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 [H69], [RS72] on PL-topology and [Z95], [G03] on the theory of polytopes.
An overview of the more recent developments in the field of combinatorial topology can be found in [L05] and [D07].
A convex d-polytope is the convex hull of n points p_i in E^d in the d-dimensional euclidean space:
\[P= conv \{v_1,\dots,v_n\}\subset E^d, \]
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, Pcap h is called a k-face of P if dim(Pcap 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.
The set of all k-faces of P is called the k-skeleton of P, written as skel_k(P).
A polytopal complex is a finite collection of polytopes P_i, 1 <= i <= 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.
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 <= 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.
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).
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.
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).
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.
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 [BL00].
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:
The dimension of C.
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.
The Euler characteristic as the alternating sum over the Betti numbers / the f-vector.
The simplicical (co-)homology groups and Betti numbers.
Whether C is strongly connected, path connected, has a boundary or not.
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.
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.
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 -> R with f|_Delta is linear for any Delta and f(v) <> f(w) whenever v <> w is called regular simplexwise linear (rsl) function or simplicial Morse function. We call a level set f^-1 (alpha), alpha in operatornameIm(f) subset R of an rsl-function that does not hit any vertex on M a slicing of M.
See [K95] for an introduction to the theory of polyhedral Morse functions.
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