chap2.txt

  
  2 Theoretical foundations
  
  The  purpose  of  this chapter is to recall some basic definitions regarding
  polytopes,   triangulations,   polyhedral   Morse  theory,  discrete  normal
  surfaces,  slicings, tight triangulations and simplicial blowups. The expert
  in these fields may well skip to the next chapter.
  
  For  a more detailed look the authors recommend the books [Hud69], [RS72] on
  PL-topology and [Zie95], [Gr{03] on the theory of polytopes.
  
  An  overview  of  the more recent developments in the field of combinatorial
  topology can be found in [Lut05] and [Dat07].
  
  
  2.1 Polytopes and polytopal complexes
  
  A  convex  d-polytope  is  the  convex  hull  of  n  points p_i ∈ 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  supporting  hyperplane  h  ⊂  E^d, P∩ h is called a k-face of P if
  dim(P∩  h)=k.  The  0-faces  are  called vertices, the 1-faces edges and the
  (d-1)-faces are called facets of P.
  
  A  d-polytope  P  for which all facets are congruent regular (d-1)-polytopes
  and  for  which  all  vertex  links are congruent regular (d-1)-polytopes is
  called regular, where the regular 2-polytopes are regular polygons.
  
  The  set  of  all  k-faces  of  P  is called the k-skeleton of P, written as
  skel_k(P).
  
  A polytopal complex C is a finite collection of polytopes P_i, 1 ≤ i ≤ n for
  which  the  intersection of any two polytopes P_i ∩ P_j is either empty or a
  common  face  of  P_i and P_j. The polytopes of maximal dimension are called
  the  facets  of  C. The dimension of a polytopal complex C is defined as the
  maximum over all dimensions of its facets.
  
  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(∂
  P) of the polytope P.
  
  
  2.2 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 ≤ 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 (i. e.,
  the  faces  that  are  not  included  in any other face of the complex) 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 weak
  pseudomanifold property if every (d-1)-face is part of exactly two facets.
  
  Since simplices are polytopes and, hence, simplicial complexes are polytopal
  complexes  all  of  the  terminology  regarding  simplicial complexes can be
  transfered from polytope theory.
  
  
  2.3 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 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  not  containing v is called link of v in C, written as
  lk_C(v).
  
  A  combinatorial  d-manifold  is  a  d-dimensional  simplicial complex whose
  vertex  links  are  all triangulated (d-1)-dimensional spheres with standard
  PL-structure.  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:
  
  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.
  
  (Co-)Homology.
        The simplicical (co-)homology groups and Betti numbers.
  
  Euler characteristic
        The Euler characteristic as the alternating sum over the Betti numbers
        / the f-vector.
  
  Connectedness and closedness.
        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.
  
  
  2.4 Discrete Normal surfaces
  
  The concept of normal surfaces is originally due to Kneser [Kne29] and Haken
  [Hak61]:  A  surface S, properly embedded into a 3-manifold M, is said to be
  normal,  if  it  respects  a  given cell decomposition of M in the following
  sense: It does not intersect any vertex nor touch any 3-cell of the manifold
  and  does  not  intersect with any 2-cell in a circle or an arc starting and
  ending  in a point of the same edge. Here we will look at normal surfaces in
  the  case that M is given as a combinatorial 3-manifold and we will call the
  corresponding  objects  discrete normal surfaces. In order to do this let us
  first define:
  Definition
  A  polytopal  manifold  is  a  polytopal  complex M such that there exists a
  simplicial  subdivision  of  M  which is a combinatorial manifold. If M is a
  surface we will call it a polytopal map. If, in addition M entirely consists
  of m-gons, we call it a polytopal m-gon map.
  Definition (Discrete Normal surface, [Spr11b])
  Let  M  be  a  combinatorial 3-manifold (3-pseudomanifold), ∆ ∈ M one of its
  tetrahedra  and  P  the intersection of ∆ with a plane that does not include
  any  vertex  of  ∆.  Then P is called a normal subset of ∆. Up to an isotopy
  that respects the face lattice of ∆, P is equal to one of the triangles P_i,
  1 ≤ i ≤ 4, or quadrilaterals P_i, 5 ≤ i ≤ 7, shown in Figure 7.
  A  polyhedral map S ⊂ M that entirely consists of facets P_i such that every
  tetrahedron  contains at most one facet is called discrete normal surface of
  M.
  The  second  author has recently investigated on the combinatorial theory of
  discrete normal surfaces, see [Spr11b].
  
  
  2.5 Polyhedral Morse theory and slicings
  
  In  the  field  of  PL-topology  Kühnel  developed  what  one  might  call a
  polyhedral  Morse  theory (compare [K{\95], not to be confused with Forman's
  discrete Morse theory for cell complexes which is decribed in Section 2.6):
  Let  M  be a combinatorial d-manifold. A function f:M -> R is called regular
  simplexwise  linear (rsl) if f(v) ≠ f(w) for any two vertices w ≠ v and if f
  is linear when restricted to an arbitrary simplex of the triangulation.
  A  vertex  x ∈ M is said to be critical for an rsl-function f:M -> R, if H_⋆
  (M_x , M_x backslash { x } , F) ≠ 0 where M_x := { y ∈ M | f(y) ≤ f(x) } and
  F is a field.
  It  follows that no point of M can be critical except possibly the vertices.
  In arbitrary dimensions we define:
  Definition (Slicing, [Spr11b])
  Let  M  be  a  combinatorial  pseudomanifold  of dimension d and f:M -> R an
  rsl-function.  Then we call the pre-image f^-1 (α) a slicing of M whenever α
  ≠ f(v) for any vertex v ∈ M.
  By construction, a slicing is a polytopal (d-1)-manifold and for any ordered
  pair  α  ≤  β  we  have f^-1 (α) ≅ f^-1 (β) whenever f^-1([α,β]) contains no
  vertex of M. In particular, a slicing S of a closed combinatorial 3-manifold
  M is a discrete normal surface: It follows from the simplexwise linearity of
  f  that  the  intersection of the pre-image with any tetrahedron of M either
  forms  a  single  triangle  or  a  single quadrilateral. In addition, if two
  facets  of  S  lie  in adjacent tetrahedra they either are disjoint or glued
  together  along  the  intersection  line  of  the  pre-image  and the common
  triangle.
  Any  partition  of  the  set  of  vertices  V  =  V_1  dot∪ V_2 of M already
  determines a slicing: Just define an rsl-function f: M -> R with f(v) ≤ f(w)
  for  all  v  ∈  V_1  and  w  ∈  V_2 and look at a suitable pre-image. In the
  following  we  will  write S_(V_1,V_2) for the slicing defined by the vertex
  partition V = V_1 dot∪ V_2.
  Every  vertex  of  a  slicing  is  given  as  an  intersection  point of the
  corresponding  pre-image with an edge ⟨ u,w ⟩ of the combinatorial manifold.
  Since  there  is  at  most  one such intersection point per edge, we usually
  label  this  vertex  of  the  slicing  according  to  the  vertices  of  the
  corresponding edge, that is binomuw with u ∈ V_1 and w ∈ V_2.
  Every  slicing  decomposes  the surrounding combinatorial manifold M into at
  least  2  pieces  (an  upper part M^+ and a lower part M^-). This is not the
  case  for  discrete  normal  surfaces (see 2.4) in general. However, we will
  focus  on  the  case where discrete normal surfaces are slicings and we will
  apply the above notation for both types of objects.
  Since  every combinatorial pseudomanifold M has a finite number of vertices,
  there  exist  only  a  finite number of slicings of M. Hence, if f is chosen
  carefully,  the  induced  slicings  admit  a useful visualization of M, c.f.
  [SK11].
  
  
  2.6 Discrete Morse theory
  
  For  an introduction into Forman's discrete Morse theory see [For95], not to
  be  confused with Banchoff and Kühnel's theory of regular simplexwise linear
  functions which is described in Section 2.5).
  
  
  2.7 Tightness and tight triangulations
  
  Tightness is a notion developed in the field of differential geometry as the
  equality  of the (normalized) total absolute curvature of a submanifold with
  the  lower  bound  sum  of  the  Betti numbers [Kui84], [BK97]. It was first
  studied  by  Alexandrov,  Milnor,  Chern  and  Lashof  and  Kuiper and later
  extended  to  the  polyhedral  case  by Banchoff [Ban65], Kuiper [Kui84] and
  Kühnel  [K{\95].  From  a  geometrical  point  of  view,  tightness  can  be
  understood  as  a generalization of the concept of convexity that applies to
  objects  other  than topological balls and their boundary manifolds since it
  roughly  means  that  an  embedding  of  a  submanifold  is  ``as  convex as
  possible'' according to its topology. The usual definition is the following:
  Definition (Tightness, [K{\95])
  Let  F  be  a  field.  An  embedding M → E^N of a compact manifold is called
  k-tight  with  respect  to  F if for any open or closed halfspace h⊂ E^N the
  induced homomorphism
  
  
        H_i(M\cap h;\mathbb{F})\longrightarrow H_i(M;\mathbb{F}) 
  
  
  is  injective  for all i≤ k. M is called F-tight if it is k-tight for all k.
  The  standard  choice  for the field of coefficients is F_2 and an F_2-tight
  embedding is called tight.
  With  regard  to  PL  embeddings  of PL manifolds tightness of combinatorial
  manifolds  can  also  be  defined  via  a  purely combinatorial condition as
  follows. For an introduction to PL topology see [RS72].
  Definition (Tight triangulation [K{\95])
  Let  F  be  a field. A combinatorial manifold K on n vertices is called (k-)
  tight  w.r.t.  F  if  its  canonical  embedding K⊂ ∆^n-1⊂ E^n-1 is (k-)tight
  w.r.t. F, where ∆^n-1 denotes the (n-1)-dimensional simplex.
  In  dimension  d=2 the following are equivalent for a triangulated surface S
  on  n  vertices: (i) S has a complete edge graph K_n, (ii) S appears as a so
  called  regular case in Heawood's Map Color Theorem [Rin74], compare [K{\95]
  and  (iii)  the  induced  piecewise  linear  embedding  of  S into Euclidean
  (n-1)-space has the two-piece property [Ban74], and it is tight [K{\95].
  Kühnel   investigated  the  tightness  of  combinatorial  triangulations  of
  manifolds also in higher dimensions and codimensions, see [K{\94]. It turned
  out  that  the tightness of a combinatorial triangulation is closely related
  to  the  concept of Hamiltonicity of a polyhedral complexes (see [K{\95]): A
  subcomplex A of a polyhedral complex K is called k-Hamiltonian if A contains
  the  full k-dimensional skeleton of K (not to be confused with the notion of
  a  k-Hamiltonian  graph). This generalization of the notion of a Hamiltonian
  circuit  in  a  graph  seems  to  be  due to C.Schulz [Sch94]. A Hamiltonian
  circuit  then  becomes  a  special  case  of a 0-Hamiltonian subcomplex of a
  1-dimensional graph or of a higher-dimensional complex.
  A  triangulated  2k-manifold  that  is  a  k-Hamiltonian  subcomplex  of the
  boundary complex of some higher dimensional simplex is a tight triangulation
  as   Kühnel   [K{\95]   showed.   Such   a   triangulation  is  also  called
  (k+1)-neighborly  triangulation  since  any  k+1 vertices in a k-dimensional
  simplex  are  common neighbors. Moreover, (k+1)-neighborly triangulations of
  2k-manifolds  are  also referred to as super-neighborly triangulations -- in
  analogy  with neighborly polytopes the boundary complex of a (2k+1)-polytope
  can  be  at  most  k-neighborly  unless  it  is  a simplex. Notice here that
  combinatorial  2k-manifolds  can  go  beyond  k-neighborliness, depending on
  their topology.
  Whereas  in the 2-dimensional case all tight triangulations of surfaces were
  classified  by  Ringel  and  Jungerman  and Ringel, in dimensions d≥ 3 there
  exist  only  a  finite number of known examples of tight triangulations (see
  [KL99]  for  a  census)  apart  from  the  trivial case of the boundary of a
  simplex  and an infinite series of triangulations of sphere bundles over the
  circle due to Kühnel [K{\95], [K{\86].
  
  
  2.8 Simplicial blowups
  
  The  blowing up process or Hopf σ-process can be described as the resolution
  of  nodes or ordinary double points of a complex algebraic variety. This was
  described  by  H.~Hopf  in  [Hop51],  compare  [Hir53] and [Hau00]. From the
  topological  point of view the process consists of cutting out some subspace
  and  gluing  in some other subspace. In complex algebraic geometry one point
  is  replaced  by the projective line CP^1 ≅ S^2 of all complex lines through
  that  point. This is often called blowing up of the point or just blowup. In
  general  the process can be applied to non-singular 4-manifolds and yields a
  transformation  of  a manifold M to M # (+CP^2) or M # (-CP^2), depending on
  the choice of an orientation. The same construction is possible for nodes or
  ordinary  double  points  (a  special  type  of singularities), and also the
  ambiguity  of  the orientation is the same for the blowup process of a node.
  Similarly it has been used in arbitrary even dimension by Spanier [Spa56] as
  a so-called dilatation process.
  A PL version of the blowing up process is the following: We cut out the star
  of  one of the singular vertices which is, in the case of an ordinary double
  point,  nothing  but  a  cone  over a triangulated RP^3. The boundary of the
  resulting   space  is  this  triangulated  RP^3.  Now  we  glue  back  in  a
  triangulated  version  mathbfC  of  a complex projective plane with a 4-ball
  removed  where  antipodal  points of the boundary are identified. mathbfC is
  called  a  triangulated mapping cylinder and by construction its boundary is
  PL homeomorphic to RP^3.
  For  a  combinatorial version with concrete triangulations, however, we face
  the  problem  that these two triangulations are not isomorphic. This implies
  that  before  cutting out and gluing in we have to modify the triangulations
  by bistellar moves until they coincide:
  Definition (Simplicial blowup, [SK11])
  Let  v  be  a  vertex  of  a  combinatorial 4-pseudomanifold M whose link is
  isomorphic  with  the  particular  11-vertex  triangulation of RP^3 which is
  given  by  the boundary complex of the triangulated mathbfC given in [SK11].
  Let  ψ  :  operatornamelk(v)  →  ∂mathbfC  denote  such  an  isomorphism.  A
  simplicial  resolution  of  the  singularity  v  is  given  by the following
  construction M ↦ widetildeM := (M ∖ operatornamestar(v)^∘) ∪_ψ mathbfC.
  The  process is described in more detail in [SK11]. In particular it is used
  to transform a 4-dimensional Kummer variety into a K3 surface.