge
$e_{r}\in\mathcal{E}_{v}^{x_{r}^{\prime}}\setminus\mathcal{E}_{v}^{x_{r}}\subset\mathcal{E}_{v}^{x}.$
Since $K$ is $k$-full, there is $y\in K^{[r+1]}$ such that
$\mathcal{E}_{v}^{y}=\mathcal{E}_{v}^{x_{r}}\cup\\{e_{r}\\}.$ Therefore we
have that $\left|\mathcal{E}_{v}^{x\cap y}\right|=r+1$ and this implies
$\operatorname{rk}(x\cap y)\geq r+1$ by induction hypothesis. Hence $x\cap
y=x=y$ and we have
$\left|\mathcal{E}_{v}^{x}\right|=r+1=\operatorname{rk}(x).$ Similarly, the
uniqueness part is proven as follows. Let $y\in K^{[r+1]}$ such that
$\mathcal{E}_{v}^{y}=\mathcal{E}_{v}^{x}.$ By the previous argument
$\left|\mathcal{E}_{v}^{x\cap y}\right|=\operatorname{rk}(x\cap y)$ and
$\left|\mathcal{E}_{v}^{x\cap y}\right|=\left|\mathcal{E}_{v}^{x}\right|=r+1$
and this implies that $x\cap y=x=y.$ ∎

The following then serves as our definition of a simple cc.

######  Definition 2.9 (Simple cc).

A $R$-cc such that $R\geq 2$ is simple if it is closed and $R$-full. A $R$-cc
such that $R\leq 1$ is simple if and only if it is closed.

The next lemma provides an alternative definition of a simple cc as a graph-
based $R$-full $(R+1)$-regular $R$-cc.

######  Lemma 2.10.

A connected $R$-full $R$-cc is closed if and only if it is graph-based and
$(R+1)$-regular.

###### Proof.

Let $K$ be a connected $R$-full cell complex of rank $R.$ By Lemma 2.7 $K$ is
$n$-regular and Lemma 2.8 implies that $n\geq R$ and that for all $1\leq
r<s\leq R$ the number of $s$-cells containing a given $r$-cell $x$ is given by

$\left|\\{S\subset\mathcal{E}_{v}\setminus\mathcal{E}_{v}^{x_{r}}\leavevmode\nobreak\
|\leavevmode\nobreak\ \left|S\right|=s-r\\}\right|=\binom{n-r}{s-r},$

where $v\in x_{r}.$ For the case $s=R,\leavevmode\nobreak\ r=R-1$ and for any
$y\in K^{[R-1]},$ we obtain that

$\left|\bigtriangledown\left(y\right)\right|=\binom{n-r}{s-r}=\binom{n-R+1}{1}=n-R+1=2\quad\text{if
and only if}\quad n=R+1.$

∎

Using the previous lemma, we are then able to prove the main result of this
section.

######  Proposition 2.11.

A connected closed cc $K$ is simple if and only if $\overline{K}$ is
simplicial.

###### Proof.

Suppose that $K$ is a connected simple $R$-cc. In order to show that
$J=\overline{K}$ is simplicial it is sufficient to prove that any subset of a
maximal cell $z$ of $J$ is a cell of $J.$ Since $K$ is $(R+1)$-regular by
Lemma 2.10, $\overline{z}\in K^{[0]}$ is contained in $(R+1)$ edges, which
implies that $\left|\triangle\left(z\right)\right|=R+1.$ By Lemma 2.8 we have
that

$\left|z\right|=\left|\overline{\overline{z}}\right|=\left|\\{y\in
K^{[R]}\leavevmode\nobreak\ |\leavevmode\nobreak\ \overline{z}\in
y\\}\right|=\binom{R+1}{R}=R+1.$

As a consequence of Lemma 2.8, we also have that $\left|x\right|=R$ for all
$x\in\triangle\left(z\right),$ since each vertex $v\in x\subset z$ corresponds
to a configuration $\overline{z}\in\overline{x}\subset\overline{v}\in
K^{[R]},$ which is bijectively mapped to a set
$S\subset\mathcal{E}_{\overline{z}}$ such that
$\left|S\right|=\operatorname{rk}_{K}(\overline{v})=R$ and $\overline{x}\in S$
and there are $\binom{R}{R-1}=R$ such sets. Therefore we have

$\triangle\left(z\right)=\\{z\setminus\\{v\\}\leavevmode\nobreak\
|\leavevmode\nobreak\ v\in z\\}.$

Using that $J\cup\\{\emptyset\\}$ is closed under intersection, this implies
that $J\cap z=\mathcal{P}\left(z\right)$ and thus $J$ is simplicial.

Conversely, if $J=\overline{K}$ is a connected closed simplicial $R$-cc and
$v\in K^{[0]},$ let $S\subset\mathcal{E}_{v},\leavevmode\nobreak\
\left|S\right|=r.$ If we set $z:=\overline{v}$ we obtain the following
inclusions

$\\{\overline{e}\leavevmode\nobreak\ |\leavevmode\nobreak\ e\in
S\\}:=\\{y_{1},\dots,y_{r}\\}\subset\triangle\left(z\right)\subset J^{[R-1]}.$

Since $J$ is simplicial, for all $1\leq i\leq r$ we have
$y_{i}=z\setminus\\{x_{i}\\}$ for some (distinct) $x_{i}\in z$ and therefore

$y_{1}\cap\dots\cap y_{r}=z\setminus\\{x_{1},\dots,x_{r}\\}\in J^{[R-r]},$

which implies that $\overline{y_{1}\cap\dots\cap y_{r}}$ is the $r$-cell of
$K$ containing all edges in $S.$ ∎

###  2.3 Digression on polytopes and shellability

Now that we introduced the notion of simple cell complex, we would like to
discuss some links between the cell complexes considered in this thesis and
some common notions of discrete geometry that could be relevant when stepping
into physical considerations: polytopes and the related notion of shellability
for cell complexes. Polytopes or polyhedral complexes appear as a natural
choice to generalize simplices or simplicial complexes. For example, in [25]
three dimensional polyhedra are used as natural generalizations of a
tetrahedron to define a model of three-dimensional quantum geometry related to
the spin foam or Loop Quantum Gravity approach. The notion of shellability is
an assumption one can make on some discretizations to ensure that it satisfies
the Euler-Poincaré formula. In spirit, shellable objects are iteratively
defined as being constructed using pieces of a given dimension $d$ glued
together in a given order, satisfying that the next piece is intersecting the
previous ones on a shellable object of dimension $d-1.$

Convex polytopes in $\mathbb{R}^{n}$ (or equivalently polyhedra), can be
defined in many equivalent ways (see e.g. Theorem 2.15 [77]). We sometimes
designate such convex polytopes in $\mathbb{R}^{n}$ as embedded polytopes, as
opposed to the notion of abstract polytopes we introduce below. The notion of
face of an embedded polytope (defined on p.51 of [77]) is more general than
the notion of face used in this work. Polytopes can be arranged into a
polyhedral complex: a finite collection $\mathcal{C}$ of polyhedra in
$\mathbb{R}^{n}$ containing every face of each of its polyhedron and such that
the intersection of two polyhedra $P,Q\in\mathcal{C}$ is a face of both $P$
and $Q,$ with the convention that the empty polyhedron is a face of any
polyhedron.

Within their combinatorial theory, convex polytopes are often reduced to their
face lattice, i.e. the poset of their faces ordered by inclusion. The boundary
of a polytope forms a polyhedral complex and the face lattice of a polytope is
characterized by the face lattice of its boundary. The face lattices obtained
in this way constitute examples of cell complexes as defined here. The term
”simple” is used in this context to denote polytopes such that (the face
lattice of) their boundary is dual to a simplicial complex. Therefore our
notion of simplicity for cell complexes is equivalent to the corresponding
notion for polytopes when restricted to cc that can be realized as such.
Typical examples of cc that cannot be realized as a face lattice of a
polyhedral complex are given using cc having cells with non-trivial topology.
We will come back to this point in the discussion given after the Proposition
2.20 on the Euler-Poincaré formula.

In [58], McMullen and Schulte introduced the notion of abstract polytopes, a
notion that is designed to possess the main characteristics of the face
lattice of polytopes, but without any a priori notion of embedding in
Euclidean space nor convexity and thus share some properties with our
definition of cell complexes. Their definition of polytopes is however more
general than the definition we give next, as it does not require faces to be
characterized by their vertices nor to be stable under intersection (including
$\emptyset$ as a face).

In order to introduce the definition of polytopes, we need the following
notion of section.

######  Definition 2.12.

A section in a cc $K$ is a sub-poset of $K$ of the form

$\interval{x}{y}:=\\{w\in K\leavevmode\nobreak\ |\leavevmode\nobreak\ x\subset
w\subset y\\},$

where $x\in K\cup\\{\emptyset\\}$ and $y\in K$ such that $x\subset y.$ A
section $\interval{x}{y}$ is connected if
$\operatorname{rk}(y)-\operatorname{rk}(x)\leq 2,$ where we set
$\operatorname{rk}(\emptyset)=-1,$ or, for all $x\subsetneq
w,w^{\prime}\subsetneq y,$ there exists a sequence
$w_{0}=w,w_{1},\dots,w_{k}=w^{\prime}$ such that $x\subsetneq w_{i}\subsetneq
y$ and $w_{i}\subset w_{i+1}$ or $w_{i+1}\subset w_{i}$ for all
$i=0,\dots,k-1.$

######  Definition 2.13 (Polytope).

An pure $R$-cc $K$ is a polytope of rank $R$ if $\left|K^{[R]}\right|=1$ and
every section in $K$ is connected.

For example, if $K$ is a polytope of rank $R$ then in particular $K$ has no
$\partial K$-pinch $x$ such that $\operatorname{rk}(x)\geq R-2,$ as it would
imply that the section $\interval{x}{z}$ is not connected, where $z$ is the
unique maximal cell of $K.$ In [58], the property that each section is
connected is called strong-connectedness. We already use this terminology here
to denote the characteristic property of a pseudo-manifold.

The following result gives sufficient conditions for the cells of a local cell
complex to be seen as abstract polytopes. It is the only result in this work
that do not hold for cell complexes of arbitrary rank.

######  Proposition 2.14.

Let $K$ be a local simple cell complex of rank $R\leq 4.$ Then $K\cap x,$ and
thus $\overline{K}\cap\overline{x},$ are polytopes for all $x\in K.$

###### Proof.

It is sufficient to prove the case $R=4.$ Let $\interval{a}{b}$ be a section
in $K\cap x.$ If $\operatorname{rk}(b)-\operatorname{rk}(a)\leq 2$ then
$\interval{a}{b}$ is connected by definition and if $a=\emptyset,b\in K$ then
$\interval{a}{b}$ is connected since $K$ is cell-connected.

In the case $a\in K^{[0]},b\in K^{[3]}\cup K^{[4]},$ $\interval{a}{b}$ is
connected since $K$ is full and therefore for all
$e_{1},e_{2}\in\mathcal{E}_{v}$ there is a 2-cell $C\in K^{[2]}$ containing
$v$ as well as $e_{1}$ and $e_{2}.$

Finally, for the case $a\in K^{[1]},b\in K^{[4]},$ the section
$\interval{a}{b}$ is connected if for all $C_{1},C_{2}\in K^{[2]}$ such that
$a\subset C_{i}\subset b,i=1,2,$ there is $B\in K^{[3]}$ such that
$C_{i}\subset B\subset b,\leavevmode\nobreak\ i=1,2.$ Finding such $B$ amounts
to find an edge $e\in\overline{K}^{[1]}$ such that

$\overline{K}^{[0]}\ni\overline{b}\in
e\subset\overline{a}\in\overline{K}^{[3]}$

and such that $e\in\overline{C_{1}}\cap\overline{C_{2}}.$ Such an edge exists
as $\overline{K}$ is a simplicial complex by Proposition 2.11 and therefore
$\overline{a}$ is a tetrahedron and $\overline{C_{i}}$ is a triangle of
$\overline{a}$ for $i=1,2.$ ∎

We now turn to the notion of shelling of a cc and the proof the Euler-Poincaré
formula for shellable complexes in Proposition 2.20. Both the proof of the
latter result and the following definition of shelling are closely inspired by
the content of Sections 8.1 & 8.2 of [77], containing also many examples of
shellable polyhedral complexes that also constitute examples for our
framework.

######  Definition 2.15 (Shelling and shellable cell complex).

A shelling of a non-singular $R$-cell complex $K$ is defined in the following
recursive manner.

  1. 1.

For $R=0,$ $K$ is a point and has a shelling which is the trivial linear order
on the 1-element set $K^{[0]}=K.$

  2. 2.

For $R\geq 1,$ a shelling of $K$ is a linear ordering of its facets
$K^{[R]}=\\{z_{1},\dots,z_{N}\\}$ such that $\partial(K\cap z_{1})$ has a
shelling and for all $1\leq k\leq N,$ the cc

$K\cap\left(z_{k}\cap\left(\bigcup_{j=1}^{k-1}z_{j}\right)\right)$ (4)

is a non-singular sub-$(R-1)$-cell complex of $K$ which has a shelling that
can be completed into a shelling of $\partial(K\cap z_{k})$ and such that (4)
has non-empty boundary for $1<k<N$ and empty boundary for $k=N$ (i.e. (4) is
equal to $\partial(K\cap z_{N})$) only if $k=N$ and $K$ is closed.

A cc is shellable if it is non-singular and has a shelling.

We make the following remark about what are 1-dimensional shellings according
to our definition, which will be used in the proof of the Euler-Poincaré
formula 2.20.

######  Remark 2.16.

The previous definition differs slightly from the one given in [77, p.233] as
in our setup a shellable 1-cell complex is either a tree, if it has a non-
empty boundary, or a cycle, if it is closed, as opposed to being any connected
graph as allowed by Ziegler’s definition. This allows to dervie the Euler-
Poincaré formula also for shellable cell complexes that are not the boundary
complex of a polytope. Also, we require the cc (4) to have a non-empty
boundary for all $1<k<N,$ as it is not clear a priori that it is the case in
general. This allows to prove that the reverse order of a shelling is also a
shelling.

The barycentric subdivision of a shellable cell complex is shellable, as shown
in the next lemma. This is a relatively basic result that we will use in
Corollary 2.21. It is also for example shown in [1] that the second derived
subdivision of any convex polytope is shellable and that a triangulated
manifold is a PL-sphere or PL-ball if and only if for some $m\geq 0$ its
$m$-th derived subdivision is shellable. There exists however an example of a
non-shellable triangulation of a PL-ball ([53]) or even simple examples of
convex polytopes such as a tetrahedra ([70]) admitting non-shellable
triangulations.

######  Lemma 2.17.

The barycentric subdivision of a shellable cell complex is shellable.

###### Proof.

Let $K$ be a shellable cell complex. We prove the statement by induction on
the rank $R=\operatorname{Rk}(K).$ We first argue that it is sufficient to
prove that the statement holds for each sub-cell complex $K\cap z$ where $z\in
K^{[R]}.$ This follows from the observation that if we have a shelling order
of $K\cap z$ for each maximal cell $z$ then we can obtain a shelling of
$\widecheck{K}$ by combining the shellings of each sub-cell complex $K\cap z$
according to the order on $K^{[R]}$ defined by the shelling of $K.$ For this
we use that $\widecheck{K}$ is a simplicial complex, therefore the cell
complex defined at each step using (4), with $\widecheck{K}$ instead of $K$,
is simply the restriction of $\widecheck{K}$ to a $(R-1)$-simplex. By the
previous observation, the proof for the cases $R=0,1$ is clear. This same
observation also implies the induction step relatively directly, by the
following argument. By Definition 2.15 a maximal cell $z\in K^{[R]}$ satisfies
that $\partial(K\cap z)=K\cap\triangle\left(z\right)$ is shellable and by
induction we also have that its barycentric subdivision
$\widecheck{\partial(K\cap z)}$ is shellable. A shelling of the latter cell
complex directly provides a shelling of $K\cap z,$ as the map between
$\left(\widecheck{\partial(K\cap z)}\right)^{[R-1]}$ and $\widecheck{(K\cap
z)}^{[R]}$ defined by

$\\{x_{0}\subsetneq\dots\subsetneq
x_{R-1}\\}\mapsto\\{x_{0}\subsetneq\dots\subsetneq x_{R-1}\subsetneq z\\}$

is clearly a poset isomorphism. Hence the image of a shelling of
$\widecheck{\partial(K\cap z)}$ through this map is a shelling of
$\widecheck{(K\cap z)}.$ Therefore we obtain that $\widecheck{K\cap z}$ is
shellable for all $z\in K^{[R]}$ and we can conclude that $\widecheck{K}$ is
also shellable. ∎

For our purposes, the Euler characteristic is defined as follows.

######  Definition 2.18 (Euler characteristic).

The Euler characteristic of a cell complex $K$ of rank $R$ is defined by

$\chi(K)=\sum_{r=0}^{R}(-1)^{r}\left|K^{[r]}\right|.$

As a direct consequence of the latter definition, we can make the following
observation, used later in Corollary 2.26.

######  Remark 2.19.

Since $\left|\overline{K}^{[R-r]}\right|=\left|K^{[r]}\right|$ we have
$\chi(K)=(-1)^{R}\chi(\overline{K}),$ for $K$ closed.

We are now able to state the Euler-Poincaré formula for cc, which proof is
essentially identical to the proof of Corollary 8.17 of [77].

######  Proposition 2.20 (Euler-Poincaré formula).

Let $K$ be a shellable $R$-cell complex. Then if $\partial K\neq\emptyset,$ we
have

$\chi(K)=1,$

otherwise, i.e. if $K$ is closed, we have

$\chi(K)=1+(-1)^{R}.$

###### Proof.

We prove the claim by induction on $R\geq 0$ and $N=\left|K^{[R]}\right|,$
using that for any sub-cell complex $J,J^{\prime}$ of $K,$ $\chi$ satisfies

$\chi(J)+\chi(J^{\prime})=\chi(J\cup J^{\prime})+\chi(J\cap J^{\prime}).$ (5)

The claim for $R\leq 1$ and all $N\geq 0$ is clear by Remark 2.16. For $R\geq
2,$ let $K^{[R]}=\\{z_{1},\dots,z_{N}\\}$ be a shelling of $K.$ For $1\leq
k\leq N,$ let $J_{k}:=K\cap\left(\bigcup_{j=1}^{k}z_{j}\right).$ By (5), we
get

$\chi\left(J_{k}\right)=\chi(K\cap
z_{k})+\chi\left(J_{k-1}\right)-\chi\left((K\cap z_{k})\cap J_{k-1}\right).$

By the induction hypothesis, we have that

$\chi\left(J_{k-1}\right)=1,$

as well as

$\chi(K\cap z_{k}))=\chi(\\{z_{k}\\})+\chi(\partial(K\cap
z_{k}))=(-1)^{R}+1+(-1)^{R-1}=1,$

since $\partial(K\cap z_{k})$ is a closed $(R-1)$-cell complex and the
shellability of $K$ implies in particular that the boundary of each of its
facet is shellable.

For $1\leq k<N,$ the shelling property and the induction hypothesis also
implies

$\chi\left((K\cap z_{k})\cap J_{k-1}\right)=1.$

For $k=N,$ we have the following two cases:

$\displaystyle\chi\left((K\cap z_{N})\cap
J_{N-1}\right)=\begin{cases}\chi\left(\partial(K\cap
z_{N})\right)=1+(-1)^{R-1},&\text{if }K\text{ is closed,}\\\ 1,&\text{
otherwise.}\end{cases}$

This concludes the proof. ∎

We can for example consider the boundary of a $R$-simplex, which clearly
defines a closed shellable simplicial complex $K$ of rank $R-1$ and
Proposition 2.20 therefore implies that $\chi(K)=1+(-1)^{(R-1)}.$ As a
consequence, the Euler characteristic of a simplex (seen as a polytope) is
$1.$

As noted in the next corollary, the previous results imply that the Euler
characteristic is invariant under taking the barycentric subdivision, provided
that the complex we started with is shellable.

######  Corollary 2.21.

If $K$ is a shellable cell complex then $\chi(\widecheck{K})=\chi(K).$

###### Proof.

This is a direct consequence of Lemma 2.17 and Proposition 2.20. ∎

In our context, cc are not shellable in general and their Euler characteristic
can therefore take various values. For example, a 3-cc $T$ could be defined so
that $\partial T$ is a discretization of a torus, seen as a surface of genus
1, and $\left|T^{[3]}\right|=1,$ hence $\chi(T)=-1$ (such an cc is given in
Figure 17 as an example of cobordism in Chapter 3). It is a standard result
from Bruggesser and Mani (Theorem 8.11 in [77]) that convex polytopes are
shellable. Therefore $T$ also provides a case of cc that cannot be realized as
the face lattice of an embedded convex polytope, although it would satisfy the
conditions of (abstract) polytope defined in Definition 2.13 (the homogeneity
of $\partial T$ in particular implies that $\partial T$ is local and non-
pinching, which implies the connectedness of all sections in $T$). Similarly,
a $R$-cc $K$ having a facet $z\in K^{[R]}$ such that $K\cap z=T$ cannot be
realized as the face lattice of a polyhedral complex. In such an example, the
$3$-cell $z$ would be a center of an non-homogeneous vertex in $\widecheck{K}$

In the remainder of this section we introduce a definition of 2-shellability
that applies to cc of rank higher or equal to 2 and prove that the
2-shellability of a simple cc is equivalent to the shellability of its dual.
2-shellability is characterized only using the cells of the 2-skeleton and is
inspired by the notion of ”weak-shellability” given by Vince in [74].

######  Definition 2.22 (2-shellable cell complex).

A cell complex $K$ is 2-shellable if there is a linear ordering on
$K^{[0]}:=\\{v_{1},\dots,v_{N}\\}$ such that for all $C\in K^{[2]}$ the graph

$K_{j}\cap(C\setminus\\{v_{j+1}\\})$

is connected, where $K_{j}:=K\cap\\{v_{1},\dots,v_{j}\\}$ is a cell complex.
Such linear ordering is called a 2-shelling of $K.$

######  Remark 2.23.

We notice that Definition 2.22 implies that the 2-skeleton of a 2-shellable
cell complex can be constructed by successively adding each 2-cell in such a
way that at each step of the construction, the new 2-cell intersects the
previous ones on a simple path. Indeed, using the notation of Definition 2.22,
a 2-shelling satisfies the following property. At step $j+1,$ if $C\in
K^{[2]}$ contains $v_{j+1}$ and $C\cap\\{v_{1},\dots,v_{j}\\}\neq\emptyset$
then there is no gap between $v_{j+1}$ and the previous vertices in $C,$ in
the sense that there exists $1\leq k\leq j$ such that
$C\supset\\{v_{j+1},v_{k}\\}\in K^{[1]}.$ The reason for this is that if there
were a gap and if we take the first $l>j+1$ such that

$v_{l}\in C\setminus\\{v_{1},\dots,v_{j+1}\\}$

we have that $K_{l}\cap(C\setminus\\{v_{l}\\})$ is disconnected, contradicting
that $\\{v_{1},\dots,v_{N}\\}$ is a 2-shelling. We can therefore suppose,
without loss of generality, that $l=j+2,$ i.e. the shelling order continues by
adding all the vertices in $C$ first, before completing another 2-cell.

This following technical result will be used in the proof of the next Theorem
2.25.

######  Lemma 2.24.

Let $K$ be a simple cc such that $\overline{K}$ is shellable. Let
$\\{\sigma_{1},\dots,\sigma_{N}\\}$ be a shelling of $\overline{K}$ and define
$v_{i}:=\overline{\sigma_{i}},$ for $i=1,\dots N.$ We also set
$K_{j}:=K\cap\\{v_{1},\dots,v_{j}\\}$ and define the following simplicial
complex:

$\overline{K}_{j}:=\overline{K}\cap\left(\bigcup_{i=1}^{j}\sigma_{i}\right).$

Then we have that $K_{j}\cap\overline{\omega}$ is connected for all
$\omega\in\overline{K}_{j}$ and all $1\leq j\leq N.$

###### Proof.

Suppose by contradiction that there exists $1<j\leq N$ such that
$K_{j}\cap\overline{\omega}$ is not connected for some
$\omega\in\overline{K}_{j},$ which implies
$\operatorname{rk}(\overline{\omega})\geq 2.$ Since $\overline{\omega}\in
K=K_{N}$ is connected, we can take $j$ maximal with the property that
$K_{j}\cap\overline{\omega}$ is not connected, so that
$K_{j+1}\cap\overline{\omega}$ is connected. Hence there are at least two
vertices $v_{k},v_{l}$ in $\overline{\omega}\cap\mathcal{N}(v_{j+1})$ that
belong to two different connected components of $K_{j}\cap\overline{\omega}.$
Since $K$ is full, there is a 2-cell $C\in K^{[2]}$ containing the edges
$\\{v_{j+1},v_{k}\\}$ and $\\{v_{j+1},v_{l}\\}$ and thus
$K_{j}\cap(C\setminus\\{v_{j+1}\\})$ is not connected, which contradicts that
$\\{v_{1},\dots,v_{N}\\}$ is a 2-shelling of $K.$ ∎

The following theorem is analogous to Theorem 7 given in [74] stated in terms
of $n$-graphs. Our proof is novel as the definition of $n$-graph allows to use
some equivalence relations between edges that do not exist in general for
simple cell complexes. This result given in terms of simple cc is nevertheless
not a generalization of Vince’s result since $n$-graphs are more general than
our notion of graphs, as in particular they allow multiple edges to contain
the same two vertices.

######  Theorem 2.25.

A simple cell complex $K$ of rank $R\geq 2$ is 2-shellable if and only if
$\overline{K}$ is shellable.

###### Proof.

Let $K$ be a simple cell complex such that $\overline{K}$ is shellable with
shelling $\\{\sigma_{1},\dots,\sigma_{N}\\}$ and define $v_{i},$ $K_{j}$ and
$\overline{K}_{j}$ as in Lemma 2.24.

We show that $\\{v_{1},\dots,v_{N}\\}$ is a 2-shelling of $K.$ For this, let
us assume by contradiction that there is $C\in K^{[2]}$ such that
$K_{j}\cap(C\setminus\\{v_{j+1}\\})$ is not connected. We consider $j$ to be
maximal with this property for $C$ fixed ($j$ is well defined since $K\cap C$
is connected). The maximality of $j$ implies that $K_{j+1}\cap C$ is
connected, therefore $v_{j+1}$ has two neighbours in $K_{j}\cap C,$ linked to
$v_{j+1}$ by the edges $e,e^{\prime}.$ Since $\sigma_{j+1}$ is a simplex, the
two faces $\overline{e},\overline{e^{\prime}}$ of $\sigma_{j+1}$ share the
$(R-2)$-cell $\overline{C}.$ But $K_{j+1}\cap C\neq C$ by definition of $j,$
therefore $\overline{C}\notin\overline{K}_{j+1},$ contradicting the
intersecting property ii) of the $(R-1)$-cell complex

$\overline{K}_{j+1}\cap\left(\sigma_{j+1}\cap\left(\bigcup_{i=1}^{j}\sigma_{i}\right)\right).$

Conversely, assume that $\\{v_{1},\dots,v_{N}\\}$ is a 2-shelling of $K.$ We
show that $\\{\sigma_{1},\dots,\sigma_{N}\\}$ is a shelling of $\overline{K}$
where $\sigma_{i}=\overline{v_{i}}.$

Since any subset of the faces $\triangle\left(\sigma\right)$ of an $R$-simplex
$\sigma\in\overline{K}$ is a $(R-1)$-sub-cell complex that can be completed
into a shelling of $\partial(\overline{K}\cap\sigma),$ it is sufficient to
show that for all $1<j\leq N,$ if
$\omega:=\sigma_{k}\cap\sigma_{j}\neq\emptyset$ for some $1\leq k<j$ then
there exists $1\leq l<j$ such that

$w\subset\sigma_{l}\cap\sigma_{j}\in\triangle\left(\sigma_{j}\right).$

The latter implies that $\overline{K}_{j-1}\cap\sigma_{j}$ is indeed a
$(R-1)$-cc that has a shelling which can be completed in to a shelling of
$\partial(\overline{K}\cap\sigma_{J}).$ By Lemma 2.24, such $\omega$ satisfies
that $K_{j}\cap\overline{\omega}$ is connected, therefore there is $1\leq k<j$
such that $\sigma_{k}\in\overline{\omega}$ and $\\{\sigma_{k},\sigma_{j}\\}\in
K_{j}^{[1]},$ i.e.
$\operatorname{rk}_{\overline{K}}(\sigma_{k}\cap\sigma_{j})=R-1,$ and this
completes the proof. ∎

######  Corollary 2.26.

If $K$ is a local simple 2-shellable $R$-cell complex then $K$ satisfies
Euler-Poincaré formula

$\chi(K)=1+(-1)^{R}.$

###### Proof.

By Proposition 2.11 and Theorem 2.25, $\overline{K}$ is a shellable simplicial
complex, which satisfies Euler-Poincaré formula by Proposition 2.20. As
noticed in Remark 2.19 and since $K$ is in particular closed, we have

$\chi(K)=(-1)^{R}\chi(\overline{K})=(-1)^{R}+1.$

∎

###  2.4 Simply connected cc

In this section, we introduce a combinatorial notion of homotopy in cell
complexes of rank higher or equal to 2 defined using $2$-cells and edges.
Although the idea is simply to allow paths to be ”deformed through” $2$-cells,
the exact formulation of these notion is somewhat technical. For example, in
order to define such deformation on every paths, we have to deal with the case
were a path goes several times through a given edge. This can create
ambiguities on which part of the path is to be deformed through a given
$2$-cell. Two avoid this, we added a condition in the definition of such ”path
move” so that it acts on the part of the path which first goes through the
$2$-cell that the move is based on. Similar questions arise when defining the
move of paths through an edge $e,$ which acts non-trivially on paths that
contains a sub-path of the form $(e,e)$ (using the notations from Section
1.3), i.e. when a path goes through the edge $e$ in both directions in a row.

Let $K$ be a graph-based 2-cc and let $v,w\in K^{[0]}.$ We denote by
$\mathcal{P}$ the set of paths in $K^{(1)}$ and $\mathcal{P}(v,w)$ the set of
paths from $v$ to $w$ in $K^{(1)}.$ For a 2-cell
$C\in\overline{v}\cap\overline{w},$ a simple path $p\in\mathcal{P}(v,w)$ in
$K^{(1)}\cap C$ and $C_{p}$ the connected component of $C$ containing $p,$ we
define the complementary path ${p}^{C}$ of $p$ in $C$ to be the sub-path of
$l_{C_{p}}(v)$ from $v$ to $w$ such that

$l_{C}(v)=p\ast({p}^{C})^{-1},$

as in Figure 9, where $l_{C_{p}}(v)$ was introduced in Definition 1.13.

Figure 9: On this illustration we have a simple path $p$ of length 2 from the
vertex $v$ to the vertex $w$ contained in a 2-cell $C.$ The path $p$ can be
seen as as sub-path of a longer path going through other edges before $v$ and
after $w$ (including the edges in $C$). Since we do not a priori assume cells
to be connected, a $2$-cell can have more than one connected component,
$C_{p}$ is therefore the connected component of $C$ containing $p.$ By Lemma
1.12 we have that the edges in $C_{p}$ form a loop $l_{C}(v)$ that we can
define as based at the vertex $v$ and such that $p\leq l_{C}(v).$ The
complementary path $p^{C}$ of $p$ through $C_{p}$ is then defined as the sub-
path of $l_{C}(v)$ going from $v$ to $w$ and disjoint from $p.$

######  Definition 2.27 (Moves of a path).

Let $C\in K^{[2]}$ and let $p$ be a simple path in $K^{(1)}\cap C.$ The 2-cell
move of $p$ through $C$ is a function
$m_{C}^{p}:\mathcal{P}\longrightarrow\mathcal{P}$ defined by

$\displaystyle m_{C}^{p}(q)=\begin{cases}q_{1}\ast{p}^{C}\ast q_{2},&\text{if
}q=q_{1}\ast p\ast q_{2}\text{ with }q_{1},q_{2}\in\mathcal{P},p\not\leq
q_{1};\\\ q&\text{otherwise.}\end{cases}$

We can see that $m_{C}^{p}$ is a bijection with inverse $m_{C}^{{p}^{C}}.$

Let $e\in K^{[1]}$ and $v\in e.$ The edge-move through $e$ based at $v$ is a
function $m_{e,v}:\mathcal{P}\longrightarrow\mathcal{P}$ defined by

$\displaystyle m_{e,v}(q)=\begin{cases}q_{1}\ast q_{2},&\text{if
}q=q_{1}\ast(e,e)\ast q_{2}\text{ with }q_{i}\in\mathcal{P},e\cap
q_{i}^{[0]}={v},i=1,2,(e,e)\not\leq q_{1};\\\ q&\text{otherwise.}\end{cases}$

In this case, the inverse of $m_{e,v}(q)$ is the maps $m^{-1}_{e,v}$ which
adds $e$ to a path $q$ if $e\cap q^{[0]}=\\{v\\}$ and leaves $q$ invariant
otherwise. We denote by $\mathcal{M}_{1}$ the set of edge-moves and their
inverses and by $\mathcal{M}_{2}$ the set of 2-cell-moves and their inverses
and define the set of moves to be
$\mathcal{M}=\mathcal{M}_{1}\cup\mathcal{M}_{2}.$ Note that if $q$ is a path
from $v$ to $w$ then so is $m(q)$ for all $m\in\mathcal{M}.$ Therefore moves
of paths can be seen as functions on $\mathcal{P}(v,w).$

######  Definition 2.28 (Homotopy, homotopy class of paths and contractible
cycle).

An homotopy is a map $H:\mathcal{P}(v,w)\longrightarrow\mathcal{P}(v,w)$ given
by a finite composition of moves in $\mathcal{M}.$ Two paths
$p_{1},p_{2}\in\mathcal{P}(v,w)$ are (homotopically) equivalent, noted
$p_{1}\sim p_{2},$ if there exists an homotopy $H$ such that $H(p_{1})=p_{2}.$
The homotopy class of a path $p$ is denoted by $[p].$ A cycle $p$ based at $v$
is contractible if $[p]=[\emptyset_{v}].$

######  Definition 2.29 (Simply connected cc).

A graph-based cc $K$ is simply connected if every cycle in $K$ is contractible
in $K^{(2)}.$

The next remarks on simple connectedness will be used in the discussion at the
end of this chapter.

######  Remark 2.30.

It seems clear that the notion of homotopy used here is equivalent to the
usual notion of homotopy of paths on a manifold, in the sense that if $K$ is a
cell complex which geometrical realization is a discretization of a cellular
manifold (implying that $K\cap x$ is geometrically realized as a ball for all
$x\in K$) then $K^{(2)}$ is simply connected (in the sense of Definition 2.28)
if and only if $M$ is simply connected in the usual topological sense. A
sketch of an argument proving this can be formulated as follows, using that in
this case $\overline{K}$ also admits a geometrical realization discretizing
$M$ (this can be argued using the fact that
$\widecheck{K}\cong\widecheck{\overline{K}}$ by the Proposition 1.24). A path
$p$ in $M$ can be projected onto $K^{(1)}$ by decomposing $p$ into sub-paths
such that there is one sub-path for each crossing of $p$ from one maximal cell
of $\overline{K}$ to another. One can then continuously deform each sub-path
as to be included into the edge of (the geometric realization of) $K$ composed
of the two corresponding maximal cells of $\overline{K}$ and the combination
of these deformations defines a usual homotopoy of paths on $M.$ The resulting
path can be interpreted as a path on the graph $K^{(1)}$ and an homotopy as
defined in these notes also corresponds to a usual homotopy on $M.$

######  Remark 2.31.

It is a well-known result ([27]) that shellable pseudomanifolds are
homeomorphic to a sphere (using the notion of geometrical realization
mentioned in Section 1.6). As a consequence of Theorem 2.25, a 2-shellable
simple local cell complex $K$ satisfies that $\overline{K}$ is simply
connected.

###  2.5 Monodromy-free and even cc

We now turn to the notions of monodromy-freedom and evenness in a cc. The
latter corresponds simply to the property of having an even number of vertices
in each (connected component of each) $2$-cell. However we do not yet fully
understand how evenness can be interpreted in the context of the
reconstruction results of Section 2.6, although we suspect that it has
something to do with some notion of orientation. Monodromy-freedom essentially
means that the connection does not produce a monodromy around any $2$-cell and
it is the notion we focus on first.

In the case of full graph-based cc, the notation for the connection introduced
in Section 2.1 can be extended to paths in the following way.

######  Definition 2.32 ($\nabla_{p}$).

For a full graph-based 2-cc $K,$ we have that, for all $\\{v,w\\}\in K^{[1]},$

$\mathcal{E}_{v}^{\overline{w}}=\\{e\in\mathcal{E}_{v}\leavevmode\nobreak\
|\leavevmode\nobreak\
\overline{e}\cap\overline{\\{v,w\\}}\neq\emptyset\\}=\mathcal{E}_{v}.$

Therefore if $p=(e_{1},\dots,e_{k})$ is a path in $K^{(1)},$ the map

$\nabla_{p}:=\nabla_{v_{k}}^{v_{k-1}}\dots\nabla_{v_{1}}^{v_{0}}$

is a bijection from $\mathcal{E}_{v_{0}}$ to $\mathcal{E}_{v_{k}},$ where
$e_{i}=\\{v_{i-1},v_{i}\\}$ for $1\leq i\leq k.$ Also,
$\nabla_{p}^{-1}=\nabla_{p^{-1}},$ $\nabla_{p\ast
p^{\prime}}=\nabla_{p^{\prime}}\nabla_{p}$ and
$\nabla_{\emptyset_{v}}=\operatorname{id}_{\mathcal{E}_{v}}.$ We say that
$\nabla_{p}e\in\mathcal{E}_{v_{k}}$ is associated to $e\in\mathcal{E}_{v_{0}}$
by $p.$

If $K$ is a full graph-based cc and $p$ is a path in $K^{(1)}$ then the
connection $\nabla_{p}$ in $K$ is defined as the corresponding connection in
$K^{(2)}.$

Let $C$ be a connected component of a 2-cell of a full graph-based cc $K.$
Recall that

$\mathcal{E}_{C}:=\\{e\in K^{[1]}\leavevmode\nobreak\ |\leavevmode\nobreak\
\left|e\cap C\right|=1\\}.$

For $\\{v,w\\}\subset C,$ $\nabla_{w}^{v}$ maps
$\mathcal{E}_{v}\cap\mathcal{E}_{C}$ bijectively onto
$\mathcal{E}_{w}\cap\mathcal{E}_{C},$ since if $e\not\subset C$ then
$\nabla_{w}^{v}e\not\subset C$ by Lemma 2.1 2). Therefore $\nabla_{l_{C}(v)}$
restricted to $\mathcal{E}_{v}\cap\mathcal{E}_{C}$ is a bijection from
$\mathcal{E}_{v}\cap\mathcal{E}_{C}$ onto itself.

We can now introduce our definition of monodromy-free cell, illustrated in
Figure 10 and define the notion of monodromy-free cc.

######  Definition 2.33 (Monodromy-free cc).

A connected component $C$ of a 2-cell in a full graph-based cc $K$ is said to
be monodromy-free if $\nabla_{l_{C}(v)}$ restricted to
$\mathcal{E}_{v}\cap\mathcal{E}_{C}$ is the identity map on
$\mathcal{E}_{v}\cap\mathcal{E}_{C}$ for some $v\in C$ (and hence for all
$v\in C$). We say that a full graph-based cc $K$ is monodromy-free if every
connected component of every 2-cell in $K^{[2]}$ is monodromy-free.

Figure 10:  In this figure we see an example of a monodromy free (connected
component of a) 2-cell $C,$ seen as a cell in a 3 dimensional cc. We also
pictured the loop $l_{C}(v)$ based at the vertex $v$ and how the connexion
$\nabla_{l_{C}(v)}$ maps the edges of $\mathcal{E}_{C}$ around $C.$ In this
case, since there are only two edges in
$\mathcal{E}_{v^{\prime}}\cap\mathcal{E}_{C}$ for all $v^{\prime}\in C,$ it is
sufficient to check that say $e$ as on the figure is mapped to itself by
$\nabla_{l_{C}(v)}$ to deduce that $C$ is monodromy-free.

The following Lemma 2.34 shows that the monodromy-freedom condition naturally
arises when considering $R$-full cell complexes for $R\geq 3.$ This fact
motivates our definition of monodromy-free cell complexes, as it is a key
assumption when considering the reconstruction of simple cell complexes from
their 2-skeleton in Section 2.6. We recently found that a similar notion was
introduced in the last chapter of [38], where several conditions are discussed
in order to establish whether some space defined as ”simplicial glueings” in
dimensions 3 and 4 define a manifold. A simplicial glueing is simply a set of
simplices of a given dimension where faces are identified according to some
face-pairing. One of these conditions, called ”$\mathsf{Cycl}$” is stated in
dimension 4 and gives a condition on each triangle (i.e. 2-simplices) that is
precisely the condition of monodromy-freedom as defined here. It is suggested
that one can easily devise an extension to all dimensions and we believe that
the definition of monodromy-free cell complex indeed corresponds to the
extensions of the condition $\mathsf{Cycl}$ to dimensions higher or equal to
3.

######  Lemma 2.34.

A 3-full graph-based cell complex is monodromy-free.

###### Proof.

Let $[v_{1}\dots v_{N}]$ be a representation of the connected component of a
2-cell $C\in K^{[2]}$ and let $l:v_{1}\rightsquigarrow v_{1}$ be the loop
associated with this representation based at $v_{1}$ and starting with the
edge $\\{v_{1},v_{2}\\}.$ By the diamond property iv), if
$e\in\mathcal{E}_{v_{1}}\cap\mathcal{E}_{C},$ there are 2 edges
$e_{0},e_{1}\in\mathcal{E}_{v}$ included in $C$ and since $K$ is 3-full, there
is a 3-cell $B$ containing $e_{0},e_{1}$ and $e.$ Since $K$ is full, there is
a unique 2-cell $C_{e}$ containing $e$ and $e_{1}$ and by Axiom ii) since
$e,e_{1}\subset B\cap C_{e}$ we have $C_{e}\subset B.$ Therefore
$\nabla_{v_{2}}^{v_{1}}e\subset B.$ By the same reasoning, there is a unique
edge $e_{2}\in\mathcal{E}_{v_{2}}\setminus\\{e_{1}\\}$ included in $C$ and
there is a unique 2-cell $C_{e}^{\prime}$ containing $\nabla_{v_{2}}^{v_{1}}e$
and $e_{2}$ and contained in $B,$ which implies $\nabla_{v_{3}}^{v_{2}}e\in
B.$ By repeating this argument $N$ times we get $\nabla_{l}e\in B$ and this
implies that $\nabla_{l}e=e,$ by Lemma 2.8, since $K$ is 3-full. ∎

The notion of evenness of a $2$-cell and the corresponding notion for cc,
introduced in the following definition, will also be crucial for the
reconstruction in Section 2.6. The idea is that the assumption of evenness
leads to the invariance of the connection $\nabla_{p}$ under homotopies, as
shown in Lemma 2.38. One can better picture the effect of even $2$-cell versus
odd (non-even) $2$-cell on the connection using Figure 11.

######  Definition 2.35 (Even $2$-cell and even cc).

A connected component of a 2-cell is said to be even if it contains an even
number of vertices. A cc $K$ is said to be even if every connected component
of every 2-cell in $K^{[2]}$ is even.

######  Remark 2.36.

If $C$ is even and monodromy-free then for all $v\in C$ we have

$\nabla_{l_{C}(v)}=\operatorname{id}_{\mathcal{E}_{v}},$

since $\nabla_{l_{C}(v)}$ induces a pairing of the edges contained in $C.$

Figure 11: In these pictures, we see how the connection acts on the edges
within an even 2-cell (on the left) versus within an odd 2-cell (on the
right). The even case then leads to a pairing of the edges induced by the
connection.

An important example of even cc is given in the following lemma.

######  Lemma 2.37.

The dual of the barycentric subdivision of a closed cc is an even cc.

###### Proof.

Let $J$ be a closed $R$-cc and let $K=\overline{\widecheck{J}}.$ A 2-cell
$C\in K^{[2]}$ is then of the form $C=\overline{y}$ where

$y=\\{x_{1}\subsetneq\dots\subsetneq x_{R-1}\\}\in\widecheck{J}^{[R-2]}.$

We can then see that elements in $\overline{y}$ come in pairs by the following
observation. Since $\widecheck{J}$ is a simplicial complex, every
$z\in\overline{y}\subset\widecheck{J}^{[R]}$ satisfies $\left|z\setminus
y\right|=2.$ Since $z$ defines a totally ordered subset of $J,$ the minimum
rank of an element in $z\setminus y=:\\{x,x^{\prime}\\},$ say
$\operatorname{rk}(x),$ is strictly less than $R.$ This means that
$z=y\cup\\{x\subsetneq x^{\prime}\\}$ and by the diamond property iv) (and
since $J$ is graph-based for the case $\operatorname{rk}(x)=0$), there exists
a unique $x^{\prime\prime}$ such that $\\{y\cup\\{x\subsetneq
x^{\prime\prime}\\},z\\}\subset\overline{y}.$ ∎

The following lemma states under what assumptions the connection $\nabla_{p}$
is invariant under homotopies, which is a key property in order to prove the
Extension Lemma 2.40 in the next section.

######  Lemma 2.38.

Let $K$ be a full, graph-based, monodromy-free, even cc and let
$q\in\mathcal{P}$ and $m\in\mathcal{M}.$ Then we have

$\nabla_{q}=\nabla_{m(q)}.$

In particular if $p\sim q$ then $\nabla_{p}=\nabla_{q}.$

###### Proof.

If $m(q)=q$ it is trivial. If not then either $m=m_{C}^{p}$ for some 2-cell
$C$ and a simple path $p$ in $K^{(1)}\cap C$ and $q=q_{1}\ast p\ast q_{2},$ or
$m=m_{e,v}$ (or $m=m_{e,v}^{-1}$) for some edge $e$ and vertex $v$ such that
$v\in e\cap q^{[0]}$ and $q=q_{1}\ast(e,e)\ast q_{2}.$

In the first case $\nabla_{l_{C}(v)}=\operatorname{id}_{\mathcal{E}_{v}}$ for
all $v\in C,$ by Remark 2.36, therefore we can choose a representation of $C$
and some $v\in C$ such that $l_{C}(v)=({p}^{C})^{-1}\ast p.$ One then obtains
the following relations

$\displaystyle\nabla_{m(q)}$
$\displaystyle=\nabla_{q_{1}}\nabla_{{p}^{C}}\nabla_{q_{2}}$
$\displaystyle=\nabla_{q_{1}}\nabla_{l_{C}(v)}\nabla_{{p}^{C}}\nabla_{q_{2}}$
$\displaystyle=\nabla_{q_{1}}\nabla_{p}\nabla_{{p}^{C}}^{-1}\nabla_{{p}^{C}}\nabla_{q_{2}}=\nabla_{q}.$

In the second case, we have that
$\nabla_{(e,e)}=\nabla_{v}^{w}\nabla_{w}^{v}=\operatorname{id}_{\mathcal{E}_{v}}$
assuming $e=\\{v,w\\}$ and $(e,e)$ in $q=q_{1}\ast(e,e)\ast q_{2}$ is a path
from $v$ to $v,$ so that we have

$\displaystyle\nabla_{m(q)}$ $\displaystyle=\nabla_{q_{1}}\nabla_{q_{2}}$
$\displaystyle=\nabla_{q_{1}}\nabla_{(e,e)}\nabla_{q_{2}}=\nabla_{q}.$

The case $m=m_{e,v}^{-1}$ is similar to the second case. ∎

###  2.6 Extension Lemma and reconstruction

In this section, we prove that local even full monodromy-free simply connected
2-cell complexes can be associated a corresponding ”ambient” cell complex of
rank $R\geq 3$ having the original 2-cell complex as its 2-skeleton. The main
step for this construction, carried out in Theorem 2.42, is to define a
collection of ”induced” sub-2-cell complexes whose vertices then serve as
higher dimensional cells of the ambient cell complex. In particular, Corollary
2.46 shows that the ambient cell complex associated to the 2-skeleton of a
local even simple simply connected cell complex is equal to the original cell
complex. This result is in line with a number of results about the
reconstruction of the face lattice of (embedded) polytopes, see e.g. [22] for
a recent review on this topic. A well-known result from [28] and [51] states
that the face lattice of a simple polytope, i.e. a polytope which dual is
simplicial, is determined by its 1-skeleton. As we discussed in Section 2.3,
here we are dealing with local cell complexes that are more general than the
face lattice of complexes of polytopes but we have the additional assumptions
of simple connectedness and evenness. We are not aware of an equivalent result
in the literature for complex of polytopes.

The first key result of this section is the next ”Extension Lemma” 2.40 which
will be essential for the proof of Theorem 2.42. The assumptions of fullness,
evenness, monodromy-freedom and simple connectedness introduced in this
chapter can therefore be considered as chosen so that this lemma holds. The
proof of the Extension Lemma is efficiently forumulated using covariant edge
fields. One reason for this is that when defined on an appropriate domain,
covariant edge fields satisfy the property that their value over their entire
domain is determined by their value at one vertex. We formulate this fact more
precisely in the following remark on which the proof of the Extension Lemma
relies.

######  Remark 2.39.

If $\phi$ is a covariant edge field in a full graph-based cc $K$ and if $p$ is
a path in $K^{(1)}\cap\mathcal{D}_{\phi},$ from $v$ to $w$ then
$\nabla_{p}\phi(v)=\phi(w).$ In particular if $K^{(1)}\cap\mathcal{D}_{\phi}$
is connected then $\phi$ is determined by its value at any given vertex in
$\mathcal{D}_{\phi}.$

We now formulate the Extension Lemma and its proof in terms of growing domains
of covariant edge fields as it incorporates well the idea that will be used in
our formulation of the proof of Theorem 2.42. Considering that we assume the
entire cc $K$ to be simply connected, one can also read the result simply
using $\mathcal{D}_{\phi}=\\{v\\}$ for any $v\in K^{[0]}$ and $D=K^{[0]}.$

######  Lemma 2.40 (Extension Lemma).

Let $K$ be a graph-based, full, even, monodromy-free, simply connected cc and
let $\phi:\mathcal{D}_{\phi}\longrightarrow K^{[1]}$ be an covariant edge
field in $K$ such that $K^{(1)}\cap\mathcal{D}_{\phi}$ is connected. If
$D\subset K^{[0]}$ such that $K^{(1)}\cap(\mathcal{D}_{\phi}\cup D)$ is
connected then there exists a unique covariant edge field
$\hat{\phi}:\mathcal{D}_{\hat{\phi}}\longrightarrow K^{[1]}$ such that
$\mathcal{D}_{\hat{\phi}}=\mathcal{D}_{\phi}\cup D$ and
$\hat{\phi}|_{\mathcal{D}_{\phi}}=\phi.$

###### Proof.

Let $v\in D,$ $w\in\mathcal{D}_{\phi}$ and let $p$ be a path from $w$ to $v$
in $K^{(1)}\cap(\mathcal{D}_{\phi}\cup D),$ as in Figure 12. The main idea of
the proof is to define $\hat{\phi}(v)=\nabla_{p}\phi(w)$ and to show that it
is well defined. This amounts to show that if
$w^{\prime}\in\mathcal{D}_{\phi}$ and $p^{\prime}$ is a path from $w^{\prime}$
to $v$ then there is a path $q$ from $w$ to $w^{\prime}$ in
$\mathcal{D}_{\phi}$ and we have

$\displaystyle\nabla_{p}\phi(w)$ $\displaystyle=\nabla_{p^{-1}\ast q\ast
p^{\prime}}\nabla_{p}\phi(w)$
$\displaystyle=\nabla_{p^{\prime}}\nabla_{q}\phi(w)=\nabla_{p^{\prime}}\phi(w^{\prime}).$

This is indeed the case by the following argument. Since $K$ is simply
connected and even, Lemma 2.38 implies that every cycle $q^{\prime}$ in $K$
satisfies
$\nabla_{q^{\prime}}=\nabla_{\emptyset_{v}}=\operatorname{id}_{\mathcal{E}_{v}}$
for any $v\in(q^{\prime})^{[0]}.$ The first equality then follows from the
fact that $p^{-1}\ast p\ast p^{\prime}$ is a cycle. For the second equality,
we recall that $\nabla_{p_{1}\ast p_{2}}=\nabla_{p_{2}}\nabla_{p_{1}}$ and the
last equality is a consequence of $\nabla_{q}\phi(w)=\phi(w^{\prime})$ as
noticed in Remark 2.39. The latter remark also gives an argument for the
uniqueness of $\hat{\phi}.$ Finally $\hat{\phi}$ defined thereby is a
covariant edge field since if $\\{v,v^{\prime}\\}$ is an edge of $K^{(1)}\cap
D$ and $p$ is a path linking $v$ with $w\in\mathcal{D}_{\phi}$ then
$p^{\prime}=p\ast(\\{v,v^{\prime}\\})$ is a path linking $v^{\prime}$ with $w$
therefore

$\hat{\phi}(v^{\prime})=\nabla_{p^{\prime}}\phi(w)=\nabla_{v^{\prime}}^{v}\nabla_{p}\phi(w)=\nabla_{v^{\prime}}^{v}\hat{\phi}(v).$

∎

Figure 12: Configuration of Lemma 2.40.

######  Remark 2.41.

As a consequence of Lemma 2.40, a full, even, monodromy-free, simply connected
and connected cc $K$ is properly $(R+1)$-edge-colorable, meaning that one can
color each edge of $K$ with $R+1$ colors so that no two edges sharing a vertex
have the same color. Another way to say this is that $K^{(1)}$ is an
$(R+1)$-graph, where the notion of $n$-graph is a tool used in works on so-
called crystallizations of topological manifolds (see e.g. in [54], [74] or
[6]), introduced in the research related to minimal atlases of topological
manifold initiated by Pezzana and al. (see [42] for a review on this topic).
However, not all $n$-regular graphs are $n$-graph, i.e. properly $n$-edge-
colorable.

Now comes the main result of this chapter, in which we define the notion of
induced sub-cell complex.

######  Theorem 2.42.

Let $K$ be a local even full monodromy-free simply connected 2-cell complex.
Then for all $v\in K^{[0]}$ and all $S_{v}\subsetneq\mathcal{E}_{v}$ such that
$r:=\left|S_{v}\right|\geq 2$ there is a unique sub-cell complex $J(v,S_{v})$
of $K$ such that $J(v,S_{v})$ is a $r$-regular local even full monodromy-free
connected 2-cell complex and

$J(v,S_{v})^{[1]}\cap\mathcal{E}_{v}=S_{v}.$ (6)

Moreover, $J(v,S_{v})$ satisfies the following properties:

  1. 1)

$J(v,S_{v})=J(v^{\prime},S^{\prime}_{v^{\prime}})$ if and only if
$v^{\prime}\in J(v,S_{v})^{[0]}$ and
$S^{\prime}_{v^{\prime}}=J(v,S_{v})^{[1]}\cap\mathcal{E}_{v^{\prime}},$

  2. 2)

$S_{v}\subsetneq S^{\prime}_{v}$ if and only if $J(v,S_{v})\lneq
J(v,S^{\prime}_{v}).$

  3. 3)

If $\left|S_{v}\right|=2$ then $S_{v}=\mathcal{E}_{v}^{C}$ for some $C\in
K^{[2]}$ and $J(v,S_{v})=K\cap C.$

We call such $J(v,S_{v})$ an induced sub-cell complex of $K.$

###### Proof.

By Lemma 2.7, $K$ is $n$-regular and $(n-1)$-edge-regular for some $n\geq 2.$
Let $v\in K^{[0]}$ and $S_{v}\subset\mathcal{E}_{v}.$ By the Extension Lemma
2.40, for any given $e_{v}\in S_{v}$ and for all $w\in K^{[0]}$ there is an
edge $e_{w}$ in $\mathcal{E}_{w}$ associated to $e_{v},$ meaning that the map
$w\mapsto e_{w}$ is an induced edge field. Hence for all $w\in K^{[0]}$ we can
define the set $S_{w}\subset\mathcal{E}_{w}$ of edges in $\mathcal{E}_{w}$
associated to the edges in $S_{v}.$ Moreover, for each $w\in K^{[0]},$ we can
define the following set of 2-cells

$\mathcal{C}_{w}:=\\{C\in\overline{w}\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathcal{E}_{w}^{C}\subset S_{w}\\},$ (7)

where we used again the notation
$\mathcal{E}_{w}^{C}:=\\{e\in\mathcal{E}_{w}\leavevmode\nobreak\
|\leavevmode\nobreak\ e\subset C\\}$ and $\overline{w}$ is the dual set of $w$
in the 2-cc $K.$

The strategy of the proof is to inductively define a finite sequence of
connected domains $\mathcal{D}_{k}\subset K^{[0]},\leavevmode\nobreak\ 1\leq
k\leq N$ such that for all $1\leq k\leq N$ we have

$\displaystyle\mathcal{E}^{k}_{w}$
$\displaystyle:=\mathcal{E}_{w}\cap\left(K\cap\mathcal{D}_{k}\right)^{[1]}\subset
S_{w},$ (8) $\displaystyle\mathcal{C}_{w}^{k}$
$\displaystyle:=\\{C\in\overline{w}\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathcal{E}_{w}^{C}\subset\mathcal{E}_{w}^{k}\\}\subset\mathcal{C}_{w},$ (9)

for all $w\in\mathcal{D}_{k}$ and such that
$\mathcal{D}_{N}=J(v,S_{v})^{[0]}.$ Since the inclusion in (8) implies the
inclusion on line (9), we only need to check that the first inclusion holds.

Define $\mathcal{D}_{1}=\\{v\\},$ let $k\geq 1$ and suppose that we defined
the domains $\mathcal{D}_{j}$ for $1\leq j\leq k$ such that (8) is satisfied
for all $w\in\mathcal{D}_{j}$ and for all $1\leq j\leq k.$ Let us fix
$w_{0}\in\mathcal{D}_{k}$ and a 2-cell

$C_{0}\in\mathcal{C}_{w}\setminus\mathcal{C}_{w}^{k}.$

We then define $\mathcal{D}_{k+1}:=\mathcal{D}_{k}\cup C_{0},$ which implies
that for all $w\in\mathcal{D}_{k+1},$

$\mathcal{E}_{w}^{k+1}=\mathcal{E}_{w}^{k}\cup\mathcal{E}_{w}^{C_{0}},$

where $\mathcal{E}_{w}^{C_{0}}=\emptyset$ if $w\notin C_{0}.$ The definition
of the sets $S_{w},\leavevmode\nobreak\ w\in K^{[0]}$ (relying on the induced
edge field $w\longrightarrow\mathcal{E}_{w}$) and the fact that
$C_{0}\in\mathcal{C}_{w_{0}}$ imply that $C_{0}\in\mathcal{C}_{w}$ for all
$w\in C_{0}.$ We then have that $\mathcal{E}_{w}^{C_{0}}\subset S_{w}$ for all
$w\in C_{0}$ and therefore (8) is satisfied for all $w\in\mathcal{D}_{k+1}.$

We can then define $N\in\mathbb{N}^{*}$ to be the first integer such that

$\mathcal{C}_{w}^{N}=\mathcal{C}_{w}\quad\forall w\in\mathcal{D}_{N}.$ (10)

An consquence of (10) we also get

$\mathcal{E}_{w}^{N}=S_{w}\quad\forall w\in\mathcal{D}_{N},$ (11)

by the following argument. Since $\left|S_{w}\right|\geq 2$ for all $w\in
K^{[0]},$ if $e\in S_{w}$ then there exists $e^{\prime}\in
S_{w}\setminus\\{e\\}.$ Since $K$ is full, we can also pick
$C\in\overline{e}\cap\overline{e^{\prime}}.$ Hence the definition of
$\mathcal{C}_{w}$ implies that $C\in\mathcal{C}_{w}$ and (10) in turn implies
that $C\in\mathcal{C}_{w}^{N}.$ The definition of $\mathcal{C}_{w}^{k}$ in (9)
for $k=N$ yields $\mathcal{E}_{w}^{C}\subset\mathcal{E}_{w}^{N}$ which in
particular implies that $e\in\mathcal{E}_{w}^{N}$ since
$\mathcal{E}_{w}^{C}=\\{e,e^{\prime}\\}.$

We therefore have that $J=J(v,S_{v}):=(J^{[0]},J^{[1]},J^{[2]})$ defined by

$J^{[0]}:=\mathcal{D}_{N},\quad J^{[1]}:=\bigcup_{w\in J^{[0]}}S_{w},\quad
J^{[2]}:=\bigcup_{w\in J^{[0]}}\mathcal{C}_{w},$

is a sub-2-cell complex of $K.$ Axiom i) is clearly satisfied for $J,$ as well
as Axioms ii), iii) and iv) since they are satisfied for $K$ and if $C\in
J^{[2]}$ then the cc $K\cap C\subset J.$ The cc $J$ is clearly connected,
cell-connected, even, full and monodromy-free. The property (11) then implies
that

$J^{[1]}\cap\mathcal{E}_{w}=S_{w}\quad\forall w\in J^{[0]}.$ (12)

Let’s first note that point 3) is clear as it corresponds to the case where
$N=2$ in the above construction.

We can now argue that property 1) is satisfied. Fix $(v,S_{v})$ and
$J=J(v,S_{v})$ as defined above. Assume that we pick $v^{\prime}\in K^{[0]}$
and $S^{\prime}_{v^{\prime}}\subset\mathcal{E}_{v^{\prime}}$ and let
$J^{\prime}=J(v^{\prime},S^{\prime}_{v^{\prime}})$ be defined by the
construction above, using $(v^{\prime},S^{\prime}_{v^{\prime}})$ instead of
$(v,S_{v}).$ It is clear that if $J=J^{\prime}$ then $v^{\prime}\in J^{[0]}$
and $S^{\prime}_{v^{\prime}}=J^{[1]}\cap\mathcal{E}_{v^{\prime}}$ by property
(12) applied to $J^{\prime}=J.$ Therefore it suffices to show to converse
statement. For this we assume that $v^{\prime}\in J^{[0]}$ and
$S^{\prime}_{v^{\prime}}=J^{[1]}\cap\mathcal{E}_{v^{\prime}}$ and we show that
$J^{\prime}=J(v^{\prime},S^{\prime}_{v^{\prime}})=J.$ First we notice that
$S^{\prime}_{v^{\prime}}=S_{v^{\prime}},$ the set of edges in
$\mathcal{E}_{v^{\prime}}$ associated to $S_{v}.$ Therefore, for all $w\in
K^{[0]},$ the set of edges in $\mathcal{E}_{w}$ associated to
$S^{\prime}_{v^{\prime}}$ is identical to $S_{w},$ the set of edges in
$\mathcal{E}_{w}$ associated to $S_{v}.$ The corresponding sets of 2-cells
also remain unchanged, i.e.

$\mathcal{C}^{\prime}_{w}=\\{C\in\overline{w}\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathcal{E}^{C}_{w}\subset
S^{\prime}_{w}\\}=\mathcal{C}_{w},\quad\forall w\in K^{[0]}.$

Let’s define an equivalence relation on $K^{[0]}$ from the set of edges in
$K^{[1]}$ associated to a given set $S\subset\mathcal{E}_{v}$ as follows. Two
vertices $v_{0}$ and $v_{1}$ are called ”$S$-equivalent” if they can be linked
by a sequence of 2-cells in the following manner: there are sequences
$\\{C_{i}\\}_{i=1}^{M}\subset K^{[2]}$ and $\\{w_{i}\\}_{i=0}^{M}\subset
K^{[0]}$ such that $v_{0}:=w_{0}\in C_{1},$ $v_{1}=w_{M}$ and

$C_{i}\in\mathcal{C}_{w_{i}}\cap\mathcal{C}_{w_{i-1}}\quad\forall
i=1,\dots,M,$

where $\mathcal{C}_{w}$ for all $w\in K^{[0]}$ is defined by the formula (7)
choosing $S_{w}$ to be the set of edges in $\mathcal{E}_{w}$ associated to
$S.$ We then have that $v_{0}$ and $v_{1}$ are vertices of $J$ if and only
$v_{0}\in J$ and $v_{1}$ is $S_{v}$-equivalent to $v_{0}.$ By our assumptions,
$v$ and $v^{\prime}$ are $S_{v}$-equivalent and, as we noticed before,
$S^{\prime}_{v^{\prime}}$-equivalence classes are identical to
$S_{v}$-equivalence classes and this indeed implies that $J^{\prime}=J.$

Finally, the implication that $S_{v}\subsetneq S^{\prime}_{v}$ if
$J(v,S_{v})\lneq J(v,S^{\prime}_{v})$ of point 2) is clear and the reversed
implication is a consequence of the following observation: the inclusion
$S_{v}\subsetneq S^{\prime}_{v}$ implies that for each vertex $w\in K^{[0]}$
the set $S_{w}$ of edges associated to $S_{v}$ is also strictly included in
the set $S^{\prime}_{w}$ of edges associated to $S^{\prime}_{v}.$ Similarly,
the corresponding sets of 2-cells satisfy
$\mathcal{C}_{w}\subsetneq\mathcal{C}^{\prime}_{w}$ and this implies that
$J(v,S_{v})^{[i]}\subsetneq J(v,S^{\prime}_{v})^{[i]}$ for $i=0,1,2.$ ∎

Our formulation of the proof of Theorem 2.42 directly provides an algorithm to
construct the induced sub-cell complexes associated to a local even full
monodromy-free simply connected 2-cell complex. Using the same notations as in
the proof, the algorithm simply consists in the following steps, starting with
all couples of the form $(v,S_{v})$ marked as ”unselected” and running until
all such couples have been selected:

  * •

select one unselected couple $(v,S_{v}),$

  * •

construct $J(v,S_{v})$ by iteratively adding 2-cells to a domain
$\mathcal{D}_{k}$ as in the proof,

  * •

at each step, mark each couple $(w,S_{w})$ as ”selected” where
$w\in\mathcal{D}_{k+1}$ and $S_{w}$ are the edge in $\mathcal{E}_{w}$
associated to $S_{v}.$

We introduce the following notion in order to simplify the formulation of the
next proofs. The idea is simply to connect the induced cell complex defined in
Theorem 2.42 to the Definition 1.1 of a cell complex, where a cell is a subset
of vertices.

######  Definition 2.43 (Collection of induced $r$-cells).

Let $K$ be a local even full monodromy-free simply connected 2-cell complex.
$K$ is $n$-regular for some $n\in\mathbb{N}^{*}$ by Lemma 2.7) and let us also
fix an integer $2\leq r\leq n-1.$ We denote by $\mathcal{C}^{K}_{r}$ the
collection of vertex sets of all $r$-regular induced sub-cell complexes of $K$
given by Theorem 2.42. $\mathcal{C}^{K}_{r}$ is called the set of induced
$r$-cells of $K.$ By convention we set $\mathcal{C}_{r}^{K}=\emptyset$ for all
$r>n.$

Note that the point 3) of Theorem 2.42 implies in particular that
$\mathcal{C}_{2}^{K}=K^{[2]}.$ We also need the following condition on the
induced cells of a local even full monodromy-free simply connected 2-cell
complex in order to define an higher dimensional cell complex from these
cells.

######  Definition 2.44 (Non-singular collection of cells).

Let $K$ be a local cell complex and let $\mathcal{C}$ be a collection of
subsets of $K^{[0]}.$ $\mathcal{C}$ is non-singular if $K\cap(x\cap y)$ is
connected or empty for all $x,y\in\mathcal{C}.$

We are now ready to formulate the following two corollaries of Theorem 2.42.
The Corollary 2.45 gives the sufficient conditions on a $2$-cc $K$ for the
existence of an ambient cell complex $K_{amb}$ such that the $2$-skeleton of
$K_{amb}$ is $K.$ Corollary 2.46 gives the uniqueness of $K_{amb}$ and can be
interpreted as reconstructing a local even simply connected simple cc of rank
higher or equal to 3 from the data of its 2-skeleton.

######  Corollary 2.45.

Let $K$ be a local even full monodromy-free simply connected 2-cell complex
such that $\mathcal{C}^{K}_{r}$ is non-singular for all integer $2\leq r\leq
R,$ where $R$ is the largest integer $r$ such that
$\mathcal{C}_{r}^{K}\neq\emptyset.$ Then there exists a local even simply
connected simple $R$-cc $K_{amb}$ such that

$(K_{amb})^{(2)}=K.$

###### Proof.

By Lemma 2.7, $K$ is $n$-regular for some $n\geq 2.$ Define
$(K_{amb},\operatorname{rk}_{K_{amb}})$ by $K_{amb}^{[r]}=K^{[r]}$ for $0\leq
r\leq 2$ and $K_{amb}^{[r]}=\mathcal{C}^{K}_{r}$ for $r\geq 3.$ By Theorem
2.42, $\operatorname{Rk}(K_{amb})=\left|\mathcal{E}_{v}\right|-1=n-1,$ where
$v\in K^{[0]}.$ The Axiom i) is satisfied for $K_{amb}$ by point 2) of Theorem
2.42. In order to show Axiom ii), consider $x,y\in K_{amb},$ where
$x\supsetneq x\cap y\subsetneq y$ and, say,
$\operatorname{rk}(x)\leq\operatorname{rk}(y).$

We first consider the case where $x\cap y$ contains no 2-cell. Since
$\mathcal{C}^{K}_{r}$ is non-singular for all $1\leq r\leq n-1$ then $x\cap y$
is either a point or contains a unique edge by the following arguments. If
there are two or more edges in $x\cap y$ then $\operatorname{rk}(y)\geq 3$ (as
otherwise it would contradict ii) for $K$) and the non-singularity of induced
cells implies in particular that $x\cap y$ is connected, therefore there exist
at least two edges $e$ and $e^{\prime}$ in $x\cap y$ incident to the same
vertex $v.$ Since $K$ is full and by definition of the induced cc
$J(v,\\{e,e^{\prime}\\}),$ the induced cell $y$ contains the 2-cell $C$ such
that $\mathcal{E}_{v}^{C}=\\{e,e^{\prime}\\}.$ If $\operatorname{rk}(x)\geq 3$
then we also have $C\subset x$ which contradicts our assumption and if $x\in
K^{[2]}$ then there are two edges in $x\cap C$ and this contradicts Axiom ii)
for $K.$

We then consider the case were there exists $C\in\left(K\cap(x\cap
y)\right)^{[2]}$ and let $v\in C.$ By Theorem 2.42 there is an induced cell
$w$ such that
$\mathcal{E}^{C}_{v}\subset\mathcal{E}_{v}^{w}=\mathcal{E}_{v}^{x\cap
y}=\mathcal{E}_{v}^{x}\cap\mathcal{E}_{v}^{y}.$ Suppose by contradiction that
$w\subsetneq x\cap y.$ By the non-singularity of induced cells $x\cap y$ is
connected and this implies that there is a path $p:v\rightsquigarrow
v^{\prime}$ in $x\cap y$ to a vertex $v^{\prime}$ such that
$\mathcal{E}_{v^{\prime}}^{w}\subsetneq\mathcal{E}_{v^{\prime}}^{x\cap y}.$
This is a contradiction since
$\left|\mathcal{E}_{v}^{x^{\prime}}\right|=\left|\mathcal{E}_{v^{\prime}}^{x^{\prime}}\right|$
for all $x^{\prime}\in K$ by the bijectivity of the connection
$\nabla_{p}:\mathcal{E}_{v}\longrightarrow\mathcal{E}_{v^{\prime}}.$ Therefore
$w=x\cap y$ and ii) is satisfied. Properties iii) and iv) are direct
consequences of the point 2) from Theorem 2.42. $K_{amb}$ is local and full
since so is $K.$ By Lemma 2.10, $K_{amb}$ is simple since it is $n$-regular
with $n=\operatorname{Rk}(K_{amb})+1.$ $K_{amb}$ is clearly even, cell-
connected and simply connected. ∎

######  Corollary 2.46.

$K_{amb}$ given by Corollary 2.45 is unique, i.e. if $L$ is a local even
simply connected simple $R$-cell complex such that $R\geq 3$ then

$(L^{(2)})_{amb}=L.$

###### Proof.

Let $K=L^{(2)}.$ By Lemma 2.34 $K$ is monodromy-free. By Lemma 2.8 the induced
cells of $K$ are exactly the cells of $L$ of rank greater or equal to $3.$ ∎

######  Remark 2.47.

By Theorem 1.28 on the determination of local cell complexes by their
barycentric subdivision and Lemma 2.37, a consequence of Corollary 2.45 is
that local simply connected cell complexes are determined by the 2-skeleton of
the dual of their barycentric subdivision.

Physicists are sometimes looking at discrete spaces as ”Feynman diagrams” of
some generalized matrix models called group field theories (see e.g. [45] for
a review about this topic, [38] for a discussion in dimension 4 and [37] with
a definition of 2-cell complexes similar to ours). Here, we would like to
point out that, as a consequence of the results of these first two chapters,
the set of 2-shellable even local full 2-cell complexes appears to be an
interesting class of cell complexes to consider in this context. By Remark
2.23 we can see this class of 2-cc as being constructed by successively adding
2-cells while respecting some natural connectedness assumptions that involves
only local conditions. On the other hand Corollary 2.45 allows to uniquely
associate a higher dimensional ”ambient” cell structure to such 2-cell
complexes, which also satisfy the axioms of a cc if they fulfil the condition
of non-singularity from Definition 2.44. If this is the case, Theorem 2.25 and
Remark 2.31 imply that the dual of the resulting ambient cell complex is a
shellable simplicial complex which is then a triangulation of an Euclidean
ball or sphere.

## 3 Duality map for cobordisms

The goal of this chapter if to introduce the definition of a duality map that
applies to non-singular cell complexes with arbitrary boundary and which is
equivalent to the Definition 1.15 for closed cell complexes. Since we are
mainly interested in the case of local cell complexes and, as noticed in
Remark 1.22, the dual of a closed local cc (lcc) is local only if the original
lcc is also non-pinching, we will restrict our treatment to the set, noted
$\mathsf{C},$ of non-singular non-pinching lcc. We also denote by
$\mathsf{C}^{R}$ the elements of rank $R$ in $\mathsf{C}.$ Boundaries of
elements in $\mathsf{C}$ will also play an important role in what follows and,
by Lemma 1.21, these consist of closed but not necessarily connected cell
complexes. We therefore also introduce the notations $\mathsf{B}$ for the set
of closed non-pinching cell-connected cc and $\mathsf{B}^{R}$ for the set of
elements in $\mathsf{B}$ of rank $R.$ Since $\mathsf{C}$ also includes the
case of closed cell complexes, we assume by convention that the empty cell
complex $\\{\emptyset\\}$ is included in $\mathsf{B}$ and $\mathsf{B}^{R}$ for
all $R\geq 0.$

As we mentioned in the introduction, our goal is to mimic the behaviour of a
Fourier-like transformation in certain physical models that provides a duality
between two equivalent formulations of the same physical data. In this case,
one has in particular that the double dual of a model is equivalent to the
original model, an involution property we obtained for the cases of closed cc
that one would like to preserve when the duality map acts on cell complexes in
$\mathsf{C},$ or at least for a certain sub-class thereof. As we will explain
here, this idea is what lead us to consider the concept of cobordism, a
geometrical space seen as interpolating between its boundary components,
originally defined in the context of differentiable manifolds by Thom in [72]
and defining a category used in the formulation of the axioms of TQFT by
Atiyah in [10]. The parallel with TQFT motivated us to look for a suitable
category of cobordisms defined from our combinatorial notion of cell
complexes. This is the aim of the next Chapter 4, where we define a
composition operation for cobordisms leading us to the introduction of the
category of causal cobordisms.

In order to obtain an involutive duality map on a certain sub-class of cc in
$\mathsf{C}$, we have to focus on how this duality acts on the cells of the
boundary. This property is indeed already satisfied for the other cells, as a
consequence of Lemma 1.16. For this purpose, we introduce a different type of
dual set, noted $\widetilde{A}$ for $A\subset K^{[0]}.$ If
$K\in\mathsf{C}^{R}$ and $y\in(\partial K)^{[R-1]}$ is a maximal cell in the
boundary, it must be in particular that the dual of $y$ is an edge of the dual
of $K,$ as it becomes therefore again a sub-maximal cell of the double dual of
$K.$ Since $\widetilde{y}$ therefore has to contain at least two elements, it
is natural to define

$\widetilde{y}=\\{y,z\\},$

where $z$ is the unique element of $K^{[R]}$ containing $y.$ In fact, the
”generalized dual” of a cell $x\in\partial K$ will be defined as

$\widetilde{x}:=\\{y\in(\partial K)^{[R-1]}\leavevmode\nobreak\
|\leavevmode\nobreak\ x\subset y\\}\sqcup\\{z\in K^{[R]}\leavevmode\nobreak\
|\leavevmode\nobreak\ x\subset z\\}=\overline{x}^{\partial
K}\sqcup\overline{x}^{K},$

where this definition also applies to any set $A\subset K^{[0]},$ with the
convention that $\overline{A}^{\partial K}=\emptyset$ whenever
$A\not\subset\partial K^{[0]}.$

The first issue we encounter then is that $\widetilde{K}$ is not a cell
complex, as for example in the case of $y\in(\partial K)^{[R-1]},$
$\widetilde{K}$ does not contain the vertex $\\{y\\}=\overline{y}^{\partial
K}.$ One can easily fix this by considering the dual of $K$ to be
$\widetilde{K}\sqcup\overline{\partial K}$ but then this notion of dual does
not satisfy any involution property as it ”adds” more cells to the original
cell complex: for example the dual of a graph made of one edge, therefore
having two boundary components each made of one vertex, is a graph made of two
edges forming a simple path of length 2. The double dual would then be a graph
forming a simple path of length 3, etc. The radically opposite alternative
would be to ”ignore” the cells in $\partial K$ and define the dual of $K$ to
be $\overline{K\setminus\partial K},$ with obvious drawbacks when it comes to
the involution property, as this definition ”removes” cells. It turns out that
alternating between these two definitions is essentially the idea behind the
construction of the duality of this chapter: the dual of a path of length 1 is
a path of length 2 and the double dual is again a path of length 1.

But how can we signal when to use the definition that adds cells around the
boundary versus the definition that removes cells? We chose to do this by
defining our duality map on ”relative cell complexes”, i.e. couples of the
type $(K,J)$ where $J\leq K,$ in the particular case where $K\in\mathsf{C}$
and $J\in B^{[R-1]},\leavevmode\nobreak\ J\leq\partial K.$ The dual of such a
relative cell complex would use the definition that adds cells on the cells in
$\partial K\setminus J$ and use the definition that removes cell on the cells
in $J.$

This is where the picture with cobordisms arises: an element $(K,J)$ where
$K\in\mathsf{C}^{R}$ and $J\in\mathsf{B}^{[R-1]},\leavevmode\nobreak\
J\leq\partial K,$ can be interpreted as a cobordism from some ”ingoing
components” $J$ to some ”outgoing components” $L:=\partial K\setminus J.$ In
this picture, the duality maps the ingoing components to the outgoing
components of the dual. For example, the two-dimensional lattice we used to
define the Ising model in Section 0.1 can be seen as a cobordism $(K,J)$ where
$J=\partial K$ is the only boundary component of $K,$ for which we indeed
defined the dual using the definition that ”ignores” the cells in $\partial
K.$

To make this idea precise, we will need the notion of midsection, defined in
the next Section 3.1, a crucial notion for this chapter and Chapter 4. In
Section 3.2, we introduce the notions of non-degenerate, local and exactly
collared relative cell complexes. Non-degenerate and local relative cc are
used in Section 3.3 to introduce our definition of cobordism and the
corresponding notion of duality. Exactly collared relative cc will single out
a sub-class of cobordisms having additional regularity around the ingoing
boundary component. We state our main result in Section 3.3 implying in
particular that the dual of a cobordism is also a cobordism. As a corollary of
this result we show that the double dual of an exactly collared cobordism is
isomorphic to the original cobordism. In Section 3.4, we illustrate some
examples of duals of cobordisms, before giving the proof of the main theorem
in Section 3.5.

###  3.1 Midsections

We recall that the notation $\mathsf{C}$ is used for the set of non-singular
non-pinching lcc and $\mathsf{C}^{R}$ for the set of elements in $\mathsf{C}$
of rank $R.$ $\mathsf{B}^{R}$ is the set of closed non-pinching cell-connected
$R$-cc and by convention contains the empty cc $\\{\emptyset\\},$ which is a
sub-cc of every cc. Similarly $\mathsf{B}$ denotes the union of all elements
in $\mathsf{B}^{R},$ for $R\geq 0.$ As a consequence, the boundary of an
element in $\mathsf{C}^{R}$ is an element of $\mathsf{B}^{R-1}.$

Let us introduce a generalization of the notations previously used, named
”collar”. This tool is a central element of the definition of a midsection and
will be used throughout the rest of this work.

######  Definition 3.1 (Collar).

Let $A$ be a finite set and $\mathcal{A}$ be a collection of subsets of $A.$
If $B\subset A$ we define the collar of $B$ in $\mathcal{A},$ by

$\mathcal{A}_{B}:=\\{x\in\mathcal{A}\leavevmode\nobreak\ |\leavevmode\nobreak\
\emptyset\neq x\cap B\neq x\\}.$

Moreover, if $C\in\mathcal{A},$ we use the following definition and notation
for the collar of $B$ in $\mathcal{A}$ restricted to $C:$

$\mathcal{A}_{B}^{C}:=\\{x\in\mathcal{A}_{B}\leavevmode\nobreak\
|\leavevmode\nobreak\ x\subset C\\}.$

By abuse of notation, in case $\mathcal{B}$ is also a collection of subsets of
$A,$ the notation $\mathcal{A}_{\mathcal{B}}$ is used to denote
$\mathcal{A}_{B},$ where $B:=\cup_{x\in\mathcal{B}}x$ is the set of elements
of $A$ contained in a set in $\mathcal{B}.$

For example if $K\in\mathsf{C}$ and $J\leq K$ we have

$K_{J}=\\{x\in K\leavevmode\nobreak\ |\leavevmode\nobreak\ \emptyset\neq x\cap
J^{[0]}\neq x\\}.$

Since we use the notation $\mathcal{E}$ for the set of edges in $K$ when there
is no ambiguity on what the cell complex $K$ is, we will often use the
following notation for $x\in K:$

$\mathcal{E}_{J}^{x}=\\{e\in K^{[1]}\leavevmode\nobreak\ |\leavevmode\nobreak\
\emptyset\neq e\cap J^{[0]}\neq e,\quad e\subset x\\}.$

The following lemma is an important first step in order to endow the notion of
midsection with the structure of cc.

######  Lemma 3.2.

Let $K$ be a local cell complex and $J\leq K.$ Then the map

$x\longmapsto\mathcal{E}_{J}^{x},\quad x\in K_{J}$

is an injective poset homomorphism such that the inverse is a poset
homomorphism.

###### Proof.

We first notice that since $K$ is cell-connected we have that
$\mathcal{E}^{x}_{J}\neq\emptyset$ for all $x\in K_{J}.$ Let $x,y\in K_{J}$ be
such that $\mathcal{E}_{J}^{x}=\mathcal{E}_{J}^{y}=\mathcal{E}_{J}^{x\cap y},$
where the second equality is simply a consequence of the definition of a
collar. By this equality, in order to show that
$\mathcal{E}_{J}^{x}=\mathcal{E}_{J}^{y}$ implies $x=y,$ it is sufficient to
consider the case where say $x\subset y.$ For this we proceed by induction on
$r=\operatorname{rk}(x)\geq 1.$

If $\operatorname{rk}(x)=1$ then $x=\\{v,w\\}\in\mathcal{E}_{J},$ where $v\in
J^{[0]},\leavevmode\nobreak\ w\in K^{[0]}\setminus J^{[0]}$ and if we suppose
by contradiction that $y\neq x$ then by axiom iii) we can without loss of
generality suppose that $\operatorname{rk}(y)=\operatorname{rk}(x)+1=2,$ so
that $y\in K^{[2]}_{J}.$ By the diamond property iv) there exists
$x^{\prime}\in K^{[1]}$ such that
$\bigtriangledown\left(v\right)\cap\triangle\left(y\right)=\\{x,x^{\prime}\\}.$
The edge $x^{\prime}$ is not in $\mathcal{E}_{J}$ since otherwise it would
imply that $\\{x,x^{\prime}\\}\subset\mathcal{E}_{J}^{y}$ which contradicts
$\mathcal{E}_{J}^{x}=\mathcal{E}_{J}^{y}.$ By Lemma 1.12 the 2-cell $y$ admits
a representation such that $y=[x,x^{\prime},e_{1},\dots,e_{k}]$ where
$e_{k}\cap x=\\{w\\}$ thus in particular $e_{k}\notin J^{[1]}.$ Therefore
there is $1\leq i\leq k$ such that $e_{i}\in\mathcal{E}_{J}^{y}$ which is also
a contradiction with $\mathcal{E}_{J}^{x}=\mathcal{E}_{J}^{y}$ and this shows
that $x=y.$

Let’s assume the claim true for all $x,y$ such that $x\subset y$ and
$\operatorname{rk}(x)\leq r-1.$ Take $x\in K_{J}^{[r]}$ and $y\in K_{J}$ such
that $\mathcal{E}_{J}^{x}=\mathcal{E}_{J}^{y}$ and $x\subsetneq y.$ Like
before we can without loss of generality assume that
$\operatorname{rk}(y)=r+1.$ By induction we have that
$\mathcal{E}_{J}^{w}\subsetneq\mathcal{E}_{J}^{x}$ for all $w\subsetneq x,$
and, using this inclusion with $w=x\cap x^{\prime},$ this implies in
particular that $\mathcal{E}_{J}^{x^{\prime}}\not\subset\mathcal{E}_{J}^{x}$
for all $x^{\prime}\in K_{J}\cap K_{x}$ (since
$\mathcal{E}_{J}^{x^{\prime}}\neq\emptyset$ by locality of $K$). Since
$\mathcal{E}_{J}^{x}\neq\emptyset$ there exists
$w\in\triangle\left(x\right)\cap K_{J}.$ By the diamond property iv), there
exists $x^{\prime}\in K^{[r]}$ such that
$\bigtriangledown\left(w\right)\cap\triangle\left(y\right)=\\{x,x^{\prime}\\}.$
But $x^{\prime}\in K_{J}$ since so does $w$ and $w\subset x^{\prime},$
therefore
$\emptyset\neq\mathcal{E}_{J}^{x^{\prime}}\subset\mathcal{E}_{J}^{y}=\mathcal{E}_{J}^{x}.$
This is a contradiction with our induction hypothesis and therefore $x=y.$

The definition of $\mathcal{E}_{J}^{x}$ and the injectivity proven above imply
that if $\mathcal{E}_{J}^{x}\subset\mathcal{E}_{J}^{y}$ then

$\mathcal{E}_{J}^{(x\cap
y)}=\mathcal{E}_{J}^{x}\cap\mathcal{E}_{J}^{y}=\mathcal{E}_{J}^{x}$

and therefore $x=x\cap y$ or equivalently $x\subset y$ and this concludes the
proof. ∎

The previous lemma ensures that the rank function of a midsection given in the
following definition is well defined. This definition can be seen as a
generalization of the notion of midsection used in [40] to classify ”causal
slices” of causal triangulations and which was initially introduced in the
form of a coloured graph dual to a causal slice in [3].

######  Definition 3.3 (Midsection).

Let $K$ be a local cell complex and $J\leq K.$ We define the midsection of $J$
in $K$ to be $(M_{J}^{K},\operatorname{rk}_{M_{J}^{K}})$ where

$M_{J}^{K}:=\\{\mathcal{E}_{J}^{x}\leavevmode\nobreak\ |\leavevmode\nobreak\
x\in
K_{J}\\}\quad\text{and}\quad\operatorname{rk}_{M_{J}^{K}}(\mathcal{E}_{J}^{x})=\operatorname{rk}_{K}(x)-1.$

Figure 13:  Midsection $M_{J}^{K}$ associated to the boundary part $J$ of a
3-cc $K$ composed of three maximal cells (the center one having dots on its
vertices).

We give an illustration of a midsection in Figure 13. The next proposition
shows that midsections are cc under rather general assumptions.

######  Proposition 3.4.

Let $K$ be a local cell complex and $J\lneq K$ such that $x\not\subset
J^{[0]}$ for all $x\in K\setminus J$ and such that $K_{J}\neq\emptyset.$ Then
$(M_{J}^{K},\operatorname{rk}_{M_{J}^{K}})$ is a cc and if $K$ is pure then so
is $M_{J}^{K}$ and $\operatorname{Rk}(M_{J}^{K})=\operatorname{Rk}(K)-1.$
Moreover if $J\cap x$ is connected for all $x\in K^{[2]}_{J}$ then $M_{J}^{K}$
is graph-based.

###### Proof.

Since $K$ is cell-connected, $J\neq K,$ and by the condition $x\not\subset
J^{[0]}$ for all $x\in K\setminus J,$ there is at least one edge of $K$ having
one vertex in $J^{[0]}$ and one vertex in $K^{[0]}\setminus J^{[0]},$
therefore $K_{J}\neq\emptyset.$

By definition of $\mathcal{E}_{J}^{x}$ and $\operatorname{rk}_{M_{J}^{K}},$
the elements in $M_{J}^{K}$ of rank $0,$ i.e. vertices, are indeed 1-element
sets and every set $\mathcal{E}_{J}^{x}$ is indeed a set of vertices of
$M_{J}^{K}.$ Since for $x,y\in K,$ if $x\in K_{J}$ and $x\subset y$ then $y\in
K_{J},$ the axioms of the Definition 1.1 are a direct consequence of Lemma 3.2
and the axioms for $K.$

Let $R=\operatorname{Rk}(K).$ By purity of $K,$ every cell in $x\in K_{J}$ is
contained in a $R$-cell of $K,$ which must then belong to $K_{J}$ and
constitute an $R-1$ cell in $M_{J}^{K}$ containing $\mathcal{E}_{J}^{x},$
showing that $M_{J}^{K}$ is also pure.

By Lemma 1.12, and since $K$ is local, $K\cap x$ is a cycle for every 2-cell
$x$ of $K.$ Therefore, for every edge $\mathcal{E}^{x}_{J}$ of $M_{J}^{K},$
i.e. for every $x\in K^{[2]}_{J},$ if $J\cap x$ is connected then there are
exactly two edges in $K^{[1]}_{J}$ included in $K\cap x,$ and therefore there
are exactly two vertices of $M_{J}^{K}$ in $\mathcal{E}_{J}^{x}.$ ∎

###  3.2 Non-degenerate, local and exactly collared relative cc

In this section we introduce what it means in our context to consider a ”cell
complex relatively to another cell complex containing it”. We first explicitly
define relative cell complexes, homomorphism between such objects as well as
the notion of ”relative homomorphism” between relative cell complexes.
Relative homomorphism are used mostly in the next chapter, to define the
notion of relative reduction and relative collapses at the end of Section 4.1.

######  Definition 3.5 (Relative cc and homomorphism).

A couple $(K,J)$ of cc is called a relative cc if $J\leq K.$ We say that a map
$\phi:(K,J)\longrightarrow(K^{\prime},L)$ is a homomorphism of relative cc if
it is a cc-homomorphism $\phi:K\longrightarrow K^{\prime}$ such that
$\phi(J)=L.$ A relative cc-homomorphism or a homomorphism of $J$ relative to
$K$ is a homomorphism $\phi:(K,J)\longrightarrow(K^{\prime},L)$ of relative cc
such that $\phi|_{K\setminus J}:K\setminus J\longrightarrow
K^{\prime}\setminus L$ is a poset isomorphism.

Such ”relative objects” are present in many different contexts, notably in
topology with based space or algebraic topology with relative homology group.
However our definition in effect reverses the focus that is usually put on
such relative objects. While in topology one tends to ”ignore” the structure
contained in the subspace, here the cell complex $K$ in a relative cc $(K,J)$
can be interpreted merely as a ”background” cell complex structure on which a
relative homomorphism acts trivially. This point of view remains valid in the
following definitions of the specific types of relative cell complexes we
introduce next.

######  Definition 3.6 (Non-degenerate relative cc).

We say that a relative cc $(K,J)$ is non-degenerate if for all $x\in
K\setminus J$ then $x\not\subset J^{[0]}.$ We say that $(K,J)$ is degenerate
if it is not non-degenerate.

Examples of degenerate relative cell complexes $(K,J)$ can be easily
constructed when $K$ has relatively few maximal cells and $J=\partial K.$ For
instance if $K$ has only one maximal cell then such cell has automatically all
its vertices on the boundary.

The following definition extends the notion of locality to relative cell
complexes. It is the first time where the notion of midsection enters
crucially in our formalism. As will be discussed after this definition, it was
not obvious to us that this was the right notion of locality in this context.
It will however become apparent from the proof of Theorem 3.14 that this
notion of locality for relative cc contains exactly the assumptions needed for
the duality to be well-behaved around the ingoing boundary of a cobordism.

######  Definition 3.7 (Local relative cc).

We say that a relative cc $(K,J)$ is local if it is non-degenerate, $K$ is
local, every connected component $J_{0}$ of $J$ satisfies that $J_{0}\cap x$
is connected for all $x\in K_{J_{0}}$ and $M_{J_{0}}^{K}$ is cell-connected.
In particular $(K,\emptyset)$ is local for any lcc $K.$

The case in which $K\in\mathsf{C}^{R}$ and $J\in\mathsf{B}^{R-1}$ is a sub-cc
of $\partial K$ will be our main focus in what follows. In this case, the
assumption in the definition of local relative cc that the midsection is cell-
connected is in fact not redundant with the assumption that
$K\in\mathsf{C}^{R}.$ In other words, knowing that $K$ is both non-pinching
and cell-connected together with the assumption that $(K,J)$ is non-degenerate
is not sufficient to prove that $M_{J}^{K}$ is cell-connected. This apparently
remains the case even when adding an assumption on the connectedness of
2-cells: for example we could require that $\left[v,x\right]^{(2)}:=\\{y\in
K^{(2)}\leavevmode\nobreak\ |\leavevmode\nobreak\ v\in y\subset x\\}$ is
connected (as a section, c.f. definition in Section 2.3) for all $x\in
J^{[0]}.$ Also, the connectedness of the cells in $M_{J}^{K}$ seems closely
related to the connectedness of $\partial(J\cap x)$ when this cc is well-
defined, i.e. when $J\cap x$ is pure. The purity of $J\cap x$ is however not
guaranteed in general and not necessary in order for the cell in $M_{J}^{K}$
to be connected (this purity assumption will actually come up later in Section
4.2 as an assumption allowing to define a rank function on a poset called
transition, but is not required for the results of this chapter). These
observations support the interpretation of the locality assumption on a
relative cc $(K,J)$ (more specifically the connectedness of the midsection
$M_{J}^{K}$) as a condition allowing to freely choose the endpoint of any path
in $K$ having its last edge in $K_{J}.$ If $p$ is such a path, an edge in
$M_{J_{0}}^{K}$ indeed corresponds to a 2-cell $C\in K_{J_{0}},$ which defines
moves of the form $m_{p_{0}}^{C}$ where $p_{0}$ is a simple subpath of $C\cap
p$ (cf. Definition 2.27 of moves).

We now introduce the notion of exactly collared relative cc. This notion will
be used later on in Corollary 3.15 in order to prove that the duality map
defines an involution on the set of cobordism introduced in the next section.

######  Definition 3.8 (Exactly collared relative cc).

Let $(K,J)$ be a non-degenerate cc. We say that $(K,J)$ is exactly collared if
the map from $M_{J}^{K}$ to $J$ defined by $\mathcal{E}_{J}^{x}\mapsto J\cap
x$ is a cc-isomorphism. In particular an exactly collared relative cc $(K,J)$
such that $K$ is local is a local relative cc and $(K,\emptyset)$ is also
exactly collared for all lcc $K.$

We give an illustration of a simple example of an exactly collared cobordism
in Figure 16. In the next sections of this chapter, we will be particularly
interested in the case of relative cc $(K,J)$ where $K\in\mathsf{C}$ and
$J\leq K$ is a (possibly empty) union of boundary components of $\partial K.$
We therefore mention the following lemma, as it can shed light on certain
definitions given thereafter.

######  Lemma 3.9.

Let $K\in\mathsf{C}^{R}.$ Then $J\leq\partial K$ and $J\in\mathsf{B}^{R-1}$ if
and only if $J$ is a (possibly empty) union of connected components of
$\partial K.$

###### Proof.

Let $R=\operatorname{Rk}(K).$ A union of boundary components of $K$ is clearly
an element in $\mathsf{B}^{R-1},$ by Lemma 1.21. In order to prove the reverse
implication it is sufficient to prove the case where $\partial K$ is connected
and to show that $J\neq\emptyset$ implies $J=\partial K.$ If
$J\in\mathsf{B}^{R-1}$ is non-empty then it is a closed $(R-1)$-cc. If by
contradiction there exists a vertex in $\partial K\setminus J,$ the
connectedness of $\partial K$ implies that this vertex is linked to a vertex
in $J$ via a path in $(\partial K)^{(1)}.$ Hence there is an edge
$e=\\{v,w\\}$ in $\partial K$ such that $v\in J$ and $w\in\partial K\setminus
J.$ Now we have that $\overline{J}\leq\overline{\partial K}$ and
$\overline{e}^{\partial K}$ (the dual of $e$ in $\partial K$) is a
$(R-2)$-cell of $\overline{\partial K}$ contained in $\overline{v}^{\partial
K},$ an $(R-1)$-cell of $\overline{J}.$ Therefore it implies that
$\overline{e}^{\partial K}=\overline{e}^{J}\in J^{[R-2]}$ but since
$\overline{J}$ is closed we have that $\overline{e}$ is contained in two
maximal cells of $J.$ Since $\overline{w}^{\partial K}$ is a maximal cell of
$\overline{\partial K}\setminus\overline{J}$ containing $\overline{e}^{J}$
this contradicts the non-singularity of $\overline{\partial K}.$ ∎

###  3.3 Duality map for cobordisms and main results

The following set of relative cell complexes constitutes our notion of
cobordism.

######  Definition 3.10 (Cobordism).

The set of combinatorial cobordisms, or simply cobordisms, is denoted and
defined as

$\mathsf{Cob}:=\\{(K-J)\leavevmode\nobreak\ |\leavevmode\nobreak\
K\in\mathsf{C},\leavevmode\nobreak\ J\leq\partial K,\leavevmode\nobreak\
J\in\mathsf{B}^{\operatorname{Rk}(K)-1},\leavevmode\nobreak\ (K,J)\text{ non-
degenerate and local}\\}.$

We also use the notation $\mathsf{Cob}^{R}$ to denote the elements in
$\mathsf{Cob}$ such that $\operatorname{Rk}(K)=R,$ in which case $R$ is also
called the rank or dimension of the cobordisms in $\mathsf{Cob}^{R}.$ The
boundary of a cobordism is simply defined as $\partial(K-J):=\partial K.$ The
boundary components $J$ of a cobordism $(K-J)$ are called ingoing (boundary)
components or removed components of the cobordism and the cells in $J$ are
sometimes called the removed cells. The boundary components in $\partial
K\setminus J$ are called outgoing (boundary) components of $(K-J).$ We say
that $(K-J)\in\mathsf{Cob}$ is exactly collared if $(K,J)$ is exactly
collared.

We use the sign ”$-$” in $(K-J)$ to distinguish such an element from a usual
relative cc and we use the term ”removed” for $J$ and its cells because the
duality map we define later in Definition 3.13 indeed acts on $(K-J)$ as if
the cells in $J$ are removed from $K.$

There are numerous examples of cobordisms, as in particular every element
$(K-J)$ where $K$ is a non-singular (i.e. non-branching) and non-pinching
simplicial complex and $J$ is a union of connected components of $\partial K$
is an element of $\mathsf{Cob}.$ In other words, the assumptions of locality
and non-degeneracy are always satisfied for relative complexes $(K,J)$ where
$K\in\mathsf{C}$ is simplicial.

Before introducing its generalization, we recall that in Definition 1.14 we
defined the dual of a non-empty set $A\subset K^{[0]}$ for a cc $K$ by

$\overline{A}^{K}:=\\{z\in K^{[\operatorname{Rk}(K)]}\leavevmode\nobreak\
|\leavevmode\nobreak\ A\subset z\\},$

also denoted $\overline{A}$ in case there is no ambiguity on the cc $K.$ The
following definition constitutes a canonical way to extend the latter
definition in order to associate a dual cell to the cells on the boundary of a
non-singular cell complex.

######  Definition 3.11.

Let $K$ be a cc and let $A\subset K^{[0]}$ be non-empty. We define the
$\sim$-dual of $A$ to be

$\widetilde{A}^{K}:=\overline{A}^{K}\sqcup\overline{A}^{\partial K},$

where by convention $\overline{A}^{\partial K}=\emptyset$ if
$A\not\subset(\partial K)^{[0]}.$ We also write $\widetilde{x}$ for
$\widetilde{x}^{K}$ when there is no ambiguity on $K.$

As explained in the introductory paragraph of this chapter, this definition
for example implies that the $\sim$-dual of a maximal cell of the boundary
$y\in\partial K^{[R-1]}$ of a $R$-cc $K$ is given by

$\widetilde{y}=\\{y,z\\}$

where $z\in K^{[R]}$ is the unique $R$-cell of $K$ containing $y.$ Other
examples of $\sim$-dual are given in Figures 15 and 16.

For a collection of sets $\mathcal{A}\subset\mathcal{P}\left(K^{[0]}\right),$
we use the notation

$\widetilde{\mathcal{A}}^{K}:=\\{\widetilde{A}^{K}\leavevmode\nobreak\
|\leavevmode\nobreak\ A\in\mathcal{A}\\}.$

For example if $(K-J)\in\mathsf{Cob},$ we obtain $\widetilde{K\setminus
J}^{K}=\\{\widetilde{x}^{K}\leavevmode\nobreak\ |\leavevmode\nobreak\ x\in
K\setminus J\\}.$ We simply write $\widetilde{K\setminus J}$ instead of
$\widetilde{K\setminus J}^{K}.$ Note that if $K$ is closed then
$\widetilde{K}=\widetilde{K\setminus\emptyset}=\overline{K}.$

We would like to also define a rank function for elements in $\widetilde{K}$
when $K\in\mathsf{C}$ and this is done by mean of the following lemma, which
generalizes Lemma 1.16.

######  Lemma 3.12.

Let $K$ be a non-singular cc and let $x,y\in K.$ Then

$\widetilde{x}^{K}\subsetneq\widetilde{y}^{K}\quad\text{ if and only if }\quad
y\subsetneq x.$

###### Proof.

First we note that Lemma 1.6 and Lemma 1.16 imply that if $x\in
K\setminus\partial K$ then

$\overline{y}^{K}\subsetneq\overline{x}^{K}\quad\Leftrightarrow\quad
y\subsetneq x.$ (13)

We suppose first that $\widetilde{x}^{K}\subsetneq\widetilde{y}^{K},$ i.e.
$\overline{x}^{K}\subset\overline{y}^{K}$ and $\overline{x}^{\partial
K}\subset\overline{y}^{\partial K}.$ If $x\in K\setminus\partial K$ then (13)
implies that $y\subset x.$ If $x\in\partial K$ then $\overline{x}^{\partial
K}\subset\overline{y}^{\partial K}$ implies $y\subset x$ as a consequence of
Lemma 1.16 for the case of the closed cc $\partial K.$ Since our assumption
implies that $x\neq y$ we obtain $y\subsetneq x.$

Conversely, if $y\subsetneq x$ then the definition of dual set implies
$\widetilde{x}^{K}\subset\widetilde{y}^{K}.$ If $x\in K\setminus\partial K$
then the equivalence (13) implies that
$\overline{x}^{K}\subsetneq\overline{y}^{K}$ and in particular
$\widetilde{x}^{K}\neq\widetilde{y}^{K}.$ Similarly if $x\in\partial K$ then
Lemma 1.16 implies that $\overline{x}^{\partial
K}\subsetneq\overline{y}^{\partial K}$ and
$\widetilde{x}^{K}\neq\widetilde{y}^{K}.$ ∎

As a consequence of Lemma 3.12, the map $x\mapsto\widetilde{x}$ is a bijection
from the non-singular cc $K$ to $\widetilde{K}$ and the rank function

$\operatorname{rk}_{\widetilde{K}}(\widetilde{x}):=\operatorname{Rk}(K)-\operatorname{rk}_{K}(x)$

where $x\in\widetilde{K}$ is well defined for all
$\widetilde{x}\in\widetilde{K}.$

Let $K,K^{\prime}$ be disjoint posets with rank functions
$\operatorname{rk}_{K}$ and $\operatorname{rk}_{K^{\prime}}.$ We define the
disjoint union $K\sqcup K^{\prime}$ to be a poset with rank function

$\operatorname{rk}_{K\sqcup
K^{\prime}}:=\operatorname{rk}_{K}+\operatorname{rk}_{K^{\prime}}.$

For an element $(K-J)\in\mathsf{Cob},$ this definition allows us to have a
rank function on the poset defined as the disjoint union

$\widetilde{K\setminus J}\sqcup\overline{\partial K\setminus
J}=\widetilde{K\setminus J}^{K}\sqcup\overline{\partial K\setminus
J}^{\partial K}$

used below in our definition of dual cobordism. The latter union is indeed
disjoint, since by non-degeneracy of $(K,J)$ every cell $x\in\partial
K\setminus J$ satisfies $\overline{x}^{\partial
K}\subsetneq\widetilde{x}^{K},$ which implies in particular that
$\overline{x}^{\partial K}\neq\widetilde{x}^{K}$ for all such $x$ and
therefore $\widetilde{K\setminus J}^{K}\cap\overline{\partial K\setminus
J}^{\partial K}=\emptyset.$

More explicitly, if $(K,J)$ is non-degenerate our notations imply the
following expressions for the rank function on $\widetilde{K\setminus
J}\sqcup\overline{\partial K\setminus J}:$

$\operatorname{rk}_{\widetilde{K\setminus J}\sqcup\overline{\partial
K\setminus
J}}(x^{\prime})=\begin{cases}\operatorname{rk}_{\overline{K}}(x^{\prime})=R-\operatorname{rk}_{K}(x)&\textit{if
}x^{\prime}=\overline{x}^{K}\textit{ for }x\in K\setminus\partial K,\\\
\operatorname{rk}_{\widetilde{K}}(x^{\prime})=R-\operatorname{rk}_{K}(x)&\textit{if
}x^{\prime}=\widetilde{x}^{K}\textit{ for }x\in\partial K,\\\
\operatorname{rk}_{\overline{\partial
K}}(x^{\prime})=(R-1)-\operatorname{rk}_{\partial K}(x)&\textit{if
}x^{\prime}=\overline{x}^{\partial K}\textit{ for }x\in\partial K.\end{cases}$

where $R=\operatorname{Rk}(K).$ The particular case $J=\emptyset$ yields

$(\widetilde{K}\sqcup\overline{\partial
K})^{[0]}=\overline{K^{[R]}}^{K}\sqcup\overline{(\partial
K)^{[R-1]}}^{\partial K},$

for the dual sets of rank $0.$ For dual sets of rank $1\leq r\leq R-1$ we have

$(\widetilde{K}\sqcup\overline{\partial
K})^{[r]}=\overline{(K\setminus\partial
K)^{[R-r]}}^{K}\sqcup\overline{\partial K^{[R-r-1]}}^{\partial
K}\sqcup\widetilde{(\partial K)^{[R-r]}}^{K}.$

For example, if $e^{\prime}\in(\widetilde{K}\sqcup\overline{\partial
K})^{[1]}$ then $e^{\prime}$ is a dual set having one of the three following
forms:

  * •

the ”usual” dual of a sub-maximal cell of $K$ i.e.
$e^{\prime}=\overline{y}^{K}$ for $y\in K^{[R-1]},$

  * •

the ”usual” dual of a sub-maximal cell of $\partial K,$ i.e.
$e^{\prime}=\overline{y}^{\partial K}$ for $y\in(\partial K)^{[R-2]},$

  * •

the $\sim$-dual of a maximal cell in $\partial K$, i.e.
$e^{\prime}=\widetilde{y}^{K}=\\{y,z\\}$ where $y\in(\partial K)^{[R-1]}$ and
$z\in K^{[R]}$ is the unique maximal cell of $K$ containing $y.$

With this in mind, the next definition of duality map for cobordism is more
transparent.

######  Definition 3.13 (Dual cobordism).

Let $(K-J)\in\mathsf{Cob},$ we define the dual cobordism to be

$\overline{(K-J)}:=\left((\widetilde{K\setminus J}\sqcup\overline{\partial
K\setminus J})-\overline{\partial K\setminus J}\right).$

This definition, above all, raises the following question: is
$\overline{(K-J)}$ also a cobordism? This question is answered positively by
the main results of this chapter stated in the following theorem.

######  Theorem 3.14.

If $(K-J)\in\mathsf{Cob}^{R}$ then:

  * •

$\overline{(K-J)}\in\mathsf{Cob}^{R},$

  * •

$\overline{(K-J)}$ is exactly collared,

  * •

$\overline{K_{J}}\in\mathsf{B}^{R-1}$ and

$\partial\overline{(K-J)}=\overline{\partial K\setminus
J}\sqcup\overline{K_{J}}.$

###### Proof.

See Section 3.5. ∎

In the proof of Theorem 3.14 we will see that, as a consequence of Lemma 3.2,
we have $\overline{K_{J}}\cong\overline{M_{J}^{K}}.$ In other words, the dual
outgoing component $\overline{K_{J}}$ is better understood as the dual of the
midsection associated to the ingoing components $J.$ We point out that
although $\overline{K_{J}}$ is a (closed) cell complex, it is not the case of
$K_{J}$ and in particular $\overline{\overline{K_{J}}}\not\cong K_{J}.$

Definition 3.13 leads to the interpretation that the action of the duality map
on cobordisms in effect exchanges the role of the ingoing and outgoing