characters of the asymptotic symmetry group of asymptotically flat gravity,
together with some quantum corrections, when choosing a specific boundary
state with a clear geometric interpretation in the semi-classical limit. An
open problem in this context is how this result changes when considering a sum
over topologies, i.e. in the case in which we only fix the boundary graph,
equipped with the same boundary state, but sum over all manifold with torus
boundary. The last chapter of this thesis was devoted for a brief discussion
of this generalization using the formalism and techniques developed in this
thesis. In particular, we have constructed a family of triangulations of the
solid torus, which admit a coloured graph representing them, and which have a
similar structure as the solid torus triangulations used in the work in the
Ponzano-Regge model mentioned above. We then have seen that all manifolds
obtained by performing the internal connected sum of some closed $3$-manifold
with the solid torus yield the same contribution and hence, we get again a
factorization of the transition amplitude. An open problem is of course that
there are also other manifolds with torus boundary, which cannot be decomposed
as a connected sum. Such manifolds are called prime manifolds and it is
unclear if also these topologies yield the same contribution to the transition
amplitude, although we would conjecture they do, because, by the discrete
Bianchi identity, we always should recover a theory of flat boundary
connections together with possibly additional constraints coming from
uncontractible cycles of the boundary graph, which become contractible through
the bulk. Furthermore, an obvious next step would be to explicitly calculate
the transition amplitude and sum over diagrams of the coloured Boulatov model,
by choosing the same kind of boundary state used in the work regarding the
Ponzano-Regge model.
Yet another open problem in this context, which would be very interesting to
look at in some future work, is the question of how to define the asymptotic
limit in the context of the (coloured) Boulatov model and in GFT in general. A
possible way to approach this question is to study non-internal proper dipole
moves in more details. As we have seen in our discussion of crystallization
theory, these type of moves leave the topology invariant, but explicitly
change the number of boundary triangles by two. Although not discussed in this
thesis explicitly, it is in fact not too hard to prove an analogues statement
as Proposition 3.3.4 for non-internal dipole, when defining a dipole
contraction of spin-networks in a suitable way. A closer analysis of the
change of amplitudes under non-internal proper dipole moves could us provide
with some first insight in the problem mentioned above. Of course, an open
problem which still remains afterwards is how to define the asymptotic limit
on the level of the Boulatov fields. The final goal in this context would be
to calculate the full transition amplitude of the coloured Boulatov model with
respect to the same boundary state as chosen in the work regarding the
Ponzano-Regge model and to take the corresponding asymptotic limit, in order
to relate the results to other works done in this context and in order to
study the holographic dualities in this context.
Another interesting result regarding the case of a torus boundary is the
observation that we get two different contributions depending on which of the
two uncontractible cycles of the $2$-torus becomes contractible through the
bulk. When working with the Ponzano-Regge model, which is a triangulation-
independent theory, we fix both the boundary and bulk complex and hence,
taking two triangulations of the solid torus, which only differ by how they
are glued to the boundary and by which cycle becomes the contractible one
through the bulk, the corresponding transition functions are the same up to
relabelling of boundary edges. However, in the Boulatov model, in which we
only fix the boundary complex with a fixed labelling, we will get both
contributions and whenever the corresponding spin network state is not
symmetric under these changes, we will get in fact two different
contributions. In other words, the labelling of the boundary graph becomes
important in the Boulatov model. The most interesting point in this
observation is the holographic character: We a priori only fix the boundary
graph, which does not include any information about how the bulk topologies
are glued to it, but the corresponding transition amplitude encodes the
information that there are in fact two different results regarding on which
cycle becomes the contractible one, which is a pure bulk information. Of
course, there is a lot more work to do in this direction: First of all, it is
interesting to look at more complicated graphs representing the boundary
torus, i.e. graphs involving a twist. Furthermore, it would be interesting to
look at more complicated boundary topologies, like the genus $g=2$ surface. In
fact, we have already provided the basics needed in order to start analysing
these situations in this thesis: Simple boundary graphs representing surfaces
of arbitrary genus were presented in Section 3.3.7 and manifolds with boundary
given by such surfaces can be obtained by performing the boundary connected
sum of manifolds with torus boundary. A conjecture, which we would like to
make at this point, is that the transition amplitude of the coloured Boulatov
model always factorizes into several terms, where each term encodes flatness
of the boundary together with an additional constraint, which appears in the
form of a delta function containing a tricoloured path on the boundary graph,
which is independent from the other delta functions, which encode flatness of
the boundary connection. Of course, proving this claim rigorously requires
much more work, but our results regarding the case of a boundary torus seem to
support this conjecture.
A completely different direction for some further work is to study the
situation of Riemannian quantum gravity with a cosmological constant. In the
second chapter of this thesis, we have introduced the Turaev-Viro spin foam
model, which is defined using the quantum group $U(\mathfrak{su}(2))_{q}$ for
some root of unity $q$, and which describes $3$-dimensional Riemannian quantum
gravity with a positive cosmological constant $\Lambda\propto r^{-2}$, where
$r\geq 3$ is defined via $q=\mathrm{exp}(\pi i/r)$. Now, it is actually not
too hard to define a corresponding group field theory, which produces the
Turaev-Viro model as its Feynman amplitudes. In fact, the corresponding model
was already introduced in the original article introducing the Boulatov model
[Bou92]. Of course, since it is a priori not clear how to integrate over a
quantum group, we cannot use the group representation, as for the Boulatov
model. However, we can define the Turaev-Viro group field theory using the
spin-representation. An important feature of the Turaev-Viro model is that its
corresponding partition functions are finite, as opposed to the Ponzano-Regge
model. It would be interesting to define a coloured version of this theory and
to analyse the transition amplitudes in this case. Furthermore, it would be
interesting to study the relation of this theory with the Boulatov model. We
have already discussed that the limit $q\to 1$ cannot always be taken and
hence, the relation between the Ponzano-Regge model and Turaev-Viro model is
not completely well-understood. Hence, it would be interesting to study the
relation in more details on the level of group field theories.

##  Appendix A Appendix

###  A.1 Mathematical Gauge Theory and Topological BF-Theory

In the first chapter of this appendix, we review some important concepts of
mathematical gauge theory, which is the branch of mathematics studying the
geometry of principal fibre bundles and their associated vector bundles. In
physics, gauge theories are used to describe field theories, which are
invariant under local transformations, but the corresponding formalism has
also many important applications in pure mathematics, like Seiberg-Witten and
Donaldson theory. In the end of this chapter, we discuss some aspects of
$BF$-theory, which we have discussed in Section 1.2.1, like its equations of
motion and its gauge symmetries, in more details. The main references for the
following discussion are the excellent textbooks [Bau14, Ham17, RS13, RS17].
For applications in pure mathematics see for example the monographs [DK97,
Sco05].

####  A.1.1 A Glossary of Mathematical Gauge Theory

In the first part of this chapter, we review some important concepts of
mathematical gauge theory. We will not delve into a detailed discussion and
the following part should be understood as a glossary, which introduces the
most important definitions in order to fix the terminology which we have used
in Chapter 1 of this thesis as well as in our discussion of $BF$-theory in
Section A.1.2 below. We will omit citations throughout this section and for
more details the reader is guided to the literature cited above.

Principal Fibre Bundles:

The central objects in gauge theory are “principal fibre bundles”, which
combine the notion of fibre bundles from differential topology with notions
from group theory. To start with, recall that a “(smooth) fibre bundle” is a
quadruple $(E,\mathcal{M},\pi,F)$ consisting of a smooth manifold
$\mathcal{M}$, called the “base space”, a smooth manifold $E$, called “total
space”, a smooth manifold $F$, called “general fibre”, as well as a smooth and
surjective map $\pi:E\to\mathcal{M}$, such that $E$ is “locally trivial”,
which means that $\forall x\in M$, there is an open neighbourhood
$U_{x}\subset M$ of $x$ and a diffeomorphism $\Phi_{x}:\pi^{-1}(U_{x})\to
U_{x}\times F$ such that the following diagram commutes:

${\pi^{-1}(U_{x})}$${U_{x}\times
F}$${U_{x}}$$\scriptstyle{\pi}$$\scriptstyle{\Phi_{x}}$$\scriptstyle{\operatorname{pr}_{1}}$

The embedded submanifold $E_{x}:=\pi^{-1}(\\{x\\})$, which is diffeomorphic to
$F$, is called “fibre over $x$”. Furthermore, we call a pair
$(U_{x},\Phi_{x})$ “bundle chart around $x$” and a collection of bundle
charts, which cover $\mathcal{M}$, “bundle atlas”. Using this, a “principal
fibre bundle” is then defined to be a smooth fibre bundle
$(P,\mathcal{M},\pi,G)$, where $G$ is some Lie group, called the “structure
group”, together with a smooth right action $\mu:P\times G\to P$ with the
following extra properties:

  * (1)

The action $\mu$ preserves the fibres of $\pi$, i.e. $\pi(\mu(p,g))=x$ for all
$x\in\mathcal{M}$, $p\in P_{x}=\pi^{-1}(\\{x\\})$ and $g\in G$. Furthermore,
the restricted action $\mu|_{P_{x}\times G}:P_{x}\times G\to P_{x}$ is simply
transitive for all $x\in\mathcal{M}$. In the following, we will write $p\cdot
g:=\mu(p,g)$.

  * (2)

There exists a “principal bundle atlas”, i.e. a bundle atlas
$\\{(U_{i},\Phi_{i})\\}_{i\in I}$, where $I$ denotes some index set,
consisting only of “$G$-equivariant maps”, i.e. $\Phi_{i}(p\cdot
g)=\Phi_{i}(p)\cdot g$ for all $i\in I$, $p\in P_{U_{i}}:=\pi^{-1}(U_{i})$ and
$g\in G$, where the action on the right-hand side is the obvious one defined
by $(x,g)\cdot h:=(x,g\cdot h)$ for all $x\in U_{i}$ and $g,h\in G$.

Connections on Principal Bundles:

Another central object in gauge theory are “connection $1$-forms”, which
correspond to the gauge fields in physics terminology. Let $G\to
P\stackrel{{\scriptstyle\pi}}{{\to}}\mathcal{M}$ be a principal fibre bundle
with structure group $G$. A “connection $1$-form”, or “gauge field”, on this
bundle is a $\mathfrak{g}$-valued $1$-form $A\in\Omega^{1}(P,\mathfrak{g})$
satisfying the following two conditions:

  * (1)

$A$ is of type $\mathrm{Ad}$, i.e.
$(r_{g}^{\ast}A)_{p}=\mathrm{Ad}_{g^{-1}}\circ A_{p}$ for all $g\in G$ and
$p\in P$, where $r_{g}:P\to G,p\mapsto p\cdot g$ denotes the right translation
of the group action $\cdot:P\times G\to P$ and where
$\mathrm{Ad}:G\to\mathrm{Aut}(\mathfrak{g})$ denotes the adjoint
representation of $G$.

  * (2)

$A(\widetilde{X})=X$ for all $X\in\mathfrak{g}$, where $\widetilde{X}$ denotes
the “fundamental vector field” corresponding to $X$, i.e. the vector field
$\widetilde{X}\in\mathfrak{X}(P)$ defined by
$\widetilde{X}_{p}:=\frac{\mathrm{d}}{\mathrm{d}t}\big{|}_{t=0}(p\cdot\mathrm{exp}_{G}(tX))$
for all $p\in P$.

If $s\in\Gamma^{\infty}(U,P)$ is a “local gauge”, i.e. a local section of $P$,
where $U\subset\mathcal{M}$ is some open set, then we can define the form
$A_{s}:=s^{\ast}A\in\Omega^{1}(U,\mathfrak{g})$, which is now a $1$-form
living on $\mathcal{M}$. Using a basis $\\{T_{a}\\}_{a}$ of the Lie algebra
$\mathfrak{g}$, this form can be decomposed as $A_{s}=\sum_{a}A_{s}^{a}T_{a}$,
where $A_{s}^{a}\in\Omega^{1}(U)$ are $\mathbb{R}$-valued forms. Furthermore,
if $(U,\varphi=\\{x^{\mu}\\})$ is a local manifold chart, we can introduce the
maps $A_{\mu}^{a}:=A_{s}^{a}(\partial_{\mu})\in C^{\infty}(U)$ and hence we
can write $A^{a}_{s}=\sum_{\mu}A_{\mu}^{a}\mathrm{d}x^{\mu}$ for every $a$.
The fields $\\{A_{\mu}^{a}\\}_{\mu,a}$ are then called “(local) gauge boson
fields”.

If $A\in\Omega^{1}(P,\mathfrak{g})$ is a connection $1$-form on the principal
bundle $P$, then we define the vector space $H_{p}:=\mathrm{ker}(A_{p})\subset
T_{p}P$. This defines an “Ehresmann connection”, i.e. a vector subbundle
$H\subset TP$ (a “distribution on $P$”) such that $T_{p}P=H_{p}\oplus V_{p}$
for all $p\in P$, where $V_{p}:=T_{p}(P_{\pi(p)})$ denotes the “vertical
tangent space at $p$”, and such that $r_{g\ast}(H_{p})=H_{p\cdot g}$ for all
$p\in P$ and $g\in G$. We call a tangent vector contained in $H_{p}\subset
T_{p}P$ “horizontal”.

Curvature:

The “curvature” of a connection $1$-form $A\in\Omega^{1}(P,\mathfrak{g})$ is
the $2$-form $F^{A}\in\Omega^{2}(P,\mathfrak{g})$ defined via the “structure
equation”

$\displaystyle F^{A}:=\mathrm{d}A+\frac{1}{2}[A\wedge A],$ (A.1)

where
$[\cdot\wedge\cdot]:\Omega^{k}(P,\mathfrak{g})\times\Omega^{l}(P,\mathfrak{g})\to\Omega^{k+l}(P,\mathfrak{g})$
is the wedge-product defined via the Lie-bracket of $\mathfrak{g}$, i.e.

$\displaystyle[\alpha\wedge\beta]:=\sum_{a}(\alpha^{a}\wedge\beta^{b})[T_{a},T_{b}],$
(A.2)

for all $\alpha=\alpha^{a}T_{a}\in\Omega^{k}(P,\mathfrak{g})$ and
$\beta=\beta^{a}T_{a}\in\Omega^{l}(P,\mathfrak{g})$, where $\alpha^{a}$ and
$\beta^{b}$ are real-valued forms. Again, using a local gauge
$s\in\Gamma^{\infty}(U,P)$, this can locally be written as
$F_{s}^{A}:=s^{\ast}F^{A}\in\Omega^{2}(U,\mathfrak{g})$. Using a basis of the
Lie algebra $\mathfrak{g}$, we decompose this as
$F^{A}_{s}=\sum_{a}F^{a}_{s}T_{a}$ with $F_{s}^{a}\in\Omega^{2}(U)$ and using
a local chart $(U,\varphi=\\{x^{\mu}\\})$, we can further write
$F_{s}^{a}=\sum_{\mu,\nu}F_{\mu\nu}^{a}\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}$.
The collection $\\{F_{\mu\nu}^{a}\\}_{\mu,\nu,a}\in C^{\infty}(U)$ is called
“(local) field strength of the gauge boson field $A$”.

Associated Vector Bundles:

Let $(V,\rho)$ be some finite-dimensional real (resp. complex) representation
of a Lie group $G$ and $G\to P\stackrel{{\scriptstyle\pi}}{{\to}}\mathcal{M}$
be a principal $G$-bundle. Then we define a real (resp. complex) vector
bundle, called the “associated vector bundle”, as follows: First of all, we
define a free right group action $(P\times V)\times G\to(P\times V)$ of $G$ on
$P\times V$ via

$\displaystyle(p,v)\cdot g:=(p\cdot g,\rho(g^{-1})v)$ (A.3)

for all $(p,v)\in P\times V$, $g\in G$. One can show that the quotient
$E:=(P\times V)/G$ is a well-defined smooth manifold and that the manifold $E$
is a well-defined vector bundle with projection $\pi_{E}:E\to\mathcal{M}$
given by $\pi_{E}([p,v]):=\pi(p)$ for all $[p,v]\in E$. The fibres of this
bundle are given by $E_{x}:=(P_{x}\times V)/G$ for all $x\in\mathcal{M}$. This
bundle is usually denoted by

$\displaystyle P\times_{\rho}V:=E.$ (A.4)

Sections in the associated vector bundle can be described by sections of the
principal bundle $P$. More precisely, if $s\in\Gamma^{\infty}(U,P)$ with
$U\subset\mathcal{M}$ open is a local gauge, then for every smooth map $f\in
C^{\infty}(U,V)$ there is a section $\tau\in\Gamma^{\infty}(U,E)$ defined via

$\displaystyle\tau(x)=[s(x),f(x)]$ (A.5)

for all $x\in U$ and vice versa. Last but not least, if
$\langle\cdot,\cdot\rangle_{V}$ is a $G$-invariant, non-degenerate and
symmetric (resp. hermitian) bilinear (resp. sesquilinear) form on $V$, then
one can define a vector bundle (pseudo-)metric
$\langle\cdot,\cdot\rangle_{E}\in\Gamma^{\infty}(E^{\ast}\otimes E^{\ast})$ of
$E$ via

$\displaystyle\langle[p,v],[p,w]\rangle_{E}:=\langle v,w\rangle_{V}$ (A.6)

for all $[p,v],[p,w]\in E$. One can easily verify that this is indeed a bundle
metric and that the definition is independent of the choice of representatives
of $[p,v],[p,w]$. This metric is called the “bundle metric induced by
$\langle\cdot,\cdot\rangle_{V}$”. Let us remark that associated vector bundles
are central in the standard model of particle physics, since matter fields are
defined to be sections of associated bundles (possibly twisted by spinor
bundles for fermions) for some given representations.

Exterior Covariant Derivative:

Let $G\to P\stackrel{{\scriptstyle\pi}}{{\to}}\mathcal{M}$ be a principal
$G$-bundle with a connection $1$-form $A\in\Omega^{1}(P,\mathfrak{g})$ and
$(V,\rho)$ be a representation of $G$. Then, we define a differential operator
on $V$-valued differential forms, i.e. a family of maps
$D_{A}:\Omega^{k}(P,V)\to\Omega^{k+1}(P,V)$ for $k\in\mathbb{N}_{0}$, via

$\displaystyle(D_{A}\omega)_{p}(v_{1},\dots,v_{k}):=(\mathrm{d}\omega)_{p}(\mathrm{pr}_{H}v_{1},\dots,\mathrm{pr}_{H}v_{k})$
(A.7)

for all $\omega\in\Omega^{k}(P,V)$, $v_{1},\dots,v_{k}\in T_{p}P$ and $p\in
P$, where $\mathrm{pr}_{H}:T_{p}P\to H_{p}$ denotes the obvious projection to
the horizontal tangent space $H_{p}:=\mathrm{ker}(A_{p})$. Note that this
differential operator can be used to define the curvature, since

$\displaystyle D_{A}A=F^{A}\in\Omega^{2}(P,\mathfrak{g}).$ (A.8)

In order to define a covariant derivative on the base manifold $\mathcal{M}$,
let us introduce the following terminology: We define the vector space
$\Omega^{k}_{\mathrm{hor}}(P,V)^{\rho}$ of “horizontal $k$-forms of type
$\rho$” consisting of $k$-forms $\omega\in\Omega^{k}(P,V)$ with the following
two properties:

  * (1)

$\omega_{p}(v_{1},\dots,v_{k})=0$ whenever at least one of the vectors
$v_{1},\dots,v_{k}\in T_{p}P$ is vertical $\forall p\in P$.

  * (2)

$(r_{g}^{\ast}\omega)_{p}=\rho(g^{-1})\circ\omega_{p}$ for all $p\in P$ and
for all $g\in G$.

One can easily verify that the curvature $2$-form corresponding to some
connection $1$-form $A\in\Omega^{1}(P,\mathfrak{g})$ is horizontal and of type
$\mathrm{Ad}$, i.e.
$F^{A}\in\Omega_{\mathrm{hor}}^{2}(P,\mathfrak{g})^{\mathrm{Ad}}$. Now, it
turns out that there is the following isomorphism:

$\displaystyle\Omega^{k}_{\mathrm{hor}}(P,V)^{\rho}\cong\Omega^{k}(\mathcal{M},E),$
(A.9)

where $E:=P\times_{\rho}V$ denotes the associated vector bundle. In
particular, this means that we can view the curvature $F^{A}$ as an element
$F^{A}_{\mathcal{M}}$ of $\Omega^{2}(\mathcal{M},\mathrm{Ad}(P))$, where
$\mathrm{Ad}(P):=P\times_{\mathrm{Ad}}\mathfrak{g}$ denotes the “adjoint
bundle”. The differential operator $D_{A}$ maps forms of
$\Omega^{k}_{\mathrm{hor}}(P,V)^{\rho}$ into forms of
$\Omega^{k+1}_{\mathrm{hor}}(P,V)^{\rho}$ and hence we get a well defined
family of operators

$\displaystyle\mathrm{d}_{A}:\Omega^{k}(\mathcal{M},E)\to\Omega^{k+1}(\mathcal{M},E).$
(A.10)

If we restrict this map to $\Omega^{0}(\mathcal{M},E):=\Gamma^{\infty}(E)$,
then we get a connection
$\nabla^{A}:=\mathrm{d}_{A}|_{\Gamma^{\infty}(E)}:\Gamma^{\infty}(E)\to\Gamma^{\infty}(T^{\ast}\mathcal{M}\otimes
E)$ of the vector bundle $E$. This connection is usually called “covariant
derivative”. Now, it turns out that the exterior covariant derivative
$\mathrm{d}_{\nabla^{A}}$ induced by this connection is exactly
$\mathrm{d}_{A}$, i.e. it holds that $\mathrm{d}_{A}=\mathrm{d}_{\nabla^{A}}$.
As a consequence, we can compute $\mathrm{d}_{A}\omega$ for some
$\omega\in\Omega^{k}(\mathcal{M},E)$ in some local frame
$\\{e_{a}\\}\subset\Gamma^{\infty}(U,E)$, where $U\subset\mathcal{M}$ open,
via

$\displaystyle(\mathrm{d}_{A}\omega)|_{U}=\sum_{a}(\mathrm{d}\omega^{a}\otimes
e_{a}+(-1)^{k}\omega^{a}\wedge\nabla_{A}e_{a}),$ (A.11)

where $\omega^{a}$ are the real-valued coordinate forms of $\omega$, i.e.
$\omega|_{U}=\sum_{a}\omega^{a}\otimes e_{a}$. As a last remark, let us note
that if $\langle\cdot,\cdot\rangle_{V}$ is some $G$-invariant, non-degenerate
and symmetric bilinear form on $V$, then the covariant derivative $\nabla^{A}$
is compatible with the induced bundle metric $\langle\cdot,\cdot\rangle_{E}$,
i.e.

$\displaystyle X(\langle
s,t\rangle_{E})=\langle\nabla^{A}_{X}s,t\rangle_{E}+\langle
s,\nabla^{A}_{X}t\rangle_{E}$ (A.12)

for all sections $s,t\in\Gamma^{\infty}(E)$ and vector fields
$X\in\mathfrak{X}(\mathcal{M})$.

The Bianchi-Identity:

Let $G\to P\stackrel{{\scriptstyle\pi}}{{\to}}\mathcal{M}$ be a principal
$G$-bundle with a connection $1$-form $A\in\Omega^{1}(P,\mathfrak{g})$. Then,
as discussed above, we can define the corresponding curvature
$F^{A}\in\Omega^{2}(P,\mathfrak{g})$, which can equivalently be viewed as
$2$-form $F^{A}_{\mathcal{M}}\in\Omega^{2}(\mathcal{M},\mathrm{Ad}(P))$. The
“Bianchi identity” is the following formula

$\displaystyle\mathrm{d}F^{A}+[A\wedge F^{A}]=0,$ (A.13)

which follows from a straightforward calculation. Using the exterior covariant
derivative, this can also be written as $D_{A}F^{A}=0$, or equivalently, as
$\mathrm{d}_{A}F^{A}_{\mathcal{M}}=0$.

Gauge Transformations:

A “(global) gauge transformation” of some principal $G$-bundle $G\to
P\stackrel{{\scriptstyle\pi}}{{\to}}\mathcal{M}$ is defined to be a principal
bundle automorphism $f$ of $P$, i.e. a smooth diffeomorphism $f:P\to P$
satisfying the following two properties:

  * (1)

$f$ is a fibre bundle automorphism, i.e. $\pi\circ f=\pi$.

  * (2)

$f$ is $G$-equivariant, i.e. $f(p\cdot g)=f(p)\cdot g$ for all $p\in P$, $g\in
G$.

We will denote the set of all gauge transformations by $\mathcal{G}(P)$. Once
can show that there is the following group isomorphism:

$\displaystyle\mathcal{G}(P)\to C^{\infty}(P,G)^{G},f\mapsto\sigma_{f},$
(A.14)

where $C^{\infty}(P,G)^{G}$ is the subset of $C^{\infty}(P,G)$ consisting of
smooth functions $\sigma:P\to G$ satisfying $\sigma(p\cdot
g)=g^{-1}\sigma(p)g$ for all $p\in P$ and $g\in G$ and where $\sigma_{f}$ is
the map defined by $f(p)=p\cdot\sigma_{f}(p)$ for all $p\in P$. Note that the
gauge group $C^{\infty}(P,G)^{G}$ can be given the structure of an infinite-
dimensional Hilbert Lie group [Sch10, p.7]. In physics, one usually defines
gauge transformations to be maps living on the spacetime manifolds
$\mathcal{M}$. This can be related to the mathematical notion of gauge
transformations as follows: For every local gauge $s\in\Gamma^{\infty}(U,P)$
with $U\subset\mathcal{M}$ open, there is an isomorphism of groups
$C^{\infty}(P_{U},G)^{G}\cong C^{\infty}(U,G)$, where
$P_{U}:=\pi^{-1}(U)\subset P$. If $G$ is abelian, then one can show that there
is also a global isomorphism, i.e. we can choose $U=\mathcal{M}$.

Let us discuss the action of the gauge group $\mathcal{G}(P)$ on various
different objects. The proof of the claimed equalities is straightforward.

  * (1)

If $A\in\Omega^{1}(P,\mathfrak{g})$ is a connection $1$-form, then
$f\in\mathcal{G}(P)$ acts on $A$ as

$\displaystyle f^{\ast}A=\mathrm{Ad}_{\sigma_{f}^{-1}}\circ
A+\sigma_{f}^{\ast}\mu_{G},$ (A.15)

where $\mu_{G}\in\Omega^{1}(G,\mathfrak{g})$ denotes the Maurer-Cartan form on
$G$, i.e. the $1$-form defined by
$(\mu_{G})_{g}(v):=\mathrm{d}_{g}L_{g^{-1}}(v)\in T_{e}G\cong\mathfrak{g}$ for
all $g\in G$, $v\in T_{g}G$ with $L_{g}:G\to G,h\mapsto gh$.

  * (2)

If $(V,\rho)$ is some representation of $G$ and $E:=P\times_{\rho}V$ the
corresponding associated vector bundle, then $f\in\mathcal{G}(P)$ acts on
$[p,v]\in E$ as

$\displaystyle f\cdot[p,v]:=[f(p),v].$ (A.16)

One can easily verify that this is independent on the choice of representative
of $[p,v]$.

  * (3)

Let $(V,\rho)$ and $E$ as in (2). Then $f\in\mathcal{G}(P)$ acts on
$\omega\in\Omega^{k}_{\mathrm{hor}}(P,V)^{\rho}$ as

$\displaystyle f^{\ast}\omega=\rho(\sigma_{f}^{-1})\circ\omega.$ (A.17)

As an application, we see that the curvature
$F^{A}\in\Omega^{2}_{\mathrm{hor}}(P,\mathfrak{g})^{\mathrm{Ad}}$ of some
connection $1$-form $A\in\Omega^{1}(P,\mathfrak{g})$ transforms as

$\displaystyle F^{f\ast A}=f^{\ast}F^{A}=\mathrm{Ad}_{\sigma_{f}^{-1}}\circ
F^{A}.$ (A.18)

Now, recall that there is an isomorphism
$I:\Omega^{k}_{\mathrm{hor}}(P,V)^{\rho}\to\Omega^{k}(\mathcal{M},E)$. Then,
one can easily show that under this isomorphism $I(f^{\ast}\omega)=f\cdot
I(\omega)$, where the action on the right-hand side is the action as defined
in (2) applied point-wise to forms in $\Omega^{k}(\mathcal{M},E)$.

####  A.1.2 Application to Topological BF-Theory

As an application of the formalism discussed previously, let discuss
topological $BF$-theory in more details. More precisely, we will discuss
rigorously the gauge symmetries of $BF$-theory as well as a precise
mathematical derivation of the equations of motion. Let us fix the following
date:

  * (1)

A $d$-dimensional smooth, compact and oriented manifold $\mathcal{M}$ without
boundary and a principal $G$-bundle $G\to
P\stackrel{{\scriptstyle\pi}}{{\to}}\mathcal{M}$, where $G$ is some finite-
dimensional Lie group with Lie algebra $\mathfrak{g}:=\mathrm{Lie}(G)$.

  * (2)

An $\mathrm{Ad}$-invariant symmetric and non-degenerate inner product
$\langle\cdot,\cdot\rangle_{\mathfrak{g}}$ on $\mathfrak{g}$. Recall that this
inner product induces a bundle pseudo-metric
$\langle\cdot,\cdot\rangle_{\mathrm{Ad}(P)}\in\Gamma^{\infty}(\mathcal{M},\mathrm{Ad}(P)^{\ast}\otimes\mathrm{Ad}(P)^{\ast})$
on the adjoint bundle $\mathrm{Ad}(P)=P\times_{\mathrm{Ad}}\mathfrak{g}$,
where $\mathrm{Ad}:G\to\mathrm{Aut}(\mathfrak{g})$ denotes the adjoint
representation of $G$.

Recall that the bundle metric $\langle\cdot,\cdot\rangle_{\mathrm{Ad}(P)}$
induced by the inner product $\langle\cdot,\cdot\rangle_{\mathfrak{g}}$
induces a wedge product on $\mathrm{Ad}$-valued forms, i.e. a family of maps

$\displaystyle\mathrm{tr}(\cdot\wedge\cdot):\Omega^{k}(\mathcal{M},\mathrm{Ad}(P))\times\Omega^{l}(\mathcal{M},\mathrm{Ad}(P))\to\Omega^{k+l}(\mathcal{M})$
(A.19)

for all $k,l\in\mathbb{N}_{0}$. More precisely, we set for all
$\alpha\in\Omega^{k}(\mathcal{M},\mathrm{Ad}(P))$ and
$\beta\in\Omega^{l}(\mathcal{M},\mathrm{Ad}(P))$

$\displaystyle\mathrm{tr}(\alpha\wedge\beta)_{x}$
$\displaystyle(v_{1},\dots,v_{k+l}):=$ (A.20)
$\displaystyle\frac{1}{k!l!}\sum_{\sigma\in\mathfrak{S}^{k+l}}\mathrm{sign}(\sigma)\langle\alpha_{x}(v_{\sigma(1)},\dots,v_{\sigma(k)}),\beta_{x}(v_{\sigma(k+1)},\dots,v_{\sigma(k+l)})\rangle_{\mathrm{Ad}(P)_{x}}$

for all $v_{1},\dots,v_{k+l}\in T_{p}\mathcal{M}$ and $x\in\mathcal{M}$. If
$G$ is compact and simple and $\langle\cdot,\cdot\rangle_{\mathfrak{g}}$
positive definite, then $\langle\cdot,\cdot\rangle_{\mathfrak{g}}$ is
necessarily a (negative) multiple of the Killing form, which is the reason for
the notation “$\mathrm{tr}$” used above. Using this, we define “$BF$-theory”
via the action functional

$\displaystyle\mathcal{S}_{\mathrm{BF}}:\Omega^{d-2}(\mathcal{M},\mathrm{Ad}(P))\times\mathcal{C}(P)\to\mathbb{R},$
(A.21)

where $\mathcal{C}(P)\subset\Omega^{1}(P,\mathfrak{g})$ denotes the set of all
connection $1$-forms on $P$, which is an infinite-dimensional affine space
over $\Omega^{1}_{\mathrm{hor}}(P,\mathfrak{g})^{\mathrm{Ad}}$, defined for
all $B\in\Omega^{d-2}(\mathcal{M},\mathrm{Ad}(P))$ and for all connection
$1$-forms $A\in\mathcal{C}(P)$ by

$\displaystyle\mathcal{S}_{\mathrm{BF}}[B,A]:=\int_{\mathcal{M}}\\!\operatorname{tr}(B\wedge
F_{\mathcal{M}}^{A}).$ (A.22)

First of all, let us proof that the action of $BF$-theory is invariant under
gauge transformations:

(Gauge-Invariance of the $BF$-Action)
Let $f\in\mathcal{G}(P)$ be a global gauge transformation. Then

$\displaystyle\mathcal{S}_{\mathrm{BF}}[f\cdot
B,f^{\ast}A]=\mathcal{S}_{\mathrm{BF}}[B,A]$

for all $B\in\Omega^{d-2}(\mathcal{M},\mathrm{Ad}(P))$ and for all
$A\in\mathcal{C}(P)$, where $f\cdot B$ denotes the action of gauge
transformations on forms in $\Omega^{d-2}(\mathcal{M},\mathrm{Ad}(P))$, as
defined previously

###### Proof.

The claim is rather obvious and follows directly from $\mathrm{Ad}$-invariance
of $\langle\cdot,\cdot\rangle_{\mathfrak{g}}$. First of all, let
$\sigma_{f}\in C^{\infty}(P,G)^{G}$ be the map defined by
$f(p)=p\cdot\sigma_{f}(p)$ for all $p\in P$. Then, recall that

$\displaystyle F^{f^{\ast}A}=f^{\ast}F^{A}=\mathrm{Ad}_{\sigma_{f}^{-1}}\circ
F^{A}.$ (A.23)

Furthermore, recall that if we view the curvature as an element in
$\Omega^{2}(\mathcal{M},\mathrm{Ad}(P))$, then we have equivalently

$\displaystyle F_{\mathcal{M}}^{f^{\ast}A}=f\cdot F_{\mathcal{M}}^{A}.$ (A.24)

Now, using the definition of the Lagrangian of $BF$-theory, i.e.

$\displaystyle\operatorname{tr}(B$ $\displaystyle\wedge
F_{\mathcal{M}}^{A})_{x}(v_{1},\dots,v_{d})=$ (A.25)
$\displaystyle=\frac{1}{2!(d-2)!}\sum_{\sigma\in\mathfrak{S}^{d}}\mathrm{sign}(\sigma)\langle
B_{x}(v_{\sigma(1)},\dots,v_{\sigma(d-2)}),F_{\mathcal{M},x}^{A}(v_{\sigma(d-1)},v_{\sigma(d)})\rangle_{\mathrm{Ad}(P)_{x}}$

for all $x\in\mathcal{M}$ and for all $v_{1},\dots,v_{d}\in T_{x}\mathcal{M}$,
the claim follows from the fact that the $\mathrm{Ad}(P)$-bundle metric is
gauge-invariant, i.e.

$\displaystyle\langle
f\cdot[p,v],f\cdot[p,w]\rangle_{\mathrm{Ad}(P)_{x}}=\langle[p,v],[p,w]\rangle_{\mathrm{Ad}(P)_{x}}$
(A.26)

for all $x\in\mathcal{M}$, for all $p\in P_{x}$ and for all
$[p,v],[p,w]\in\mathrm{Ad}(P)$ as well as the fact that the action of $f$ on
$\mathrm{Ad}(P)$-valued forms is defined point-wise. ∎

Besides the gauge-symmetry, it turns out that $BF$-theory has also another
type of symmetry thanks to the Bianchi-identity, which can be interpreted as
some kind of translational invariance. For this, we need the following
preliminary Lemma:

(Leibniz Rule for Bundle-Valued Forms)
Let $(V,\rho)$ be a finite-dimensional real representation and
$E:=P\times_{\rho}V$ be the associated vector bundle. Furthermore, let
$\langle\cdot,\cdot\rangle_{E}$ denote the bundle metric on $E$ induced by a
$V$-invariant symmetric and non-degenerate inner product on $V$. Then

$\displaystyle\mathrm{d}(\mathrm{tr}(\alpha\wedge\beta))=\mathrm{tr}(\mathrm{d}_{A}\alpha\wedge\beta)+(-1)^{k}\mathrm{tr}(\alpha\wedge\mathrm{d}_{A}\beta)$

for all $\alpha\in\Omega^{k}(\mathcal{M},E)$ and for all
$\beta\in\Omega^{l}(\mathcal{M},E)$.

###### Proof.

We prove the claim locally: Let $\\{e_{a}\\}_{a}\subset\Gamma(U,E)$ be a local
frame of $E$ defined on some open subset $U\subset\mathcal{M}$. Then we can
write $\alpha\in\Omega^{k}(\mathcal{M},E)$ and
$\beta\in\Omega^{l}(\mathcal{M},E)$ in this frame as
$\alpha|_{U}=\sum_{a}\alpha^{a}\otimes e_{a}$ and
$\beta|_{U}=\sum_{a}\beta^{a}\otimes e_{a}$ for real-valued coordinate forms
$\alpha^{a}\in\Omega^{k}(U)$ and $\beta^{a}\in\Omega^{l}(U)$. Hence, we can
write the left-hand side of the equation we would like to prove as

$\displaystyle\mathrm{d}$
$\displaystyle(\mathrm{tr}(\alpha\wedge\beta))|_{U}=\mathrm{d}\bigg{(}\sum_{a,b}(\alpha^{a}\wedge\beta^{b})\langle
e_{a},e_{b}\rangle_{E}\bigg{)}=$ (A.27)
$\displaystyle=\sum_{a,b=1}(\mathrm{d}\alpha^{a}\wedge\beta^{b}+(-1)^{k}\mathrm{d}\alpha^{a}\wedge\mathrm{d}\beta^{b})\langle
e_{a},e_{b}\rangle_{E}+(-1)^{k+l}\sum_{a,b=1}\alpha^{a}\wedge\beta^{b}\wedge\mathrm{d}\langle
e_{a},e_{b}\rangle_{E},$

where we used the Leibniz rule for real-valued forms and the fact that
$\langle e_{a},e_{b}\rangle\in C^{\infty}(U,\mathbb{R})=\Omega^{0}(U)$. In
order to write down the left-hand side, recall that by definition of
$\mathrm{d}_{A}$, we have that

$\displaystyle\mathrm{d}_{A}\alpha|_{U}:$
$\displaystyle=\sum_{a=1}(\mathrm{d}\alpha^{a}\otimes
e_{a}+(-1)^{k}\alpha^{a}\wedge\nabla^{A}e_{a})=$ (A.28)
$\displaystyle=\sum_{a=1}(\mathrm{d}\alpha^{a}+(-1)^{k}\sum_{c=1}(\alpha^{c}\wedge{\omega^{a}}_{c}))\otimes
e_{a},$

where we introduced the local connection forms
${\omega^{a}}_{b}\in\Omega^{1}(U)$ corresponding to the connection
$\nabla^{A}$, i.e. we write $\nabla^{A}e_{a}=\sum_{b=1}{\omega^{b}}_{a}\otimes
e_{b}$. We get a similar expression for $\mathrm{d}_{A}\beta$. Using this, we
can write the right-hand side as

$\displaystyle(\mathrm{tr}(\mathrm{d}_{A}\alpha$
$\displaystyle\wedge\beta)+(-1)^{k}\mathrm{tr}(\alpha\wedge\mathrm{d}_{A}\beta))|_{U}=$
(A.29) $\displaystyle=$
$\displaystyle\sum_{a,b=1}(\mathrm{d}\alpha^{a}\wedge\beta)\langle
e_{a},e_{b}\rangle_{E}+(-1)^{k}\sum_{a,b,c=1}(\alpha^{c}\wedge{\omega^{a}}_{c}\wedge\beta^{b})\langle
e_{a},e_{b}\rangle_{E}+$
$\displaystyle+(-1)^{k}\sum_{a,b=1}(\alpha^{a}\wedge\mathrm{d}\beta^{b})\langle
e_{a},e_{b}\rangle_{E}+(-1)^{k+l}\sum_{a,b,c=1}(\alpha^{a}\wedge\beta^{c}\wedge{\omega^{b}}_{c})\langle
e_{a},e_{b}\rangle_{E}.$

Comparing (A.27) and (A.29), we see that the claimed equality is fulfilled, if
and only if

$\displaystyle\sum_{a,b=1}\alpha^{a}\wedge\beta^{b}\wedge\mathrm{d}\langle
e_{a},e_{b}\rangle_{E}\stackrel{{\scriptstyle!}}{{=}}\sum_{a,b,c=1}(\alpha^{c}\wedge\beta^{b}\wedge{\omega^{a}}_{c}+\alpha^{a}\wedge\beta^{c}\wedge{\omega^{b}}_{c}))\langle
e_{a},e_{b}\rangle_{E}.$ (A.30)

Now, recall that the covariant derivative $\nabla^{A}$ is compatible with the
induced bundle metric, i.e.

$\displaystyle\mathrm{d}(\langle
e_{a},e_{b}\rangle_{E})=\langle\nabla^{A}e_{a},e_{b}\rangle_{E}+\langle
e_{a},\nabla^{A}e_{b}\rangle_{E}.$ (A.31)

Using the connection forms ${\omega^{a}}_{b}$ of $\nabla^{A}$, this can be
written as

$\displaystyle\mathrm{d}(\langle
e_{a},e_{b}\rangle_{E})=\sum_{c=1}{\omega^{c}}_{a}\langle
e_{c},e_{b}\rangle_{E}+\sum_{c=1}{\omega^{c}}_{b}\langle
e_{a},e_{c}\rangle_{E}.$ (A.32)

Plugging this into the left-hand side of Equation (A.30), we get the right-
hand side, as required. ∎

(Translational Invariance of $BF$-Theory)
For all $B\in\Omega^{d-2}(\mathcal{M},\mathrm{Ad}(P))$, for all
$A\in\mathcal{C}(P)$ and for all
$\eta\in\Omega^{d-3}(\mathcal{M},\mathrm{Ad}(P))$ it holds that

$\displaystyle\mathcal{S}_{\mathrm{BF}}[B+\mathrm{d}_{A}\eta,A]=\mathcal{S}_{\mathrm{BF}}[B,A].$

###### Proof.

Using Lemma A.1.2 as well as the Bianchi identity
$\mathrm{d}_{A}F^{A}_{\mathcal{M}}=0$, we have that

$\displaystyle\mathcal{S}_{\mathrm{BF}}[B+\mathrm{d}_{A}\eta,A]$
$\displaystyle=\int_{\mathcal{M}}\\!\operatorname{tr}((B+\mathrm{d}_{A}\eta)\wedge
F_{\mathcal{M}}^{A})=\int_{\mathcal{M}}\\!\operatorname{tr}(B\wedge
F_{\mathcal{M}}^{A})+\int_{\mathcal{M}}\\!\operatorname{tr}(\mathrm{d}_{A}\eta\wedge
F_{\mathcal{M}}^{A})=$ (A.33)
$\displaystyle=\int_{\mathcal{M}}\\!\operatorname{tr}(B\wedge
F_{\mathcal{M}}^{A})+\int_{\mathcal{M}}\\!\mathrm{d}(\mathrm{tr}(\eta\wedge
F_{\mathcal{M}}^{A}))+(-1)^{d}\int_{\mathcal{M}}\\!\mathrm{tr}(\eta\wedge\underbrace{\mathrm{d}_{A}F_{\mathcal{M}}^{A}}_{=0})=$
$\displaystyle=\int_{\mathcal{M}}\\!\operatorname{tr}(B\wedge
F_{\mathcal{M}}^{A})=\mathcal{S}_{\mathrm{BF}}[B,A],$

where we used Stoke’s Theorem in the last to last step as well as the fact
that $\partial\mathcal{M}=\emptyset$. ∎

To end the discussion, let us derive the equations of motion of topological
$BF$-theory:

(Equation of Motions of $BF$-Theory)
The equation of motions corresponding to the $BF$-action are given by

$\displaystyle F_{\mathcal{M}}^{A}=0\hskip 28.45274pt\text{and}\hskip
28.45274pt\mathrm{d}_{A}B=0$

for $B\in\Omega^{d-2}(\mathcal{M},\mathrm{Ad}(P))$ and $A\in\mathcal{C}(P)$.

###### Proof.

Let us first calculate the critical points of the $BF$-action with respect to
the $B$-field. Let $\alpha\in\Omega^{d-2}(\mathcal{M},\mathrm{Ad}(P))$, then

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}\mathcal{S}_{\mathrm{BF}}[B+t\alpha,A]=\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}\int_{\mathcal{M}}\\!t\cdot\operatorname{tr}(\alpha\wedge
F_{\mathcal{M}}^{A})=\int_{\mathcal{M}}\\!\operatorname{tr}(\alpha\wedge
F_{\mathcal{M}}^{A})\stackrel{{\scriptstyle!}}{{=}}0,$ (A.34)

which is fulfilled for all $\alpha$ if and only if $F_{\mathcal{M}}^{A}=0$, by
non-degeneracy. To derive the second equation of motion, we use the fact that

$\displaystyle
F_{\mathcal{M}}^{A+t\alpha}=F_{\mathcal{M}}^{A}+t(\mathrm{d}_{A}\alpha)+\mathcal{O}(t^{2}).$
(A.35)

for all $\alpha\in\Omega^{2}(\mathcal{M},\mathrm{Ad}(P))$ [Ham17, p.418]. As a
consequence, using Lemma A.1.2 as well as Stoke’s Theorem, we have that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}\mathcal{S}_{\mathrm{BF}}[B,A+t\alpha]=\int_{\mathcal{M}}\\!\operatorname{tr}(B\wedge\mathrm{d}_{A}\alpha)=-\int_{\mathcal{M}}\\!\operatorname{tr}(\mathrm{d}_{A}B\wedge\alpha)\stackrel{{\scriptstyle!}}{{=}}0,$
(A.36)

which is satisfies for all $\alpha\in\Omega^{1}(\mathcal{M},\mathrm{Ad}(P))$
if and only if $\mathrm{d}_{A}B=0$. ∎

###  A.2 Cellular Complexes, Triangulations and Pseudomanifolds

In the second part of this appendix, we give a brief overview of some
important mathematical tools for discretization, which is very important for
background-independent approaches of quantum gravity, but also in many other
areas of physics and mathematics. Some useful references for the following
chapter are [Lee11, Ch.5] and [Mun18, Ch.1].

####  A.2.1 Cellular Decompositions and CW-Complexes

Let us start by briefly discussing the general concept of cellular
decompositions and CW-complexes. First of all, let us introduce the following
terminology: An “open $k$-cell” is any topological space homeomorphic to the
open unit ball

$\displaystyle
B^{k}:=B_{1}(0):=\\{x\in\mathbb{R}^{k}\mid\|x\|<1\\}\subset\mathbb{R}^{k}.$
(A.37)

equipped with the standard topology. A closed $k$-cell is any topological
space homeomorphic to the closed unit ball $\overline{B^{k}}$. The number $k$
is also called the “degree” of the cell. Of course, every open ball is an open
cell, but there are many more examples: For example, every compact and convex
subset with non-empty interior of $\mathbb{R}^{k}$ is a closed $k$-cell and
its interior is an open $k$-cell. In particular, every closed interval is a
closed $1$-cell and every compact region in the plane bounded by a regular
polygon is a closed 2-cell. The idea of a “cellular decomposition” of some
topological space is that the space can be “build up” by gluing cells along
their boundaries in order to obtain cells of increasing dimensions. In
general, a cellular decomposition of some topological space is a collection of
disjoint cells, such their union is equivalent to the given space. However, we
need some more restrictions, because an arbitrary partition will tell us
nothing about the topology of our given space. Therefore, we assume that the
boundary of each cell is attached in some reasonable way to lower-dimensional
cells:

(Cellular Decompositions and Complexes)
Let $(X,\mathcal{T})$ be a topological Hausdorff space. A cellular
decomposition of $X$ is a partition (=open cover consisting of non-empty
disjoint subsets) $(c_{i})_{i\in I}$ consisting of open cells of different
dimensions, such that for every cell $c_{i}$ of dimension $k\geq 1$, there is
a continuous map $f_{i}:\overline{B^{k}}\to X$ such that $f_{i}|_{B^{k}}$ is a
homeomorphism between $B^{k}$ and $c_{i}$ and such that $f_{i}$ maps the
boundary $\partial\overline{B^{k}}$ in a union of cells in $(c_{i})_{i\in I}$
of dimension $<k$.

A topological Hausdorff space together with a cellular decomposition is called
“cellular complex”. Such a complex is called “finite”, if there are only
finitely many cells and “locally finite”, if the collection of open cells is
locally finite. A very general and important type of cellular decompositions
are CW-complexes, introduced by J. H. C. Whitehead [Whi49], which are central
in algebraic topology. They are defined as follows:

(CW-Complex)
Let $(X,\mathcal{T})$ be a topological Hausdorff space with a cellular
decomposition $(c_{i})_{i\in I}$. We call it a “CW-complex”, if the following
additional conditions are satisfied:

  * (1)

For every cell $c_{i}$ of dimension $k\geq 1$, there is a continuous map
$f_{i}:\overline{B^{k}}\to X$ such that $f_{i}|_{B^{k}}$ is a homeomorphism
between $B^{k}$ and $c_{i}$ and such that $f_{i}$ maps the boundary
$\partial\overline{B^{k}}$ into a finite union of cells in $(c_{i})_{i\in I}$
of dimension $<k$.

  * (2)

A subset $M\subset X$ is closed if and only if $M\cap f_{i}(\overline{B^{k}})$
is closed for all $i\in I$.

The name “CW-complex” comes from “closure-finiteness” (condition (1)) and
“weak topology” (condition (2)). Note that every locally finite cellular
complex is in particular a CW-complex, as proven in [Lee11, p.132f.].

####  A.2.2 Abstract Simplicial Complexes and Their Geometric Realization

Simplicial complexes are a particular class of CW-complexes consisting only of
“simplices”, which are higher-dimensional analogue of triangles and
tetrahedra. These complexes can either be defined abstractly or embedded in
Euclidean space. Let us start with the abstract definition:

(Abstract Simplicial Complex)
Let $\mathcal{V}$ be a finite set. Furthermore, let
$\Delta\subset\mathcal{P}(\mathcal{V})\textbackslash\emptyset$ be a collection
of non-empty finite subsets of $\mathcal{V}$ satisfying the following two
properties:

  * (1)

$\Delta$ contains all singletons, i.e. $\\{v\\}\in\Delta$ for all
$v\in\mathcal{V}$.

  * (2)

For any non-empty $\tau\subset\sigma$ for $\sigma\in\Delta$ it holds that
$\tau\in\Delta$.

We call the pair $(\mathcal{V},\Delta)$ “(abstract) simplicial complex with
vertex set $\mathcal{V}$”.

If it is clear from the context, a simplicial complex $(\mathcal{V},\Delta)$
is often simply denoted by $\Delta$. In addition to the above definition, let
us fix the following terminology:

  * •

An element $v\in\mathcal{V}$ is called “vertex” and an element
$\sigma\in\Delta$ is called “simplex”. Any non-empty subset
$\tau\subset\sigma$ is called “face of $\sigma$”.

  * •

The “dimension” of a simplex $\sigma\in\Delta$ is the number
$d\in\mathbb{N}_{0}$ defined by $d:=|\sigma|-1$. A $d$-dimensional simplex is
also called $d$-simplex. A $k$-dimensional face of some $d$-simplex $\sigma$
is called “$k$-face of $\sigma$”. Let us denote the set of all $d$-simplices
by $\Delta_{d}$. Vertices are by definition $0$-simplices, i.e.
$\mathcal{V}=\Delta_{0}$. The “dimension of a simplicial complex $\Delta$” is
the maximal number $d\in\mathbb{N}_{0}$ such that $\Delta_{d}\neq\emptyset$.

  * •

The collection of all simplices with dimension smaller equal to some
$k\in\\{0,\dots,d\\}$ is called the “$k$-skeleton” of the complex $\Delta$.

  * •

A “subcomplex of $\Delta$” is a subset $\mathcal{S}\subset\Delta$ such that
$(\bigcup\mathcal{S},\mathcal{S})$ is again an abstract simplicial complex.

  * •

A simplicial complex $\Delta$ is called “finite”, if $\Delta$ is a finite set.
More generally, we say that $\Delta$ is “locally finite”, if every vertex in
$\Delta$ only belongs to finitely many simplices.

Let now $S\subset\Delta$ be some subset of the abstract simplicial complex
$\Delta$, which is not necessarily a subcomplex, i.e. it does not have to be
an abstract simplicial complex by itself. Then, one often uses the following
terminology:

  * (1)

The “closure $\mathrm{Cl}_{\Delta}(S)$ of $S$” is the smallest subcomplex of
$\Delta$ containing $S$, i.e.

$\displaystyle\mathrm{Cl}_{\Delta}(S):=\\{\sigma\in\Delta\mid\exists\tau\in
S:\sigma\subset\tau\\}.$ (A.38)

As an example, the closure of a single $k$-simplex $\sigma$ is the set
consisting of the simplex itself together with all its faces. Note that $S$ is
a subcomplex of $\Delta$ if and only if $\mathrm{Cl}_{\Delta}(S)=S$.

  * (2)

The “star” of a single simplex $\sigma\in\Delta$ is defined to be set of all
simplices in $\Delta$ having $\sigma$ as face. More explicitly:

$\displaystyle\mathrm{St}_{\Delta}(\sigma):=\\{\tau\in\Delta\mid\sigma\subset\tau\\}.$
(A.39)

The “star of $S$” is then the union of the stars of all its simplices. Note
that the star is in general just a collection of sets and not a subcomplex.
Hence, one often defines the “closed star”, which is the subcomplex
$\mathrm{Cl}_{\Delta}(\mathrm{St}_{\Delta}(S))$. Note also that some authors
define the star directly in this way.

  * (3)

The “link of $S$” is defined to be
$\mathrm{Lk}_{\Delta}(S):=\mathrm{Cl}_{\Delta}(\mathrm{St}_{\Delta}(S))\textbackslash\mathrm{St}_{\Delta}(\mathrm{Cl}_{\Delta}(S))$.
If $\sigma\in\Delta$ is some single simplex, then its links is hence given by

$\displaystyle\mathrm{Lk}_{\Delta}(\sigma)=\\{\tau\in\mathrm{Cl}_{\Delta}(\mathrm{St}_{\Delta}(\sigma))\mid\tau\cap\sigma=\emptyset\\}=\\{\tau\in\Delta\mid\tau\cup\sigma\in\Delta\text{
and }\tau\cap\sigma=\emptyset\\}.$ (A.40)

The link of some subset $\mathcal{S}$ is again a subcomplex of $\Delta$.

The following figure shows a $2$-dimensional abstract simplicial complex
$\Delta$ as well as the star, closed star and link of a vertex $v$:

Figure A.1: A $2$-dimensional simplicial complex and the star
$\mathrm{St}_{\Delta}(\\{v\\})$, closed star
$\mathrm{Cl}_{\Delta}(\mathrm{St}_{\Delta}(\\{v\\}))$ and link
$\mathrm{Lk}_{\Delta}(\\{v\\})$ of a vertex $v$ in $\Delta$ drawn in blue.

Abstract simplicial complexes are in general purely combinatorial objects. In
order to equip them with a topology, we need to notion of a “geometric
realization”. Let us discuss this in more details: Let
$v_{0},\dots,v_{k}\in\mathbb{R}^{n}$ be vectors. They are called “affinely
independent”, if they are not contained in any $<k$-dimensional affine
subspace of $\mathbb{R}^{n}$, or in other words, if
$v_{1}-v_{0},\dots,v_{k}-v_{0}$ are linearly independent vectors.

(Euclidean Simplices)
Let $v_{0},\dots,v_{k}\in\mathbb{R}^{n}$ be affinely-independent vectors. The
“Euclidean $k$-simplex spanned by $\\{v_{0},\dots,v_{k}\\}$” is the convex
hull of these vectors, i.e.

$\displaystyle\Delta^{k}:=\bigg{\\{}\sum_{i=0}^{k}t_{i}v_{i}\,\bigg{|}\,\sum_{i=0}^{k}t_{i}=1\text{
and }0\leq t_{i}\leq 1\bigg{\\}}.$

Note that every Euclidean $k$-simplex is a closed $k$-cell, i.e. a topological
space (with its subspace topology induced from $\mathbb{R}^{n}$) homeomorphic
to the closed unit ball, as defined in the previous section. The vectors
$v_{0},\dots,v_{k}\in\mathbb{R}^{n}$ of a simplex $\Delta^{k}$ are called
“vertices”. A “face” of some $k$-simplex $\sigma$ is a simplex spanned by a
non-empty subset of vertices of $\sigma$. Next, let us define Euclidean
simplicial complexes in the following way [Lee11, p.149]

(Euclidean Simplicial Complex)
A Euclidean simplicial complex is a collection $\Delta$ of Euclidean simplices
in $\mathbb{R}^{n}$ such that

  * (1)

Every face of a simplex in $\Delta$ is again in $\Delta$.

  * (2)

The intersection of two simplices is either empty or the face of both
simplexes.

  * (3)

$\Delta$ is locally finite.

The set $\mathcal{P}:=\bigcup_{\Delta\in\mathcal{K}}\Delta$ is called
polyhedron with simplicial decomposition $\Delta$.

Obviously, every Euclidean simplicial complex can in particular be understood
as an abstract simplicial complex, where an abstract $k$-simplex $\sigma$
corresponding to a Euclidean $k$-simplex spanned by
$v_{0},\dots,v_{k}\in\mathbb{R}^{n}$ is just the set
$\sigma=\\{v_{0},\dots,v_{k}\\}$. The vertex set of this abstract simplicial
complex is hence given by the union of all the vertices of the Euclidean
$k$-simplices. Furthermore, every Euclidean simplicial complex is a CW-complex
of its polyhedron consisting of all the interiors of its simplices. In order
to relate an abstract simplicial complex to a Euclidean one, we need the
following notion of equivalence:

(Isomorphism of Abstract Simplicial Complexes)
Let $(\mathcal{V}_{1},\Delta_{1})$ and $(\mathcal{V}_{2},\Delta_{2})$ be two
abstract simplicial complexes. A pair of maps
$(\varphi:\Delta_{1}\to\Delta_{2},\varphi_{0}:\mathcal{V}_{1}\to\mathcal{V}_{2})$
is called “simplicial map”, if it is of the form
$\varphi(\\{v_{0},\dots,v_{k}\\})=\\{\varphi_{0}(v_{0}),\dots,\varphi_{0}(v_{k})\\}$,
i.e. $\varphi$ maps simplices of $\Delta_{1}$ into simplices of $\Delta_{2}$.
If both $\varphi$ and $\varphi_{0}$ are bijective, then
$(\varphi^{-1},\varphi_{0}^{-1})$ is also a simplicial map and we call
$(\varphi,\varphi_{0})$ “isomorphism”.

If it clear from the context, we usually just call $\varphi$ simplicial map.
Let $\Delta$ be some abstract simplicial complex. We call some Euclidean
simplicial complex, whose underlying abstract simplicial complex is isomorphic
to $\Delta$, “geometric realization of $\Delta$”. We denote the corresponding
polyhedron by $|\Delta|$. Note that $|\Delta|$ has a topology, namely the
topology induced from $\mathbb{R}^{n}$. We also say that $\Delta$ is a
“triangulation of the topological space $|\Delta|$”. It turns out that such a
geometric realization always exists:

Every finite abstract simplicial complex $\Delta$ has a geometric realization
$|\Delta|$ embedded in $\mathbb{R}^{n}$ for sufficient high $n\in\mathbb{N}$
(in fact, one can choose $n=2\mathrm{dim}(\Delta)+1$), i.e. a Euclidean
simplicial complex, which is isomorphic to it.

###### Proof.

See for example Proposition 5.41 in [Lee11, p.154].∎

######  Remarks .

  *   * (a)

For two Euclidean simplicial complexes $\Delta_{1}$ and $\Delta_{2}$, it turns
out that a map $\varphi:\Delta_{1}\to\Delta_{2}$ is a simplicial isomorphism
if and only if $\varphi$ induces a homeomorphism between $|\Delta_{1}|$ and
$|\Delta_{2}|$ (see Lemma 2.8. in [Mun18]). Hence, the geometric realization
of an abstract simplicial complex is unique up to homeomorphism.

  * (b)

The result can be extended to finite-dimensional, countable and locally finite
simplicial complexes. There also more general constructions for arbitrary
abstract simplicial complexes, however, in this case we have to extend the
meaning of geometric realization.

∎

As mentioned in the Remark above, if the underlying abstract simplicial
complexes of two Euclidean complexes are simplicial isomorphic, then there
corresponding polyhedra are homeomorphic. Since every abstract simplicial
complex has a geometric realization, we can state the following:

If two finite abstract simplicial complexes $\Delta_{1}$ and $\Delta_{2}$ are
simplicial isomorphic, then their geometric realizations are homeomorphic.

There is also a stronger notion of isomorphism for abstract simplicial
complexes, namely the concept of “piecewise-linear maps”. In order to state
the definition, we need the concept of a “subdivision”: Let $\Delta_{1}$ and
$\Delta_{2}$ be two abstract simplicial complexes. Then $\Delta_{1}$ is called
a “(simplicial) subdivision” of $\Delta_{2}$, if $|\Delta_{1}|$ and
$|\Delta_{2}|$ are homeomorphic and if every simplex in $\Delta_{1}$ is
contained in a simplex of $\Delta_{2}$. Using this, PL-homeomorphisms are
defined as follows:

(PL-Homeomorphism)
Let $\Delta_{1}$ and $\Delta_{2}$ be two abstract simplicial complexes. A map
$\varphi:\Delta_{1}\to\Delta_{2}$ is called “piecewise-linear” if there exits
subdivisions $\Delta^{\prime}_{1}$ and $\Delta^{\prime}_{2}$, such that
$\varphi:\Delta^{\prime}_{1}\to\Delta^{\prime}_{2}$ is a simplicial map. If
this induced map is in addition a simplicial isomorphism, then we say that
$\varphi$ is “PL-homeomorphism” and we call $\Delta_{1}$ and $\Delta_{2}$ “PL-
homeomorphic”.

In other words, two abstract simplicial complexes are PL-homeomorphic if and
only if there exist (simplicial) subdivisions of them, which are simplicial
isomorphic. Of course, every simplicial map is a PL-map, but not vice versa.

######  Remark .

Two PL-homeomorphic abstract simplicial complexes do always have homeomorphic
geometric realizations. However, the reverse is in general not true, i.e.
homeomorphic realizations does not imply that the corresponding complexes are
PL-homeomorphic. As we will see below, even if the geometric realization is a
manifold it turns out that in sufficient high dimensions there a many
inequivalent PL-structures.∎

Let us briefly discuss the notion of a “boundary of a simplicial complex”,
which will be used in the following when talking about a triangulation of a
manifold with boundary. The notion of a boundary only makes sense for “pure”
simplicial complexes: A simplicial complex is called “pure”, if every simplex
$\sigma\in\Delta$ of dimension $<d$ is the face of some $d$-simplex. Using
this, the boundary is defined as follows:

(Boundary of a Pure Simplicial Complex)
Let $\Delta$ be a pure abstract simplicial complex. The “boundary of $\Delta$”
is then defined to be

$\displaystyle\partial\Delta:=\mathrm{Cl}_{\Delta}(\\{\tau\in\Delta_{k}\mid
k\leq d-1\land\exists!\sigma\in\Delta_{d}:\tau\subset\sigma\\}).$

In other words, it is the smallest subcomplex consisting of all $k$-simplices
belonging to exactly one $d$-simplex.

The boundary is by construction a $(d-1)$-dimensional simplicial complex. Note
that the boundary complex does not have to be connected, even in the case
$\Delta$ is connected.

If $\sigma$ is a $d$-simplex, then we can assign an “orientation” to it by
ordering its vertices (up to even permutations). In general, there are always
two possible orientations of a $d$-simplex. A choice of orientation induces an
orientation of its faces. As an example, let $[v_{0},\dots,v_{d}]$ be the
orientation of a $d$-simplex, i.e. the equivalence class of the ordering of
its vertices $v_{0},\dots,v_{d}$ up to even permutation. Then the $(d-1)$-face
$\sigma^{\hat{i}}$ with vertices $v_{0},\dots,\hat{v_{i}},\dots,v_{d}$ gets
the orientation

$\displaystyle(-1)^{i}[v_{0},\dots,\hat{v_{i}},\dots,v_{d}],$ (A.41)

where a factor of $(-1)$ in front of an equivalence class means to take the
opposite orientation. Using this, we can define “oriented simplicial
complexes” for “non-branching” simplicial complexes, in which every
$(d-1)$-simplex is the face of exactly one or two $d$-simplices.

(Orientation of a Non-Branching Simplicial Complex)
Let $\Delta$ be a non-branching abstract simplicial complex. An “orientation
of $\Delta$” is a choice of an orientation for each $d$-simplex
$\sigma\in\Delta_{d}$, such that each $(d-1)$-simplex, which is the face of
exactly two $d$-simplices, gets different induced orientations from these two
$d$-simplices. If such a choice exists, then we say that $\Delta$ is
“orientable”, otherwise, we say that $\Delta$ is “non-orientable”.

Note that for orientable simplicial complexes there are always two choices of
an orientation. Last but not least, let us define the Euler characteristic of
an abstract simplicial complex. This quantity is for example used to classify
$2$-dimensional manifolds.

(Euler Characteristic)
Let $\Delta$ be a $d$-dimensional abstract simplicial complex. Then the “Euler
characteristic of $\Delta$” is the integer $\chi(\Delta)$ defined by

$\displaystyle\chi(\Delta)=\sum_{i=0}^{d}(-1)^{i}|\Delta_{i}|.$

The Euler characteristic is a topological invariant, meaning that the Euler
characteristic is the same for two abstract simplicial complexes with
homeomorphic and in fact even homotopy equivalent geometric realizations. More
generally, one can define the Euler characteristic for finite CW-complexes.
Also in this case, it is a topological invariant [Lee11, p.373f.].

####  A.2.3 Triangulations and Piecewise-Linear Manifolds

After having introduced the notion of simplicial complexes, we are now in the
position to define the concept of a “triangulation” of a manifold, which
abstracts the idea of dividing a space into triangles. In full generality, a
triangulation of a topological space is defined as follows:

(Triangulation)
Let $(X,\mathcal{T})$ be a topological Hausdorff space. A pair
$(\Delta,\varphi)$ consisting of an abstract simplicial complex $\Delta$ and a
homeomorphism $\varphi:|\Delta|\to X$ is called a “triangulation of $X$”.

######  Remark .

A triangulation of a $d$-dimensional manifold $\mathcal{M}$ is always pure and
non-branching. Hence, it makes sense to define the boundary complex and to
talk about orientability of the corresponding simplicial complex of the
triangulation. If $\mathcal{M}$ has a boundary, then the boundary complex
triangulates the boundary $\partial\mathcal{M}$. Furthermore, if a
triangulable manifold $\mathcal{M}$ is orientable then also its triangulation
is and a choice of orientation of the corresponding complex results into a
choice of orientation of $\mathcal{M}$.∎

For manifolds, there is a slightly stronger notion of triangulations, which is
usually called a “piecewise-linear structure”. It can be defined either using
a special type of atlas or using a special type of triangulation. Let us start
with the first definition:

(Piecewise-Linear Manifolds)
Let $\mathcal{M}$ be a $d$-dimensional topological manifold. A “piecewise-
linear structure on $\mathcal{M}$”, or short “PL-structure”, is an atlas in
which all transition functions are piecewise-linear maps.

As already mentioned, the concept of a PL-structure can also be defined via a
specific type of triangulation, which is usually called a “combinatorial
triangulation”:

(Combinatorial Triangulations)
A triangulation $(\Delta,\varphi)$ of a topological manifold possibly with
boundary is called “combinatorial”, if the link of every vertex in $\Delta$ is
PL-homeomorphic to the $(d-1)$-dimensional standard PL-sphere or PL-
homeomorphic to the $(d-1)$-dimensional standard PL-ball.

######  Remarks .

  *   * (a)

The “standard $PL$-sphere of dimension $d$ is defined to be the boundary of a
$(d+1)$-simplex and the “standard $PL$-ball” of dimension $d$ is defined to be
a single $d$-simplex.

  * (b)

Adding the detail that links of vertices are homeomorphic to the standard PL-
sphere or standard PL-ball is actually only relevant in $d=4$, since for
$d\neq 4$, one can show that the $d$-sphere has a unique PL-structure [Sma62].
The case $d=4$ is still an open problem known as “smooth $4$-dimensional
Poincaré conjecture” [FGMW10].

∎

(Combinatorial Triangulations and PL-Manifolds)
Every combinatorial triangulation of a topological manifold $\mathcal{M}$
gives rise to a PL-structure and vice versa.

###### Proof.

For the direction “$\Rightarrow$”, see Corollary 1.16 in [Hud69, p.24f.]. For
direction “$\Leftarrow$” see the Remark in [Hud69, p.26]. ∎

######  Remarks .

  *   * (a)

In order to distinguish combinatorial triangulations to the general concept
defined in Definition A.2.3, the latter one are sometimes called “simplicial
triangulations”.

  * (b)

In general, it is also not to hard to see that the geometric realization of
any simplicial complex with the property that the links of all its vertices
represents spheres or balls is a manifold and in fact, by the proposition
above, a PL-manifold. Hence, such complexes are also often called
“combinatorial manifolds”.

∎

It turns out that we do not have to make a distinction between triangulation
and combinatorial triangulations in sufficient low dimensions, as the
following proposition shows:

All triangulations of a $d$-dimensional manifold with $d\leq 4$ are
combinatorial triangulations.

###### Proof.

For $d=2,3$, the claim follows from the triangulation theorems of Radó and
Moise stated below. For $d=4$, one can show that the claim is equivalent to
the famous “Poincaré conjecture”, which states that any $3$-dimensional closed
and simply-connected $3$-manifold is homeomorphic to the $3$-sphere. This
conjecture is now known to be true, since it follows from “Thurston’s
geometrization conjecture”, which was proven by G. Perelman in 2002/3 [Per02,
Per03a].∎

######  Remark .

It is important to note that this result is in general not true in $d>4$. In
other words, not every triangulation of a manifold with dimension larger then
$4$ has the property that links are spheres. In fact, one can show, by using
“Connon’s double suspension theorem”, that there are triangulations of $S^{5}$
admitting links, which are not even manifolds. [Edw06]∎

For a long time it was an unsolved problem whether all manifolds admit a
piecewise-linear structure or a triangulation. Historically, it was
conjectured by H. Poincaré [Poi99] (translated in modern language) that every
smooth manifold admits a triangulation. In 1926, it was conjectured by H.
Kneser that also all topological manifolds are triangulable [Kne26]. Yet
another famous conjecture in this context is the so-called “Hauptvermutung”
[Ran96] (German for “main conjecture”) by E. Steinitz and H. F. F. Tietze from
1908, which states that any two triangulations of a triangulable topological
space are PL-homeomorphic. Poincaré’s conjecture turns out to be true, as
stated in the theorem below. However, the answer to Kneser’s conjecture as
well as to the Hauptvermutung is now known to be negative. Nevertheless, it
turns out that at least for low dimensions Kneser’s conjecture is true:

(Triangulation Theorems for Manifolds)

  * •

Every $2$\- and $3$-dimensional topological manifold admits a piecewise-linear
structure, which is unique up to piecewise-linear equivalence. The result for
$d=2$ was proven by T. Radó in 1924 [Rad24] and the result for $d=3$ by E. E.
Moise in 1952 [Moi52]. An alternative proof for $d=3$ was shortly after given
by R. H. Bing in 1959 [Bin59].

  * •

Every smooth manifold (and in fact every $C^{1}$-manifold) of arbitrary
dimension admits a unique piecewise-linear structure, which is unique up to
piecewise-linear equivalence. This was proven by S. S. Cairns and J. H. C.
Whitehead [Cai35, Whi40].

In higher dimensions, topological manifolds do not always admit piecewise-
linear structures or triangulations. As an example, in 1982 M. H. Freedman
showed that the $4$-dimensional compact and simply-connected manifold $E_{8}$,
which is constructed using the root lattice of the simple Lie group with the
same name, does not admit any piecewise-linear structure [Fre82]. In 1990, it
was proven by S. Akbulut and J. D. McCarthy that this manifold does not even
admit a triangulation [AM90]. There are also more general results: First of
all, there is a theorem by R. C. Kirby and L. C. Siebenmann that shows that
for every $d\geq 5$, there is a $d$-dimensional topological manifold, which
does not admit any piecewise-linear structure [KS69]. Furthermore, as recently
shown by C. Manolescu, for every $d\geq 5$ there is a $d$-dimensional
topological manifold, which does not even admit a triangulation [Man16b]. For
more details, see also [Man16a]. Last but not least, there are also manifolds,
which admit triangulations, but not PL-structures. Examples include manifolds
of the type $E_{8}\times T^{k}$, where $T^{k}$ denotes the $k$-torus with
$k\geq 1$ [Rud16].

Another nice result concerning the topology of low-dimensional manifolds is
the fact that every topological manifold of dimension $d\leq 3$ admits a
smooth structure, which is unique up to diffeomorphism. Combining this with
the triangulation theorems above shows that in $d\leq 3$, the topological,
smooth and piecewise-linear category of manifolds are equivalent. Hence, we
can work with arbitrary $3$-manifolds using their topological, smooth and
piecewise-linear structure simultaneously. Again, this is in general not true
in higher dimensions: It is a well-known fact that there are topological
manifolds in dimension $d\geq 4$, which do not admit any smooth structure
(e.g. $E_{8}$). Furthermore, it is also well known that there are topological
manifolds of dimension $d=4$ (and in fact only in this dimension as shown by
R. C. Kirby and L. C. Siebenmann [KS85]) with an infinite number of
inequivalent smooth structures and hence also with an infinite number of
inequivalent PL-structures. Examples include the “$K3$-surface” and, as more
recently discovered, the manifold $S^{2}\times S^{2}$ [AD10]. Another subtlety
in higher dimensions is the fact that a PL-structure does not have to come
from a smooth structure. An explicit example for this observation is the
$10$-dimensional “Kervaire manifold”, which admits a PL-structure, but not a
smooth structure [Ker60]. Again, this is only true in sufficient high
dimensions, since one can show that every PL-manifold of dimension $d\leq 7$
has at least one compatible smooth structure and for $d<7$, this smooth
structure is also unique up to diffeomorphism (“Munkres-Hirsch-Mazur
obstruction theory”, [Mun60, HM74]). In other words, for $d\leq 6$, the smooth
and piecewise-linear category are equivalent. Let us stress that the latter
statement does not mean that any manifold of dimension $d\leq 6$ has a unique
smooth structure, but rather that the number of inequivalent smooth structures
is the same as the number of inequivalent PL-structures for $d\leq 6$. More
details can be found in the first chapter of the review [Ben12].

####  A.2.4 Pseudomanifolds

In quantum gravity approaches including a sum over topologies in their
perturbative expansion, like tensor models and group field theories, one often
has to deal with more singular combinatorial objects, which are not manifolds.
Hence, we have to discuss some generalizations. A very important concept in
this context are pseudomanifolds, which are defined as follows:

(Pseudomanifolds)
Let $\Delta$ be a finite abstract $d$-dimensional simplicial complex. We call
its geometric realization $|\Delta|$ a “$d$-dimensional pseudomanifold”, if
the following conditions are fulfilled:

  * (1)

$\Delta$ is “pure”, i.e. every simplex $\sigma\in\Delta$ of dimension $<d$ is
the face of some $d$-simplex.

  * (2)

$\Delta$ is “non-branching”, i.e. every $(d-1)$-simplex is face of exactly one
or two $d$-simplices.

  * (3)

$\Delta$ is “strongly-connected”, i.e. for every two $d$-simplices
$\sigma,\tau\in\Delta_{d}$, there is a sequence of $d$-simplices
$\sigma=\sigma_{1},\sigma_{2},\dots,\sigma_{k}=\tau$ such that
$\sigma_{l}\cap\sigma_{l+1}$ is a $(d-1)$-simplex $\forall l$.

As before, the boundary of a pure simplicial complex is the smallest
subcomplex consisting of all the $k$-simplices with $k\leq d-1$, which belong
to precisely one $d$-simplex. Combining this with property (2), we hence see
that the boundary $\partial\Delta$ is exactly the subcomplex consisting of all
the $(d-1)$-simplices, which are the face of only on $d$-simplex as well as
all their lower-dimensional faces. We call the geometric realization of the
subcomplex $\partial\Delta$ “boundary of the pseudomanifold”. If
$\partial\Delta\neq\emptyset$, then we call $|\Delta|$ “pseudomanifold with
boundary”, otherwise we call $|\Delta|$ “pseudomanifold without boundary”.

One can easily show that every compact, connected and triangulable manifold is
a pseudomanifold. However, the converse is in general not true. As an example,
pseudomanifolds can contain some kind of singularities. An often cited example
is the “pinched torus”, which is a $2$-dimensional pseudomanifold obtained by
identifying two distinct points on the $2$-sphere.

An important class of pseudomanifolds are “normal pseudomanifolds”, which are
defined as follows:

(Normal Pseudomanifolds)
Let $|\Delta|$ be a $d$-dimensional pseudomanifold. We call it “normal” if the
links of all simplices of dimension $k\in\\{1,\dots,d-2\\}$ define themselves
pseudomanifolds.

The crucial condition in the normality-property is strongly-connectedness: In
general, every link of a pseudomanifold is pure and non-branching, but can
fail to be strongly-connected. In fact, they can even fail to be connected at
all. For a proof of this statement, see for example the appendix in [Gur10a].
An example of a non-normal pseudomanifold is the pinched torus mentioned
above, since the link of the vertex, which identifies the two distinct point
on the $2$-sphere, consists of two distinct circles and hence is not
connected. Furthermore, it is important to note that while the boundary of any
pseudomanifold is in general a pseudomanifold without boundary, the boundary
of a normal pseudomanifold does not have to be normal itself, even if the
boundary is connected. An example is discussed in Section 3.2.2.

For a more detailed discussion of pseudomanifolds with or without boundary in
general, see for example the classical textbook [ST80, p.90ff.].

###  A.3 Useful Concepts of Group Theory and Harmonic Analysis

In this chapter of the appendix, we discuss some useful definitions and
concepts of group theory and harmonic analysis. In the first part, we briefly
review the definition of the Haar measure on locally-compact Lie groups as
well as the Theorem of Peter-Weyl. Furthermore, we explain the relation
between invariant elements and intertwiners mentioned in our discussion of
spin networks. Last but not least, we introduce the non-commutative group
Fourier transform, which we have mentioned in our discussion of different
representations of the Boulatov fields, and review some of its properties.

####  A.3.1 The Haar Measure of Compact Lie Groups

Let $G$ be a locally compact Lie group. Then there is a very natural choice of
measure on $G$, which respects the group structure and is usually called the
“Haar measure”. Historically, this measure was introduced by A. Haar in 1933
[Haa33] and A. Weil [Wei40] proved a general existence and uniqueness theorem,
which states the following292929The original proof by A. Weil uses the axiom
of choice in form of Tychonoff’s theorem. However, it was shown by H. Cartan
that this can also be avoided [Car40].:

(Haar-Measure: Existence and Uniqueness)
Let $G$ be a locally-compact Lie group. Then there exists a unique (up to a
positive multiplication constant) non-trivial Radon measure
$\mu:\mathcal{B}(G)\to[0,\infty]$ on the Borel-$\sigma$-algebra
$\mathcal{B}(G)$, which is “left-invariant”, i.e. $\mu(gM)=\mu(M)$ for all
$g\in G$ and $M\in\mathcal{B}(G)$, where $gM$ denotes the set $gM:=\\{gm\mid
m\in M\\}$. This measure is usually called the “(left-invariant) Haar
measure”.

The integration with respect to the Haar measure is usually simply denoted by

$\displaystyle\int_{G}\,f(g)\,\mathrm{d}g:=\int_{G}\,f(g)\,\mathrm{d}\mu(g).$
(A.42)

Note that left-invariance of the Haar measure implies on the level of
integration that

$\displaystyle\int_{G}f(g)\,\mathrm{d}g=\int_{G}f(hg)\,\mathrm{d}g$ (A.43)

for all measurable functions $f:G\to\mathbb{C}$ and for all $h\in G$.
Similarly, one can show the existence of a right-invariant Radon measure on
locally-compact groups, i.e. a measure $\mu:\mathcal{B}(G)\to[0,\infty]$
satisfying $\mu(Mg)=\mu(M)$ for all $g\in G$ and $M\in\mathcal{B}(G)$, called
the “right-invariant Haar measure”. If the left- and right-invariant Haar
measures of some group $G$ agree, we get a well-defined “bi-invariant” Haar
measure. Groups with this property are usually called “unimodular”. For such
groups, the Haar measure satisfies in addition $\mu(M)=\mu(M^{-1})$, where
$M^{-1}:=\\{m^{-1}\mid m\in M\\}$ for all $M\in\mathcal{B}(G)$, or in other
words,

$\displaystyle\int_{G}f(g)\,\mathrm{d}g=\int_{G}f(g^{-1})\,\mathrm{d}g$ (A.44)

for all functions $f:G\to\mathbb{C}$. This is obviously the case for abelian
groups as well as for discrete groups, but it turns out that it is also the
case for compact Lie groups. Another property of the Haar measure of compact
groups is that it is finite, i.e. $\mu(G)<\infty$, and hence, we can normalize
the Haar measure such that $\mu(G)=1$, or in other words,
$\int_{G}\,\mathrm{d}g=1$.

######  Examples .

  *   * (1)

If $G$ is a finite group, then the Haar measure is given by the counting
measure.

  * (2)

The Lebesgue measure of $\mathbb{R}^{d}$ and $\mathbb{C}^{d}$ is the Haar
measure on the additive groups $(\mathbb{R}^{d},+)$ and $(\mathbb{C}^{d},+)$.
These two groups are only locally but not globally compact. However, they are
abelian and hence unimodular.

  * (3)

Recall that an element of $g$ of $\mathrm{SU}(2)$ can be parametrized as
$g=\mathrm{exp}(i\theta\vec{n}\cdot\vec{\sigma})$ with $\theta\in[0,\pi]$,
$\vec{n}\in S^{2}$ and where $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$
denote the Pauli matrices. Then the normalized Haar measure on
$\mathrm{SU}(2)$ is given by

$\displaystyle\mathrm{d}g=\frac{2}{\pi}\mathrm{d}\theta(\sin(\theta))^{2}\mathrm{d}^{2}\vec{n}$
(A.45)

where $\mathrm{d}^{2}\vec{n}$ denotes the normalized measure on the $2$-sphere
$S^{2}$. [FL04a, Appendix B]

∎

####  A.3.2 Theorem of Peter-Weyl

Consider a compact Lie group $G$ together with its bi-invariant Haar measure.
Then we can define the Hilbert space
$(L^{2}(G),\langle\cdot,\cdot\rangle_{L^{2}})$ consisting of (equivalence
classes of almost everywhere equivalent) measurable functions
$f:G\to\mathbb{C}$, which are square-integrable, i.e.

$\displaystyle\int_{G}\,|f(g)|^{2}\,\mathrm{d}g<\infty,$ (A.46)

where the inner product $\langle\cdot,\cdot\rangle_{L^{2}}$ on this space is
for two elements $f,k\in L^{2}(G)$ defined by

$\displaystyle\langle
f,k\rangle_{L^{2}}:=\int_{G}\,\overline{f(g)}k(g)\,\mathrm{d}g.$ (A.47)

In this section, we would like to briefly discuss the famous “Theorem of
Peter-Weyl”, which is central in the discussion of compact Lie groups and
harmonic analysis. Let us start by recalling some basic definitions. A
“unitary representation of $G$ ” on some Hilbert space
$(\mathcal{H},\langle\cdot,\cdot\rangle)$ is a Lie group homomorphism of the
type $\rho:G\to\mathcal{U}(\mathcal{H})$, where
$\mathcal{U}(\mathcal{H})\subset\mathrm{Aut}(\mathcal{H})$ denotes the set of
unitary operators on $\mathcal{H}$. Two unitary representations
$\rho_{1}:G\to\mathcal{U}(\mathcal{H}_{1})$ and
$\rho_{2}:G\to\mathcal{U}(\mathcal{H}_{1})$ are called “unitary equivalent”,
if there exists a unitary operator $U:\mathcal{H}_{1}\to\mathcal{H}_{2}$ such
that $U\circ\rho_{1}(g)=\rho_{2}(g)\circ U(g)$ for all $g\in G$. Last but not
least, a unitary representation $\rho:G\to\mathcal{U}(\mathcal{H})$ is called
“irreducible”, if there exists no non-trivial invariant subspace. Now, let us
denote by $\Lambda$ the set of equivalence classes of unitary and irreducible
representations. Then, the Theorem of Peter-Weyl, proven by H. K. H. Weyl and
F. Peter in 1927 [WP27], states the following [BtD85]:

(of Peter-Weyl)
Let $G$ be a compact Lie group. Then the set
$\\{\sqrt{\operatorname{dim}(\rho)}\rho_{ij}\mid\rho\in\Lambda\\}$ is an
orthonormal basis of the Hilbert space $L^{2}(G)$, where
$\rho_{ij}:G\to\mathbb{C}$ denote the “matrix elements” of a representation
$\rho\in\Lambda$, which are defined via $\rho_{ij}(g):=\langle
e_{i}|\rho(g)e_{j}\rangle$ for some orthonormal basis $\\{e_{i}\\}_{i\in I}$
of the corresponding Hilbert space $\mathcal{H}_{\rho}$.

In the context of spin networks, it is useful to state the above theorem in a
more abstract form. For this, let us first of all define the “right regular
representation” $R:G\to\mathcal{U}(L^{2}(G))$ and the “left-regular
representation” $L:G\to\mathcal{U}(L^{2}(G))$ by

$\displaystyle(R_{g}(f))(h):=f(hg)\hskip 28.45274pt\text{and}\hskip
28.45274pt(L_{g}(f))(h):=f(g^{-1}h)$ (A.48)

for all $f\in L^{2}(G)$ and for all $h,g\in G$. Note that both of these
representations are unitary precisely because the normalized Haar measure of a
compact Lie group is bi-invariant. Now, the important point is that these two
representations can be used to decompose the Hilbert space $L^{2}(G)$ in terms
of finite-dimensional subspaces, which are invariant under these
representations. To see this, first of all observe the following: For some
given $(\rho,\mathcal{H})\in\Lambda$, there are
$d_{\rho}:=\operatorname{dim}_{\mathbb{C}}(\mathcal{H})$ finite-dimensional
Hilbert spaces, namely the subspaces spanned by the rows of the
representation, i.e.

$\displaystyle\mathcal{H}_{\rho}^{i}:=\operatorname{span}\\{\rho_{ij}\mid
1\leq j\leq d_{\rho}\\}.$ (A.49)

These subspaces are clearly $R$-invariant and the restriction of $R$ to
$\mathcal{H}_{i}^{\rho}$ coincides with $\rho$, because
$R_{g}|_{\mathcal{H}_{i}^{\rho}}$ is realized by the same matrix as $\rho(g)$,
as the following short calculation shows:

$\displaystyle
R_{h}\bigg{(}\sum_{j}c_{j}\rho_{ij}(g)\bigg{)}=\sum_{j}c_{j}\rho_{ij}(gh)=\sum_{j,k}c_{j}\rho_{ij}(g)\rho_{jk}(h)$
(A.50)

and hence
$R_{h}:\mathcal{H}_{\rho}^{i}\to\mathcal{H}_{\rho}^{i},v\mapsto\rho(h)(v)$.
Now, the Theorem of Peter-Weyl is equivalent to say that there is the
following isomorphism of representations:

$\displaystyle
L^{2}(G)\cong\bigoplus_{\rho\in\Lambda}\bigoplus_{i=1}^{d_{\rho}}\mathcal{H}_{\rho}^{i}.$
(A.51)

A similar construction can be done for the left-regular representation, i.e.

$\displaystyle
L^{2}(G)\cong\bigoplus_{\rho\in\Lambda}\bigoplus_{i=1}^{d_{\rho}}(\mathcal{H}_{\rho}^{i})^{\ast},$
(A.52)

where $(\mathcal{H}_{\rho}^{i})^{\ast}$ are spanned by the complex-conjugate
rows of matrix elements. Furthermore, we can also define another
representation, namely by regarding $L^{2}(G)$ as the representation space of
a unitary representation of the group $G\times G$, i.e.

$\displaystyle\tau:G\times G\to\mathcal{U}(L^{2}(G)),(g,h)\mapsto
R_{g}L_{h}=L_{h}R_{g}$ (A.53)

which is called the “bi-regular representation”. Again, this representation is
unitary by bi-invariance of the Haar measure on $G$. The Peter-Weyl theorem
then states that

$\displaystyle
L^{2}(G)\cong\bigoplus_{\rho\in\Lambda}\mathcal{H}_{\rho}\otimes(\mathcal{H}_{\rho})^{\ast}.$
(A.54)

More details can be found in [BtD85] and [Pro04, p.32ff.].

####  A.3.3 Invariant Elements and Intertwiners

First of all, recall that the set of invariant elements of some given compact
Lie group $G$ and some given representation $(\rho,\mathcal{H})$ of $G$ is
defined by

$\displaystyle\mathrm{Inv}_{\rho}(\mathcal{H}):=\\{v\in\mathcal{H}\mid\rho(g)v=v\,\forall
g\in G\\}.$ (A.55)

Now, suppose that $(\rho_{1},\mathcal{H}_{1})$ and
$(\rho_{2},\mathcal{H}_{2})$ are two unitary representations of some given
compact Lie group $G$. We then can construct a representation of $G$ on the
set of linear maps $\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2})$

$\displaystyle\eta:G$
$\displaystyle\to\mathrm{Aut}(\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2}))$
(A.56)

by $\eta(g)(f):=\rho_{1}(g)\circ f\circ\rho_{2}(g^{-1})$ for all
$f\in\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2})$. Now, it turns out that
the set of invariant elements of this representations is related to the set of
“intertwiners” from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}$, i.e. the subset
$\mathrm{Int}(\mathcal{H}_{1},\mathcal{H}_{2})\subset\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2})$
of linear maps $f:\mathcal{H}_{1}\to\mathcal{H}_{2}$ satisfying

$\displaystyle\rho_{2}(g)\circ f=f\circ\rho_{1}(g)$ (A.57)

for all $g\in G$. More precisely, let us proof the following claim:

(Invariant Elements and Intertwiners)
It holds that

$\displaystyle\mathrm{Inv}_{\eta}(\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2}))=\mathrm{Int}(\mathcal{H}_{1},\mathcal{H}_{2}).$

###### Proof.

  * “$\subset$”

Let $f\in\mathrm{Inv}_{\eta}(\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2}))$,
i.e. $f\in\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2})$ is such that
$\eta(g)(f)=f$ for all $g\in G$. In order to show that
$f\in\mathrm{Int}(\mathcal{H}_{1},\mathcal{H}_{2})$, we have to verify the
intertwining property, i.e.

$\displaystyle\rho_{2}(g)\circ f=f\circ\rho_{1}(g).$ (A.58)

Since $f$ is an invariant element, it trivially satisfies the following
equation:

$\displaystyle f=\int_{G}\,\mathrm{d}g\,\rho_{2}(g)\circ
f\circ\rho_{1}(g^{-1}),$ (A.59)

where $\mathrm{d}g$ denotes the normalized Haar measure of $G$. As a
consequence, we have that

$\displaystyle\rho_{2}(g)\circ f$
$\displaystyle=\rho_{2}(g)\circ\int_{G}\,\mathrm{d}h\,\rho_{2}(h)\circ
f\circ\rho_{1}(h^{-1})=\int_{G}\,\mathrm{d}h\,\rho_{2}(gh)\circ
f\circ\rho_{1}(h^{-1})=$ (A.60)
$\displaystyle=\int_{G}\,\mathrm{d}h\,\rho_{2}(gh)\circ
f\circ\rho_{1}(h^{-1}g^{-1}g)=\int_{G}\,\mathrm{d}h\,\rho_{2}(h)\circ
f\circ\rho_{1}(h^{-1}g)=$
$\displaystyle=\int_{G}\,\mathrm{d}h\,\rho_{2}(h)\circ
f\circ\rho_{1}(h^{-1})\circ\rho(g)=f\circ\rho_{1}(g),$

where we used that $\mathrm{d}(gh)=\mathrm{d}h$, by bi-invariance of the Haar
measure.

  * “$\supset$”

Let $f\in\mathrm{Int}(\mathcal{H}_{1},\mathcal{H}_{2})$. Then we have that

$\displaystyle\eta(g)(f)=\rho_{1}(g)\circ f\circ\rho_{2}(g^{-1})=f$ (A.61)

where we used the intertwining property. This shows that
$f\in\mathrm{Inv}_{\eta}(\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2}))$.

∎

In the case when $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are finite-
dimensional, there is the canonical isomorphism
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}^{\ast}\cong\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2})$.
As consequence, we have the following immediate Corollary:

There is the following isomorphism:

$\displaystyle\mathrm{Inv}_{\eta}(\mathcal{H}_{1}\otimes\mathcal{H}_{2}^{\ast})\cong\mathrm{Int}(\mathcal{H}_{1},\mathcal{H}_{2}).$

This can easily be seen by the fact that we not only have the isomorphism
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}^{\ast}\cong\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2})$
on the level of vector spaces, but there is in fact also a isomorphism of
representations (i.e. a bijective intertwiner) between the representation
$\eta$ on $\mathrm{Hom}(\mathcal{H}_{1},\mathcal{H}_{2})$ and the (inner)
tensor product $\rho_{1}\otimes\rho_{2}^{\ast}$ on
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}^{\ast}$, which we hence also denote by
$\eta$ and where $\rho_{2}^{\ast}$ denotes the dual representation of
$\rho_{2}$.

####  A.3.4 Non-Commutative Group Fourier Transform

The non-commutative group Fourier transform, firstly introduced in [FL06] for
the Lie group $\mathrm{SO}(3)$, is a natural choice of mapping between the
configuration space and the corresponding momentum space of a Lie group.
Furthermore, it establishes a correspondence between group field theories and
non-commutative quantum field theory. For a review of the general notion of
this type of Fourier transform see for example [Raa11, Raa12].

Let $G$ be a Lie group, usually assumed to be compact, or at least locally
compact, such that we can define a Haar measure on $G$. The cotangent bundle
$T^{\ast}G$ is diffeomorphic to $G\times\mathfrak{g}^{\ast}$, where
$\mathfrak{g}$ denotes the Lie algebra corresponding to $G$ and
$\mathfrak{g}^{\ast}$ its dual, because the tangent bundle of a Lie group is a
trivial vector bundle. The definition of the non-commutative Fourier transform
is based on maps of the type

$\displaystyle e:T^{\ast}G\cong G\times\mathfrak{g}^{\ast}$
$\displaystyle\to\mathbb{C},$ (A.62) $\displaystyle(g,X)$
$\displaystyle\mapsto e_{g}(X)$

called the “plane-waves of $G$”. These maps are defined in such a way that the
delta function corresponding to the group $G$ can be written as

$\displaystyle\delta_{G}(g)=\int_{\mathfrak{g}^{\ast}}\frac{\mathrm{d}^{d}x}{(2\pi)^{d}}\,e_{g}(x).$
(A.63)

where $d=\mathrm{dim}(G)=\mathrm{dim}_{\mathbb{R}}(\mathfrak{g})$ and where
$\mathrm{d}^{d}x$ denotes the Lebesgue-measure on $\mathfrak{g}^{\ast}$.
Another very important object in the definition of the non-commutative group
Fourier transform is the so-called “star product”, denoted by $\star$, which
is a product respecting the group structure. For two plane waves it is defined
such that

$\displaystyle(e_{g}\star e_{h})(x):=e_{g}(x)\star e_{h}(x):=e_{gh}(x)$ (A.64)

for all $g,h\in G$ and for all $x\in\mathfrak{g}^{\ast}$. Note that this
product is associative, but in general not commutative. Furthermore, a unit
with respect to this product if given by $e_{e}(x)$, where the lower $e$
denotes the neutral element of $G$. Now let $\varphi\in L^{2}(G)$. Then we
define the non-commutative group Fourier transform via

$\displaystyle\mathcal{F}(\varphi)(x):=\hat{\varphi}(x):=\int_{G}\,\mathrm{d}g\,e_{g^{-1}}(x)\varphi(g).$
(A.65)

Let us denote the image of this transformation by
$L^{2}_{\star}(\mathfrak{g}^{\ast}):=\mathcal{F}(L^{2}(G))$. Using the above
definition, we can extend the $\star$-product to all of
$L^{2}_{\star}(\mathfrak{g}^{\ast})$. Furthermore, we can equip the space
$L^{2}_{\star}(\mathfrak{g}^{\ast})$ with an inner product defined via

$\displaystyle\langle\varphi,\psi\rangle_{L^{2}_{\star}(\mathfrak{g}^{\ast})}:=\int_{\mathfrak{g}^{\ast}}\,\frac{\mathrm{d^{d}x}}{(2\pi)^{d}}\,\overline{\varphi(x)}\star\psi(x)$
(A.66)

for all $\varphi,\psi\in L^{2}_{\star}(\mathfrak{g}^{\ast})$. The inverse
transform is given by

$\displaystyle\mathcal{F}^{-1}(\hat{\varphi})(g)=\varphi(g)=\int_{\mathfrak{g}^{\ast}}\,\frac{\mathrm{d^{d}}x}{(2\pi)^{d}}\,e_{g}(x)\star\varphi(x)$
(A.67)

as the following short calculation shows:

$\displaystyle\int_{\mathfrak{g}^{\ast}}\,\frac{\mathrm{d^{d}}x}{(2\pi)^{d}}\,e_{g}(x)\star\hat{\varphi}(x)$
$\displaystyle=\int_{G}\,\mathrm{d}h\,\int_{\mathfrak{g}^{\ast}}\,\frac{\mathrm{d^{d}}x}{(2\pi)^{d}}\,e_{g}(x)\star
e_{h^{-1}}(x)\varphi(h)=$ (A.68)
$\displaystyle=\int_{G}\,\mathrm{d}h\,\int_{\mathfrak{g}^{\ast}}\,\frac{\mathrm{d^{d}}x}{(2\pi)^{d}}\,e_{gh^{-1}}(x)\varphi(h)=$
$\displaystyle=\int_{G}\,\mathrm{d}h\,\delta_{G}(gh^{-1})\varphi(g)=\varphi(g).$

If the plane-waves satisfy the condition $e_{g^{-1}}(x)=\overline{e_{g}(x)}$,
then the non-commutative group Fourier transform is actually an isometry, i.e.
it satisfies

$\displaystyle\langle\varphi,\psi\rangle_{L^{2}(G)}=\langle\mathcal{F}(\varphi),\mathcal{F}(\psi)\rangle_{L^{2}_{\star}(\mathfrak{g}^{\ast})}=\langle\hat{\varphi},\hat{\psi}\rangle_{L^{2}_{\star}(\mathfrak{g}^{\ast})},$
(A.69)

which can easily be shown by a straightforward calculation:

$\displaystyle\langle\hat{\varphi},\hat{\psi}\rangle_{L^{2}_{\star}(\mathfrak{g}^{\ast})}$
$\displaystyle=\int_{\mathfrak{g}^{\ast}}\,\frac{\mathrm{d}^{d}x}{(2\pi)^{d}}\,\overline{\hat{\varphi}(x)}\star\hat{\psi}(x)=$
(A.70)
$\displaystyle=\int_{\mathfrak{g}^{\ast}}\,\frac{\mathrm{d}^{d}x}{(2\pi)^{d}}\,\int_{G}\,\mathrm{d}g\,\int_{G}\,\mathrm{d}h\,\overline{\varphi(g)}\psi(h)\,\overline{e_{g^{-1}}(x)}\ast
e_{h^{-1}}(x)=$
$\displaystyle=\int_{\mathfrak{g}^{\ast}}\,\frac{\mathrm{d}^{d}x}{(2\pi)^{d}}\,\int_{G}\,\mathrm{d}g\,\int_{G}\,\mathrm{d}h\,\overline{\varphi(g)}\psi(h)e_{gh^{-1}}(x)=$
$\displaystyle=\int_{G}\,\mathrm{d}g\,\int_{G}\,\mathrm{d}h\,\delta_{G}(gh^{-1})\overline{\varphi(g)}\psi(h)=$
$\displaystyle=\int_{G}\,\mathrm{d}g\,\overline{\varphi(g)}\psi(h)=\langle\varphi,\psi\rangle_{L^{2}(G)}$

In other words, the non-commutative Fourier transform is a unitary operator
$\mathcal{F}:L^{2}(G)\to L^{2}_{\star}(\mathfrak{g}^{\ast})$, which also
establishes the analogy with the standard Fourier transform from functional
analysis. Note also that there are some slightly different conventions in the
literature: As an example, some authors define the non-commutative group
Fourier transform with $e_{g}(x)$ in Formula (A.65) and hence with
$e_{g^{-1}}(x)$ in Formula (A.67), e.g. in [BO10].

The explicit form of the plane-waves $e_{g}(X)$ depends on the specific group
and can be highly non-trivial. Usually they take the form

$\displaystyle
e_{g}(x)=\mathrm{exp}\bigg{(}i\sum_{i=1}^{d}Z_{i}(g)x^{i}\bigg{)},$ (A.71)

where $Z(g)$ denote some coordinates of the group manifold $G$. As an example,
for the group $\mathrm{SO}(3)$ one could use
$Z^{i}(g)=-\frac{i}{2}\operatorname{tr}(g\sigma^{i})$, where $\sigma^{i}$
denote the Pauli-matrices. The case of $\mathrm{SU}(2)$ is more subtle. One
has to be careful in order to avoid the same problem, which we had in the
derivation of the Ponzano-Regge model, namely that using plane-waves defined
in a similar manner as for $\mathrm{SO}(3)$ would lead to the ambiguity
$Z^{i}(g^{-1})=Z^{i}(-g)$ and hence, we would not get a bijective Fourier
transform in this case. In order to avoid this issue, one has to multiply the
above formula by a polarization vector keeping track of the hemisphere to
which $g$ belongs. Also the star-product has to be modified accordingly.
Alternatively, one can also work with $\mathrm{SO}(3)$ instead and multiply
the exponent by a sign factor $\operatorname{sign}(\operatorname{tr}(g))$. In
other words, we basically use the isomorphism
$\mathrm{SO}(3)\cong\mathrm{SU}(2)/\mathbb{Z}_{2}$. For more details about the
specific case of $G=\mathrm{SU}(2)$ see for example [FM08, Liv09, DGL12].

Last but not least, the delta function303030For general locally compact Lie
groups, the delta function of $L^{2}_{\star}(\mathfrak{g}^{\ast})$ is a
distribution, however, it turns out to be a regular function for compact Lie
groups [Raa11, p.3]. of the space $L^{2}_{\star}(\mathfrak{g}^{\ast})$ is
given by

$\displaystyle\delta(x):=\int_{G}\,\mathrm{d}g\,e_{g_{-1}}(0)e_{g}(X).$ (A.72)

This can easily be seen by the following short calculation:

$\displaystyle\int_{\mathfrak{g}^{\ast}}\frac{\mathrm{d}^{d}x}{(2\pi)^{d}}\,\delta(x)\star\hat{\varphi}(x)$
$\displaystyle=\int_{\mathfrak{g}^{\ast}}\frac{\mathrm{d}^{d}x}{(2\pi)^{d}}\,\int_{G}\,\mathrm{d}g\,e_{g^{-1}}(0)e_{g}(x)\star\hat{\varphi}(x)=$
(A.73)
$\displaystyle=\int_{G}\,\mathrm{d}g\,e_{g^{-1}}(0)\varphi(g)=\hat{\varphi}(0).$

The generalization of the non-commutative Fourier transform for field in
several variables is straightforward: One uses a plane wave $e_{g}(x)$ for
each variable in the definition of the non-commutative group Fourier transform
above. The star-product is then defined component-wise.

###  A.4 Some Additional Remarks on Topological Equivalence of Coloured Graphs

In the last chapter, we discuss some additional topics concerning dipole moves
in coloured graph with boundary. First of all, we construct an explicit
example of a dipole move, which does not fulfil the properties described in
Theorem 3.2.5, but which still turns out to be proper. Secondly, we discuss
some effects non-proper dipole moves can have. Furthermore, we introduce a
second set of moves, which can be used to describe topological equivalence of
coloured graph with non-empty boundaries. These moves are called “wounds” and
can be replaced by a set of four proper dipole moves. We end this discussion
be briefly mentioning a class of pseudomanifolds, which is stable under non-
proper wound moves and non-proper dipole contractions.

####  A.4.1 Non-Proper Dipole Moves and “Accidental” Proper Moves

The Theorem of Ferri-Gagliardi (Theorem 3.2.5) gives two conditions for
general dipoles in open graphs to be proper. More precisely, we have seen that
a $k$-dipole $d_{k}$ in some open $(d+1)$-coloured graph
$\mathcal{G}\in\mathfrak{G}_{d}$ is proper, whenever it fulfils one of the
following two properties:

  * (A)

At least one of the $(d+1-k)$-bubbles separated by the dipole represents a
$(d-k)$-sphere.

  * (B)

Both vertices involved in $d_{k}$ are touching the boundary and at least one
of the $(d+1-k)$-bubbles separated by the dipole represents a $(d-k)$-ball.

Reversing this statement, we also see that whenever a $k$-dipole is non-
proper, then it cannot have one of the properties (A) and (B). However, note
that the Theorem of Ferri-Gagliardi is not an “if-and-only-if-statement”, i.e.
there are also dipoles, which do not fulfil (A) and (B), but which still turn
out to be proper. However, one should note that these “accidental proper
dipole moves” can always be replaced by a finite sequence of dipole moves of
type (A) and (B), since by the Theorem of Casali (Theorem 3.2.5), moves of
type (A) and (B) are enough to describe topological equivalence of coloured
graphs. As an example of such an accidental proper dipole, consider the
following dipole move in an open $(3+1)$-coloured graph
$\mathcal{G}\in\mathfrak{G}_{3}$:

Figure A.2: An open $(3+1)$-coloured graph $\mathcal{G}$ and a proper dipole
move not satisfying properties (A) and (B).

The edge of colour $3$ clearly defines a dipole $d_{1}$ in $\mathcal{G}$,
since it separates two $3$-bubbles of colours $012$. Furthermore, note that
both separated $3$-bubbles are open graphs and represent $2$-balls (=disks).
However, the two vertices involved in the dipole are both not touching the
boundary. Hence, this dipole does neither fulfil (A) nor (B). The boundary
graphs of $\mathcal{G}$ and $\mathcal{G}/d_{1}$ are actually colour-
isomorphic, as one can easily see: Both of them are the “pillow graph”, but
with different colouring. As a consequence, the two complexes dual to
$\mathcal{G}$ and $\mathcal{G}/d_{1}$ have the same boundaries. Now, it turns
out that they do not only have the same boundaries, but they also have the
same bulk topology, since both the graphs $\mathcal{G}$ and
$\mathcal{G}/d_{1}$ represent $3$-balls. This follows from the fact that both
of them are related to the elementary melonic $3$-ball by a finite sequence of
proper moves of type (A) and (B), as can be seen below. Hence, $d_{1}$ is a
proper dipole.

Figure A.3: Both the graphs $\mathcal{G}$ and $\mathcal{G}/d_{1}$ represent
$3$-balls and hence $d_{1}$ is a proper dipole.

Next, let us discuss some effects of non-proper dipole moves. As already
mentioned, non-proper dipoles cannot have properties (A) and (B). Let
$\mathcal{G}\in\mathfrak{G}_{d}$ be an open $(d+1)$-coloured graph and
$d_{k}=\\{v,w\\}$ a dipole within $\mathcal{G}$. Then, let us distinguish
between the following types of dipoles

  * (1)

Both of the $(d+1-k)$-bubbles separated by the dipole $d_{k}$ are neither
spheres nor balls.

  * (2)

Either both $(d+1-k)$-bubbles separated by the dipole are $3$-balls, or one of
them is a $3$-ball and the other one is neither a ball nor a sphere, however,
at least one of the two vertices $v$ and $w$ is not touching the boundary.

Every dipole, which does not fulfil condition (A) and (B), does fall into one
(and only one) of the two classes written above. However, it can happen that
dipoles in these types turn out to be proper, as the accidental proper dipole
move in the figure above shows. Dipoles, which fall into class (1), are always
non-proper. To see this, recall that contracting a dipole means to perform the
(boundary) connected sum of two links in the simplicial complex. If both the
bubbles separated by the dipole are neither spheres nor balls, then
contracting the dipole does change the topology of some subcomplex within the
complex dual to $\mathcal{G}$. However, note that these dipoles can only occur
in graphs which represents pseudomanifolds, since for manifolds, all bubbles
are dual to spheres or balls. Hence, the only non-proper dipoles for graphs
representing manifolds are of type (2). Dipoles in this class can be proper,
as the example from above shows. On the other hand, non-proper dipoles in
class (2) can have several different effects. One thing, which quite often
happens, is that such a dipole produces a disconnected graph. An example is
drawn below.

Figure A.4: A non-proper $2$-dipole of type (2), which produces a disconnected
graph.

However, this must not necessarily be the case. There are also examples of
non-proper dipole moves in class (2), which do not change the connectivity of
the graph. As an example, let us look at the following $2$-dipole contraction:

Figure A.5: A non-proper $2$-dipole contraction of type (2).

The $2$-dipole of colours 13 separates two open $2$-bubbles, which both
represent $1$-balls (=closed intervals). However, one of the two vertices is
not touching the boundary. In this case, one clearly sees that the graph
$\mathcal{G}$ represents a pseudomanifold, since its unique $3$-bubbles of
colours $012$ and $023$ both have two boundary components and represent
cylinders. However, the graph $\mathcal{G}/d_{2}$ represents a manifolds (in
fact, the $3$-ball), since the all its $3$-bubbles are either $2$-spheres or
$2$-balls (=disks). Hence, this move is an example of a dipole move creating a
manifold out of a pseudomanifold. As a consequence, this dipole is of course
non-proper.

####  A.4.2 Other Type of Topological Moves: Wounds

For coloured graphs with non-empty boundary, one can also define another set
of combinatorial and topological moves, which were introduced in [Gag87] and
are called “wounds”. As we will discuss below, these moves are just a
combination of proper dipole moves and hence do not offer any new insights in
topological equivalence of coloured graphs. Nevertheless, they still could be
useful at some point, since they give another possibility to explicitly change
the boundary graph of some given open graph. They are defined as follows:

(Wounds)
Let $\mathcal{G}\in\mathfrak{G}_{d}$ be an open $(d+1)$-coloured graph with
$\partial\mathcal{G}\neq\emptyset$ and $k\in\\{0,\dots,d\\}$. Furthermore, let
$v,w$ be two internal vertices, which both admit an adjacent external leg.
Then, we call a subgraph $\mathcal{W}_{k}$ consisting of all faces containing
$v$ and $w$ “$k$-wound of colours $i_{1},\dots,i_{k}$”, if the following two
conditions are fulfilled:

  * (1)

The vertices $v$ and $w$ are contained in the same faces of colours $0i_{j}$
for all $j\in\\{1,\dots,k\\}$.

  * (2)

The $(d+1-k)$-bubbles of colours
$\mathcal{C}_{d}\textbackslash\\{i_{1},\dots,i_{k}\\}$ containing $v$ and $w$
are distinct.

Note that every non-internal proper $k$-dipole is of course also a $k$-wound,
since all the non-cyclic faces containing the external legs and the two
vertices are trivially the same and consist only of three edges. If
$\mathcal{W}_{k}$ is a $k$-wound of colours $i_{1},\dots,i_{k}$ defined via
two vertices $v,w$, then the boundary graph $\partial\mathcal{W}_{k}$, which
is a subgraph of $\partial\mathcal{G}$, is a $k$-dipole of colours
$i_{1},\dots,i_{k}$ in $\partial\mathcal{G}$ whenever $k<d$. If $k=d$, then
$\partial\mathcal{W}_{k}$ is just a connected component of
$\partial\mathcal{G}$, given by the elementary melonic $d$-sphere. If there is
a wound in $\mathcal{G}$, then we construct a new graph as follows:

(Saturating a Wound)
Let $\mathcal{G}\in\mathfrak{G}_{d}$ and $\mathcal{W}_{k}$ be a $k$-wound in
$\mathcal{G}$ defined by two vertices $v$ and $w$. Then we define the graph
$\mathcal{G}_{\overline{\mathcal{W}_{k}}}$ by connecting the two external legs
connected to $v$ and $w$. We say that
$\mathcal{G}_{\overline{\mathcal{W}_{k}}}$ is obtained from $\mathcal{G}$ by
“saturating the wound $\mathcal{W}_{k}$”. The reverse process is called
“opening the wound”.

Let us briefly mentioned the effect of saturating a wound on the boundary
graph of $\mathcal{G}$: If $k=d$, then saturating a wound results into
removing a spherical component of $\partial\mathcal{G}$. If $k<d$, then
saturating a wound results into contracting a $k$-dipole of the boundary graph
$\partial\mathcal{G}$. An example of a $2$-wound saturation together with the
effect on its boundary graph is drawn in the following figure.

Figure A.6: An open $(3+1)$-coloured graph $\mathcal{G}\in\mathfrak{G}_{3}$
together with its boundary graph $\partial\mathcal{G}$ and the contraction of
a $2$-wound $\mathcal{W}_{2}$ of colours $12$.

As for dipole moves, we say that a wound saturation is “proper”, if the
(pseudo)manifolds dual to $\mathcal{G}$ and
$\mathcal{G}_{\overline{\mathcal{W}_{k}}}$ are PL-homeomorphic. The $2$-wound
drawn in Figure A.6 above can easily be seen to be proper, since both the
graphs $\mathcal{G}$ and $\mathcal{G}_{\overline{\mathcal{W}_{k}}}$ can be
reduced to the elementary melonic $3$-ball by a sequence of proper dipole
moves and hence represent $3$-balls themselves.

Of course, not every wound saturation is proper. As an example, consider an
open graph with only two external legs. Then the collection of all non-cyclic
faces form a $d$-wound, whose saturation leads to a closed graph. More
generally, one can easily see that $d$-wounds are in general non-proper, even
if there are more than two external legs, since they generically remove one
(spherical) boundary component from the manifold. Furthermore, also for
$k$-wounds with $k<d$ one can easily construct examples of non-proper wounds.
In [Gag87, p.60], the authors found the following conditions for wounds to be
proper:

(Condition for Proper Wounds)
Let $\mathcal{G}\in\mathfrak{G}_{d}$ and $\mathcal{W}_{k}$ be a $k$-wound with
$k<d$ of colours $i_{1},\dots,i_{k}\in\mathcal{C}_{d}$ in $\mathcal{G}$
defined by two vertices $v$ and $w$. Furthermore, suppose that the following
conditions are fulfilled:

  * (1)

At least one of the two $(d+1-k)$-bubbles of colours
$\mathcal{C}_{d}\textbackslash\\{i_{1},\dots,i_{k}\\}$ separating the vertices
$v$ and $w$ represents a $(d-k)$-ball.

  * (2)

The $(k+1)$-bubble of colours $0,i_{1},\dots,i_{k}$ in
$\mathcal{G}_{\overline{\mathcal{W}_{k}}}$ containing $v$ and $w$ represents a
$k$-sphere.

Then $\mathcal{W}_{k}$ is equivalent to a sequence of four proper dipole moves
and hence proper.

###### Proof.

Consider a $k$-wound $\mathcal{W}_{k}$ in $\mathcal{G}$, whose colours are
without loss of generality $1,\dots,k$. We denote the $(d+1-k)$-bubbles of
colours $0,k+1,\dots,d$ separated by the wound by
$\mathcal{B}_{v}^{0,k+1,\dots,d}$ and $\mathcal{B}_{w}^{0,k+1,\dots,d}$
respectively, and the spherical $k+1$-bubble in the saturated graph of colours
$0,1,\dots,k$ containing the vertices $v$ and $w$ by
$\mathcal{B}_{vw}^{0,1,\dots,k}$. Without loss of generality, we assume that
the bubble $\mathcal{B}_{v}^{0,k+1,\dots,d}$ is the bubble separated by the
wound, which represents a $(d-k)$-ball. Then, let us define a graph
$\mathcal{G}^{\prime}$ by adding four new vertices to $\mathcal{G}$, which we
denote by $v^{\prime},w^{\prime},v^{\prime\prime},w^{\prime\prime}$. We
connect the vertex $v^{\prime}$ via an edge of colour $0$ to the external leg
adjacent of $v$ and similarly the vertex $w^{\prime}$ to the external leg
adjacent of $w$. Then, we connect the vertices $v^{\prime}$ and
$v^{\prime\prime}$ by $d-k$ edges of colours $k+1,\dots,d$ and similarly, we
connect $w^{\prime}$ and $w^{\prime\prime}$ by $d-k$ edges of colours
$k+1,\dots,d$. Furthermore, we connect the vertices $v^{\prime}$ and
$w^{\prime}$ by $k$ edges of colours $1,\dots,k$ and also the vertices
$v^{\prime\prime}$ and $w^{\prime\prime}$ by $k$ edges of colours $1,\dots,k$.
Last but not least, we add an external leg to $v^{\prime\prime}$ and one to
$w^{\prime\prime}$. The resulting graph is a well-defined open
$(d+1)$-coloured graph in $\mathfrak{G}_{d}$. The type of the vertices
$v^{\prime},w^{\prime},v^{\prime\prime},w^{\prime\prime}$ (black vs. white) is
such that $\mathcal{G}^{\prime}$ is bipartite. The structure of the graph
$\mathcal{G}^{\prime}$ is sketched in the following figure:

Figure A.7: The graph $\mathcal{G}$ and the graph $\mathcal{G}^{\prime}$
constructed above.

The dashed lines in the graph above sketch the bicoloured paths (=faces)
connecting the corresponding vertices. Furthermore, note that the $d$ internal
vertices connected to $v$ do not have to be distinct and similarly for $w$. By
looking at the picture, it is clear that $\mathcal{G}^{\prime}$ has the same
boundary graph as $\mathcal{G}$. In fact, $\mathcal{G}^{\prime}$ turns out to
be equivalent to $\mathcal{G}$ up to proper dipoles. To see this, observe that
$\\{v^{\prime},v^{\prime\prime}\\}$ together with the $d-k$ edges connecting
them is an internal proper $(d-k)$-dipole, precisely because of assumption
(2): The $(k+1)$-bubble of colours $0,1,\dots,k$ containing $v^{\prime}$,
which is separated by this dipole, is just the graph obtained by adding an
internal proper $k$-dipole between the vertices $v$ and $w$ in the graph
$\mathcal{B}_{vw}^{0,1,\dots,k}$, which by assumption (2) represents a
$k$-sphere. Contracting this dipole yields a graph which looks similar to
$\mathcal{G}$ up to an internal proper $d$-dipole connected to $w$.
Contracting this dipole finally yields $\mathcal{G}$. The procedure explained
above is sketched in the following figure:

Figure A.8: The graph $\mathcal{G}^{\prime}$ can be reduced to $\mathcal{G}$
by performing first an internal proper $(d-k)$-dipole and then an internal
proper $d$-dipole.

On the other hand, as a consequence of assumption (1), also
$\\{v^{\prime\prime},w^{\prime\prime}\\}$ together with the $k$-edges
connecting them is a proper dipole, namely a non-internal proper $k$-dipole.
This follows from the fact that the $(d+1-k)$-bubble of colours
$0,k+1,\dots,d$ containing the vertex $v^{\prime\prime}$ separated by this
dipole is just the graph obtained by adding an internal proper $(d-k)$-dipole
to the external leg adjacent to the vertex $v$ in the graph
$\mathcal{B}_{v}^{0,k+1,\dots,d}$, which by assumption (1), represents a
$(d+1-k)$-ball. Contracting this dipole yields a graph which looks like the
saturated graph $\mathcal{G}_{\overline{\mathcal{W}_{k}}}$, in which we have
added a $d$-dipole in-between the vertices $v$ and $w$. Contracting this
dipole yields the graph $\mathcal{G}_{\overline{\mathcal{W}_{k}}}$. This is
sketched in the figure below.

Figure A.9: The graph $\mathcal{G}^{\prime}$ can be reduced to
$\mathcal{G}_{\overline{\mathcal{W}_{k}}}$ by performing first a non-internal
proper $k$-dipole and then an internal proper $d$-dipole.

Hence, we have shown that $\mathcal{G}$ and
$\mathcal{G}_{\overline{\mathcal{W}_{k}}}$ are related by a sequence of four
proper dipole moves, which shows that $\mathcal{G}$ and
$\mathcal{G}_{\overline{\mathcal{W}_{k}}}$ represent the same
(pseudo)manifold. To sum up, saturating a wound means just to add two internal
proper dipoles of types $d$ and $(d-k)$ to $\mathcal{G}$ followed by
cancelling an non-internal proper $k$-dipole and an internal proper
$d$-dipole. ∎

As an example, we can easily verify that the two assumption of the previous
proposition are satisfied for the open $(3+1)$-coloured graph drawn in Figure
A.6 drawn above. Let us record the following immediate corollary:

Let $\mathcal{G}\in\mathfrak{G}_{d}$ be a $(d+1)$-coloured graph representing
a manifold. Then every $k$-wound in $\mathcal{G}$ is proper.

###### Proof.

Let $\mathcal{W}_{k}=\\{v,w\\}$ be a $k$-wound of colours
$i_{1},\dots,i_{k}\in\mathcal{C}_{d}$ in $\mathcal{G}$. Since $\mathcal{G}$
represents a manifold, all its $k$-bubbles represent $(k-1)$-spheres or
$(k-1)$-balls (Proposition 3.2.2). Hence, assumption (1) of the proposition
above is clearly satisfied. Let now $\mathcal{B}_{vw}$ be the $(k+1)$-bubble
in $\mathcal{G}$ of colours $0,i_{1},\dots,i_{k}$ containing $v$ and $w$. By
assumption, this bubble represents a $k$-ball, which only has two external
legs. Connecting these two external legs hence produces a $k$-bubble, which
represents a $k$-sphere.∎

In general, as explained in some previous parts, every non-internal proper
$k$-dipole move in some open $(d+1)$-coloured graph induces a $k$-dipole in
its boundary graph. However, the reverse of this statement is in general not
true. What is true instead is that every proper $k$-dipole on the boundary
graph corresponds to a $k$-wound in the corresponding open graph. This
$k$-wound does not have to be proper, since manifolds with homeomorphic
boundaries can obviously still have non-homeomorphic interiors.

####  A.4.3 Pinched Manifolds and Non-Proper Dipole Moves

In dimension $d=3$, there is a particular class of pseudomanifolds, which is
stable under any wound wove and dipole contraction (proper and non-proper).
These type of topologies are called “pinched manifolds”. In order to define
them, let us introduce the following terminology: The “$\lambda$-punctured
$2$-sphere” $S^{2}_{\lambda}$ for some $\lambda\geq 0$ is the surface obtained
from the $2$-sphere $S^{2}$ by deleting the interior of $\lambda$ $2$-balls
(=disks). In other words, $S^{2}_{\lambda}$ denotes the unique (up to
homeomorphism) surface of genus $g=0$ with $\lambda$ boundary components. Its
Euler characteristic is hence given by $\chi(S^{2}_{\lambda})=2-\lambda$. If
$\lambda=0$, then $S^{2}_{\lambda=0}$ is just the $2$-sphere, if $\lambda=1$,
then $S^{2}_{\lambda=1}$ is homeomorphic to the disk and if $\lambda=2$,
$S^{2}_{\lambda=2}$ is homeomorphic to the cylinder. Using this, we define
pinched manifolds as follows:

(Pinched Manifolds)
A $3$-dimensional simplicial complex is called a “combinatorial pinched
manifolds”, if the link of every vertex represents a punctured $2$-sphere. The
corresponding geometric realization is called a “pinched manifold”.

Of course, every PL-manifold is also a pinched manifold, since
$S^{2}_{\lambda=0}=S^{2}$ and $S^{2}_{\lambda=1}=D^{2}$. On the other hand,
not every pinched manifold is a manifold. If it is pure, non-branching and
strongly-connected, then it is, however, always a normal pseudomanifold, by
definition. As proven in [Gag87], if $\mathcal{G}$ is some open
$(3+1)$-coloured graph and $\mathcal{W}_{k}$ is a $k\in\\{1,2,3\\}$ wound in
$\mathcal{G}$, then $\mathcal{G}$ represents a pinched manifold if and only if
$\mathcal{G}_{\overline{\mathcal{W}_{k}}}$ represents a pinched manifold.
Moreover, if $d_{k}$ is a $k\in\\{1,2,3\\}$ dipole in $\mathcal{G}$, then
$\mathcal{G}/d_{k}$ represents a pinched manifold whenever $\mathcal{G}$
represents a pinched manifold. The reverse of the latter statement is in
general not true.

As an example, consider again the following open $(3+1)$-coloured graph and
the corresponding $2$-dipole contraction, which we have already discussed
previously:

Figure A.10: A non-proper $2$-dipole contraction.

The graph $\mathcal{G}$ is clearly an open $(3+1)$-graph representing a
pinched manifold, which is not a manifold: It has exactly five $3$-bubbles,
from which one represents a $2$-sphere (the unique $3$-bubble with colours
$123$), two represent the $2$-ball (the two $3$-bubbles with colours $013$)
and two represent the $2$-punctured sphere $S^{2}_{2}$ (the unique $3$-bubbles
with colours $012$ and $023$). Contracting the $2$-dipole drawn above, which
is clearly a non-proper dipole, since both of the separated $2$-bubbles are
open, but one of the vertices is internal, yields a manifold with different
topology. This contracted manifold represented by $\mathcal{G}/d_{2}$
represents clearly the $3$-ball and hence is in particular again a pinched
manifold.

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