  
  [1m[4m[31m2. Theoretical foundations[0m
  
  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].
  
  
  [1m[4m[31m2.1 Polytopes and polytopal complexes[0m
  
  A  convex  [22m[36md-polytope[0m  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 [22m[36mk-face[0m of P if dim(Pcap
  h)=k. The 0-faces are called [22m[36mvertices[0m, the 1-faces [22m[36medges[0m and the (d-1)-faces
  are called [22m[36mfacets[0m of P.
  
  A polytope P is called [22m[36mregular[0m, 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 [22m[36mk-skeleton[0m of P, written as
  skel_k(P).
  
  A  [22m[36mpolytopal  complex[0m  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 [22m[36mf-vector[0m 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 [22m[36mboundary complex
  C(partial P) of the polytope P[0m.
  
  
  [1m[4m[31m2.2 Simplices and simplicial complexes[0m
  
  A  d-dimensional  [22m[36msimplex[0m  or  [22m[36md-simplex[0m 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
  [22m[36msimplicial  complex[0m  (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  [22m[36mpure[0m  if  all  facets are of the same
  dimension.   A   pure  simplicial  complex  of  dimension  d  satisfies  the
  [22m[36mpseudomanifold condition[0m 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.
  
  
  [1m[4m[31m2.3 From geometry to combinatorics[0m
  
  Every  d-simplex  has  an [22m[36munderlying set[0m in E^d, as the set of all points of
  that  simplex.  In  the  same way one can define the [22m[36munderlying set[0m |C| of a
  simplicial  complex  C. If the underlying set of a simplicial complex C is a
  topological   manifold,   then   C   is  called  [22m[36mtriangulated  manifold[0m  (or
  [22m[36mtriangulation of |C|[0m).
  
  One  can also go the other way and assign an abstract simplicial complex (in
  form  of  a [22m[36mposet[0m) 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 [22m[36mposet[0m).
  
  Let  v be a vertex of C. The set of all facets that contain v is called [22m[36mstar
  of  v  in  C[0m  and  is denoted by star_C(v). The subcomplex of star_C(v) that
  contains all faces that does not contain v is called [22m[36mlink of v in C[0m, written
  as lk_C(v).
  
  A  [22m[36mcombinatorial  0-sphere[0m  is a 0-dimensional simplicial complex consisting
  only  of  two  (different)  vertices.  Let  us  now  come to the notion of a
  [22m[36mcombinatorial manifold[0m:
  
  A  [22m[36mcombinatorial  d-manifold[0m  is  a  d-dimensional  simplicial complex whose
  vertex   links   are   all   combinatorial  (d-1)-spheres.  A  [22m[36mcombinatorial
  pseudomanifold[0m   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 [22m[36mEdward's sphere[0m, 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:
  
  [1m[33mDimensionality.[0m
        The dimension of C.
  
  [1m[33mf, g and h-vector.[0m
        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.
  
  [1m[33mEuler characteristic[0m
        The Euler characteristic as the alternating sum over the Betti numbers
        / the f-vector.
  
  [1m[33m(Co-)Homology.[0m
        The simplicical (co-)homology groups and Betti numbers.
  
  [1m[33mConnectedness and closeness.[0m
        Whether  C  is  strongly  connected, path connected, has a boundary or
        not.
  
  [1m[33mSymmetries.[0m
        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.
  
  
  [1m[4m[31m2.4 Normal surfaces and combinatorial slicings[0m
  
  The intersection of a tetrahedron Delta with a plane that does not intersect
  any vertex of Delta is called a [22m[36mnormal subset[0m 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 [22m[36mnormal surface[0m.
  
  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 [22m[36mregular simplexwise linear (rsl)
  function[0m  or  [22m[36msimplicial  Morse  function[0m. 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 [22m[36mslicing[0m of M.
  
  See [K95] for an introduction to the theory of polyhedral Morse functions.
  
