Theoretical foundations

The purpose of this chapter is to recall some basic definitions regarding polytopes, triangulations and PL topology. The expert in this field may well skip to the next chapter.

For a more detailed look the authors recommend the books , on PL-topology and , on the theory of polytopes.

An overview of the more recent developments in the field of combinatorial topology can be found in and .

Polytopes and polytopal complexes

A convex d-polytope is the convex hull of n points p_i \in E^d in the d-dimensional euclidean space:

P=

\operatorname{conv}
conv

\{v_1,\dots,v_n\}\subset E^d,

where the v_1,\dots,v_n do not lie in a hyperplane of E^d.

From now on when talking about polytopes in this document always convex polytopes are meant unless explicitly stated otherwise.

For any hyperplane h \subset E^d, P\cap h is called a k-face of P if \dim(P\cap h)=k. The 0-faces are called vertices, the 1-faces edges and the (d-1)-faces are called facets of P.

A polytope P is called regular, if all its (d-1)-faces are congruent regular (d-1)-polytopes. A regular 1-polytope is a regular n-gon.

Figure 1 below shows the only five regular convex 3-polytopes (also known as \emph{platonic solids}).

\begin{center} \begin{tabular}{@{}lllll@{}} \includegraphics[width=2cm]{figures/platonicsolids1}& \includegraphics[width=2cm]{figures/platonicsolids2}& \includegraphics[width=2cm]{figures/platonicsolids3}& \includegraphics[width=2cm]{figures/platonicsolids4}& \includegraphics[width=2cm]{figures/platonicsolids5} \end{tabular}\\\bigskip {\small Figure 1. The \emph{platonic solids} as the five regular convex 3-polytopes.} \end{center}

The set of all k-faces of P is called the k-skeleton of P, written as

$\operatorname{skel}_k(P)$
skel_k(P).
\begin{center} \includegraphics[width=8cm]{figures/skelcube}\\\bigskip {\small Figure 2. From left to right, drawn in red: the 0-skeleton, the 1-skeleton and the 2-skeleton of the cube.} \end{center}

A polytopal complex is a finite collection of polytopes P_i, 1 \leq i \leq n, for which the intersection of any two polytopes P_i \cap P_j is either empty or a common face of P_i and P_j.

For every d-dimensional polytopal complex the (d+1)-tuple, containing its number of i-faces in the i-th entry is called the f-vector of the polytopal complex.

Every polytope P gives rise to a polytopal complex consisting of all the proper faces of P. This polytopal complex is called the boundary complex C(\partial P) of the polytope P.

Figure 2 below shows the boundary complex of the cube.

\begin{center} \includegraphics[width=8cm]{figures/cubecomplex}\\\bigskip {\small Figure 3. The 3-cube (left) and its boundary complex (right) where the 0-faces shown in black, the 1-faces dark gray and the 2-faces in light gray.} \end{center}
Simplices and simplicial complexes

A d-dimensional simplex or d-simplex for short is the convex hull of d+1 points in E^d in general position. Thus the d-simplex is the smallest (with respect to the number of vertices) possible d-polytope. Every face of the d-simplex is a m-simplex, m \leq d.

A 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, a 3-simplex a tetrahedron, and so on.

\begin{center} \includegraphics[width=10cm]{figures/simplices}\\\bigskip {\small Figure 4. From left to right: a 0-simplex, a 1-simplex, a 2-simplex, a 3-simplex and a Schlegel diagram of a 4-simplex.} \end{center}

A polytopal complex which entirely consists of simplices is called a simplicial complex (for this it actually suffices that the facets of a polytopal complex are simplices).

\begin{center} \includegraphics[height=4cm]{figures/simplcompl}\\\bigskip {\small Figure 4. A simplicial complex (left) and a collection of simplices that does not form a simplicial complex (right).} \end{center}

The dimension of a simplicial complex is the maximal dimension of a facet. A simplicial complex is said to be pure if all facets are of the same dimension. A pure simplicial complex of dimension d satisfies the pseudomanifold condition if every (d-1)-face is part of exactly two facets.

Other properties (faces, facets, etc.) are defined in the same way as for polytopes and polytopal complexes.

From geometry to combinatorics

Every d-simplex has an underlying set in E^d, as the set of all points of that simplex. In the same way one can define the underlying set |C| of a simplicial complex C. If the underlying set of a simplicial complex C is a topological manifold, then C is called triangulated manifold (or triangulation of |C|).

One can also go the other way and assign an abstract simplicial complex (in form of a poset) to a geometrical one by identifying each simplex with its vertex set. This obviously defines a set of sets with a natural partial ordering given by the inclusion (a socalled poset).

\begin{center} \includegraphics[height=3.5cm]{figures/hasse_complex}\quad \includegraphics[height=3.5cm]{figures/hasse_diagram}\\\bigskip {\small Figure 5. A geometrical simplicial complex (left) and its abstract version in form of a poset (right).} \end{center}

Let v be a vertex of C. The set of all facets that contain v is called star of v in C and is denoted by star_C(v). The subcomplex of star_C(v) that contains all faces that does not contain v is called link of v in C, written as lk_C(v).

A combinatorial 0-sphere is a 0-dimensional simplicial complex consisting only of two (different) vertices. Let us now come to the notion of a combinatorial manifold:

A combinatorial d-manifold is a d-dimensional simplicial complex whose vertex links are all combinatorial (d-1)-spheres. A combinatorial pseudomanifold is a simplicial complex whose vertex links are all combinatorial (d-1)-manifolds.

\begin{center} \includegraphics[height=.5\textwidth]{figures/moebius}\\\bigskip {\small Figure 6. A simplicial complex that is a vertex-minimal combinatorial triangulation of the torus $T^2$ (so called M\"obius' torus) -- each vertex link is a hexagon.} \end{center}

Note, that every combinatorial manifold is a triangulated manifold. The opposite is wrong: for example, there exists a triangulation of the 5-sphere that is not combinatorial, the so called Edward's sphere, see .

A combinatorial manifold carries an induced PL-structure and can be understood in terms of an abstract simplicial complex. If the complex has d vertices there exists a natural embedding of C into the (d-1) simplex and, thus, into E^{d-1}. In general, there is no canonical embedding into any lower dimensional space. However, combinatorial methods allow to examine a given simplicial complex independently from an embedding and, in particular, independently from vertex coordinates.

Some fundamental properties of an abstract simplicial complex C are the following:

Dimensionality.
The dimension of C.
f
, g and h-vector.
The f-vector (f_k equals the number of k-faces of a simplicial complex), the g- and h-vector can be obtained from the f-vector via linear transformations.
Euler characteristic
The Euler characteristic as the alternating sum over the Betti numbers / the f-vector.
(Co-)Homology.
The simplicical (co-)homology groups and Betti numbers.
Connectedness and closeness.
Whether C is strongly connected, path connected, has a boundary or not.
Symmetries.
The automorphism group, i. e. the group of all permutations on the set of vertex labels that do not change the complex as a whole.

All of those properties and many more can be computed on a strictly combinatorial basis.

Normal surfaces and combinatorial slicings

The intersection of a tetrahedron \Delta with a plane that does not intersect any vertex of \Delta is called a normal subset of \Delta. A closed PL-surface, properly embedded into a combinatorial 3-manifold M, which is equal to a finite union of normal subsets of tetrahedra of M is called normal surface.

\medskip
Let M be a closed comb. 3-mfld., \Delta \in M a tetrahedron and v,w \in V two vertices in M. A function f : M \to \mathbb{R} with f|_{\Delta} is linear for any \Delta and f(v) \neq f(w) whenever v \neq w is called regular simplexwise linear (rsl) function or simplicial Morse function. We call a level set f^{-1} (\alpha), \alpha \in \operatorname{Im}(f) \subset \mathbb{R} of an rsl-function that does not hit any vertex on M a slicing of M.

\medskip
See for an introduction to the theory of polyhedral Morse functions.

\begin{center} \includegraphics[width=.8\textwidth]{figures/haken.pdf}\\\bigskip {\small Figure 7. The seven different normal subsets of the tetrahedron.} \end{center}

\begin{center} \includegraphics[width=.8\textwidth]{figures/slicing.pdf}\\\bigskip {\small Figure 8. One dimensional slicing of the $2$-sphere (represented as the boundary of the $3$-simplex) seen as a level set of a regular point of a simplicial Morse function.} \end{center}

\begin{center} \includegraphics[width=.4\textwidth]{figures/3x3gridtorus.pdf}\\\bigskip {\small Figure 9. Handlebody decomposition of genus $1$ of a $6$-vertex $3$-sphere - a $3 \times 3$-grid torus.} \end{center}

\begin{center} \includegraphics[width=.5\textwidth]{figures/titlepicture.pdf}\\\bigskip {\small Figure 10. Separating sphere of an $8$-vertex cylinder $S_4^2 \times [0,1]$ - A cuboctahedron (drawn as a Schlegel diagram of a quadrilateral face).} \end{center}