Polyhedral Subdivisions and Projections of Polytopes Jörg Rambau Dissertation (Advisor: Günter M. Ziegler, TU Berlin)

   
Polytopal and Simplicial Complexes

We refer to ZIEGLER [75, Lecture 5] for details.

Definition 4.4.1   A (geometric)   polytopal complex $\mathcal{C}$ in $\mathbb{R} ^d$ is a collection of polytopes in $\mathbb{R} ^d$ such that

(i)

the empty set is in $\mathcal{C}$,

(ii)

for any $P \in \mathcal{C}$ all faces of $P$ are in $\mathcal{C}$,

(iii)

the intersection of any two polytopes in $\mathcal{C}$ is a face of both.

The dimension of $\mathcal{C}$ is the largest dimension of a polytope in $\mathcal{C}$, the union $\abs{\mathcal{C}}$ of all polytopes in $\mathcal{C}$ is called the   underlying set of $\mathcal{C}$. The   facets are the inclusion-maximal elements of  $\mathcal{C}$. If all facets are of the same dimension then $\mathcal{C}$ is   pure. Two polytopal complexes $\mathcal{C}$ and $\mathcal{C}'$ are   combinatorially isomorphic if there is a bijection from $\mathcal{C}$ to $\mathcal{C}'$ preserving the incidence relations.

If all polytopes in $\mathcal{C}$ are simplices then $\mathcal{C}$ is a (geometric)   simplicial complex.

As an example of a polytopal complex, consider the set of all faces of a polytope. Another example is a   polyhedral subdivision of a point configuration (see Section 1.1(a)).

Definition 4.4.2   Let $\mathcal{C}$ be a pure $d$-dimensional polytopal complex. A   shelling of $\mathcal{C}$ is an ordering $(F_1, F_2, \dots , F_r)$ of its facets such that

(i)

$\partial F_1$ has a shelling, and

(ii)

for all $i = 1, \dots , r$ the set $F_i \cap (F_1 \cup \dots \cup F_{i-1})$ is a beginning segment of a shelling of $\partial F_i$.

Here a shelling of a $0$-dimensional complex is just any linear ordering of its vertices. A polytopal complex is   shellable if it has a shelling.

  The following theorem led to the proof of the Upper Bound Theorem by MCMULLEN [49] (see Section A.3, Theorem A.3.9).

Theorem 4.4.3   (BRUGESSER & MANI [18]) Every polytope is shellable. In particular, every regular polyhedral subdivision is shellable.

In the case of simplicial complexes things are easier because of the following simple lemma.

Lemma 4.4.4   If $\mathcal{C}$ is a pure $d$-dimensional simplicial complex, then an ordering $(F_1, F_2, \dots , F_r)$ of its facets is a shelling of $\mathcal{C}$ if and only if $F_i \cap (F_1 \cup \dots \cup F_{i-1})$ is of pure dimension $(d-1)$ for all $i = 1, \dots , r$.

Since any subset of vertices of a simplex is the vertex set of a face, it is more convenient to describe a simplicial complex in terms of the associated   abstract simplicial complex in the set of its vertices.

Definition 4.4.5   Let $\mathcal{L}$ be a finite set. An   abstract simplicial complex in $\mathcal{L}$ is a non-empty family $K$ of subsets of $\mathcal{L}$ that is closed under taking subsets. That is, if $S \in K$ and $S' \subseteq S$ then $S'$ is also in $K$. The union of all sets in $K$ is the   support $\supp (K) \subseteq \mathcal{L}$ of $K$. The   dimension of $K$ is the maximal cardinality of a set in $K$ minus $1$.

There is no real difference between the concepts of geometric and abstract simplicial complexes. This is shown by the following lemma.

Lemma 4.4.6   For every $d$-dimensional abstract simplicial complex $K$ with $n$ vertices, there is a geometric simplicial complex $\Delta$ in  $\mathbb{R} ^{2d-1}$ that is combinatorially isomorphic to $K$.

This is best possible in general, and the proof requires tools. It is, however, trivial to embed an abstract simplicial complex into  $\mathbb{R} ^n$. We will need the following standard operations on abstract simplicial complexes.

Definition 4.4.7   Let $K$ be an abstract simplicial complex in $\mathcal{L}$ and $S_0 \in K$ a simplex in $K$. Then the following abstract simplicial complexes are defined.         $$\begin{align*}\st_{K}(S_0) &:= \setof{S \in K}{ S \cup S_0 \in K}, \tag*{(star... ...K * K' &:= \setof{S \cup S'}{ S \in K, S' \in K'}. \tag*{(join)} \end{align*}$$