  
  [1X2 [33X[0;0YTheoretical foundations[133X[101X
  
  [33X[0;0YThe  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.[133X
  
  [33X[0;0YFor  a more detailed look the authors recommend the books [Hud69], [RS72] on
  PL-topology and [Zie95], [Gr{03] on the theory of polytopes.[133X
  
  [33X[0;0YAn  overview  of  the more recent developments in the field of combinatorial
  topology can be found in [Lut05] and [Dat07].[133X
  
  
  [1X2.1 [33X[0;0YPolytopes and polytopal complexes[133X[101X
  
  [33X[0;0YA  convex  [13X[22Xd[122X-polytope[113X  is  the  convex  hull  of  [22Xn[122X  points [22Xp_i ∈ E^d[122X in the
  [22Xd[122X-dimensional euclidean space:[133X
  
  
        [33X[1;6Y[24X[33X[0;0YP= conv \{v_1,\dots,v_n\}\subset E^d,[133X [124X[133X
  
  
  [33X[0;0Ywhere the [22Xv_1,dots,v_n[122X do not lie in a hyperplane of [22XE^d[122X.[133X
  
  [33X[0;0YFrom  now  on  when  talking  about polytopes in this document always convex
  polytopes are meant unless explicitly stated otherwise.[133X
  
  [33X[0;0YFor  any  supporting  hyperplane  [22Xh  ⊂  E^d[122X, [22XP∩ h[122X is called a [13X[22Xk[122X-face[113X of [22XP[122X if
  [22Xdim(P∩  h)=k[122X.  The  0-faces  are  called [13Xvertices[113X, the 1-faces [13Xedges[113X and the
  [22X(d-1)[122X-faces are called [13Xfacets[113X of [22XP[122X.[133X
  
  [33X[0;0YA  [22Xd[122X-polytope  [22XP[122X  for which all facets are congruent regular [22X(d-1)[122X-polytopes
  and  for  which  all  vertex  links are congruent regular [22X(d-1)[122X-polytopes is
  called regular, where the regular [22X2[122X-polytopes are regular polygons.[133X
  
  [33X[0;0YThe  set  of  all  [22Xk[122X-faces  of  [22XP[122X  is called the [13X[22Xk[122X-skeleton[113X of [22XP[122X, written as
  skel[22X_k(P)[122X.[133X
  
  [33X[0;0YA [13Xpolytopal complex[113X [22XC[122X is a finite collection of polytopes [22XP_i[122X, [22X1 ≤ i ≤ n[122X for
  which  the  intersection of any two polytopes [22XP_i ∩ P_j[122X is either empty or a
  common  face  of  [22XP_i[122X and [22XP_j[122X. The polytopes of maximal dimension are called
  the  [13Xfacets[113X  of  [22XC[122X. The [13Xdimension[113X of a polytopal complex [22XC[122X is defined as the
  maximum over all dimensions of its facets.[133X
  
  [33X[0;0YFor  every  [22Xd[122X-dimensional  polytopal complex the [22X(d+1)[122X-tuple, containing its
  number  of [22Xi[122X-faces in the [22Xi[122X-th entry is called the [13X[22Xf[122X-vector[113X of the polytopal
  complex.[133X
  
  [33X[0;0YEvery  polytope  [22XP[122X  gives  rise to a polytopal complex consisting of all the
  proper faces of [22XP[122X. This polytopal complex is called the [13Xboundary complex [22XC(∂
  P)[122X of the polytope [22XP[122X[113X.[133X
  
  
  [1X2.2 [33X[0;0YSimplices and simplicial complexes[133X[101X
  
  [33X[0;0YA  [22Xd[122X-dimensional  [13Xsimplex[113X  or  [13X[22Xd[122X-simplex[113X for short is the convex hull of [22Xd+1[122X
  points  in [22XE^d[122X in general position. Thus the [22Xd[122X-simplex is the smallest (with
  respect  to  the  number of vertices) possible [22Xd[122X-polytope. Every face of the
  [22Xd[122X-simplex is a [22Xm[122X-simplex, [22Xm ≤ d[122X.[133X
  
  [33X[0;0YA  0-simplex  is  a  point,  a [22X1[122X-simplex is a line segment, a [22X2[122X-simplex is a
  triangle, a [22X3[122X-simplex a tetrahedron, and so on.[133X
  
  [33X[0;0YA  polytopal  complex  which  entirely  consists  of  simplices  is called a
  [13Xsimplicial  complex[113X  (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).[133X
  
  [33X[0;0YThe dimension of a simplicial complex is the maximal dimension of a facet. A
  simplicial  complex  is  said  to  be  [13Xpure[113X  if  all  facets are of the same
  dimension.  A  pure  simplicial  complex  of  dimension [22Xd[122X satisfies the [13Xweak
  pseudomanifold property[113X if every [22X(d-1)[122X-face is part of exactly two facets.[133X
  
  [33X[0;0YSince simplices are polytopes and, hence, simplicial complexes are polytopal
  complexes  all  of  the  terminology  regarding  simplicial complexes can be
  transfered from polytope theory.[133X
  
  
  [1X2.3 [33X[0;0YFrom geometry to combinatorics[133X[101X
  
  [33X[0;0YEvery  [22Xd[122X-simplex  has  an [13Xunderlying set[113X in [22XE^d[122X, as the set of all points of
  that  simplex.  In  the  same way one can define the [13Xunderlying set[113X [22X|C|[122X of a
  simplicial  complex  [22XC[122X. If the underlying set of a simplicial complex [22XC[122X is a
  topological   manifold,   then   [22XC[122X   is  called  [13Xtriangulated  manifold[113X  (or
  [13Xtriangulation of [22X|C|[122X[113X).[133X
  
  [33X[0;0YOne 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 [13Xposet[113X).[133X
  
  [33X[0;0YLet  [22Xv[122X be a vertex of [22XC[122X. The set of all facets that contain [22Xv[122X is called [13Xstar
  of  [22Xv[122X  in  [22XC[122X[113X  and  is denoted by star[22X_C(v)[122X. The subcomplex of star[22X_C(v)[122X that
  contains  all  faces  not  containing [22Xv[122X is called [13Xlink of [22Xv[122X in [22XC[122X[113X, written as
  lk[22X_C(v)[122X.[133X
  
  [33X[0;0YA  [13Xcombinatorial  [22Xd[122X-manifold[113X  is  a  [22Xd[122X-dimensional  simplicial complex whose
  vertex  links  are  all triangulated [22X(d-1)[122X-dimensional spheres with standard
  PL-structure.  A  [13Xcombinatorial pseudomanifold[113X is a simplicial complex whose
  vertex links are all combinatorial [22X(d-1)[122X-manifolds.[133X
  
  [33X[0;0YNote  that  every  combinatorial  manifold  is  a triangulated manifold. The
  opposite is wrong: for example, there exists a triangulation of the [22X5[122X-sphere
  that is not combinatorial, the so called [13XEdward's sphere[113X, see [BL00].[133X
  
  [33X[0;0YA  combinatorial  manifold  carries  an  induced  PL-structure  and  can  be
  understood  in terms of an abstract simplicial complex. If the complex has [22Xd[122X
  vertices  there  exists a natural embedding of [22XC[122X into the [22X(d-1)[122X simplex and,
  thus, into [22XE^d-1[122X. 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.[133X
  
  [33X[0;0YSome  fundamental  properties  of  an  abstract simplicial complex [22XC[122X are the
  following:[133X
  
  [8XDimensionality.[108X
        [33X[0;6YThe dimension of [22XC[122X.[133X
  
  [8X[22Xf[122X, [22Xg[122X and [22Xh[122X-vector.[108X
        [33X[0;6YThe  [22Xf[122X-vector  ([22Xf_k[122X  equals  the  number  of  [22Xk[122X-faces  of a simplicial
        complex),  the  [22Xg[122X-  and [22Xh[122X-vector can be obtained from the [22Xf[122X-vector via
        linear transformations.[133X
  
  [8X(Co-)Homology.[108X
        [33X[0;6YThe simplicical (co-)homology groups and Betti numbers.[133X
  
  [8XEuler characteristic[108X
        [33X[0;6YThe Euler characteristic as the alternating sum over the Betti numbers
        / the [22Xf[122X-vector.[133X
  
  [8XConnectedness and closedness.[108X
        [33X[0;6YWhether  [22XC[122X  is  strongly  connected, path connected, has a boundary or
        not.[133X
  
  [8XSymmetries.[108X
        [33X[0;6YThe automorphism group, i. e. the group of all permutations on the set
        of vertex labels that do not change the complex as a whole.[133X
  
  [33X[0;0YAll  of  those  properties  and  many  more  can  be  computed on a strictly
  combinatorial basis.[133X
  
  
  [1X2.4 [33X[0;0YDiscrete Normal surfaces[133X[101X
  
  [33X[0;0YThe concept of [13Xnormal surfaces[113X is originally due to Kneser [Kne29] and Haken
  [Hak61]:  A  surface [22XS[122X, properly embedded into a [22X3[122X-manifold [22XM[122X, is said to be
  [13Xnormal[113X,  if  it  respects  a  given cell decomposition of [22XM[122X in the following
  sense: It does not intersect any vertex nor touch any [22X3[122X-cell of the manifold
  and  does  not  intersect with any [22X2[122X-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 [22XM[122X is given as a combinatorial [22X3[122X-manifold and we will call the
  corresponding  objects  [13Xdiscrete normal surfaces[113X. In order to do this let us
  first define:[133X
  [33X[0;0Y[12XDefinition[112X[133X
  [33X[0;0YA  [13Xpolytopal  manifold[113X  is  a  polytopal  complex [22XM[122X such that there exists a
  simplicial  subdivision  of  [22XM[122X  which is a combinatorial manifold. If [22XM[122X is a
  surface we will call it a [13Xpolytopal map[113X. If, in addition [22XM[122X entirely consists
  of [22Xm[122X-gons, we call it a [13Xpolytopal [22Xm[122X-gon map[113X.[133X
  [33X[0;0Y[12XDefinition[112X (Discrete Normal surface, [Spr11b])[133X
  [33X[0;0YLet  [22XM[122X  be  a  combinatorial [22X3[122X-manifold ([22X3[122X-pseudomanifold), [22X∆ ∈ M[122X one of its
  tetrahedra  and  [22XP[122X  the intersection of [22X∆[122X with a plane that does not include
  any  vertex  of  [22X∆[122X.  Then [22XP[122X is called a [13Xnormal subset[113X of [22X∆[122X. Up to an isotopy
  that respects the face lattice of [22X∆[122X, [22XP[122X is equal to one of the triangles [22XP_i[122X,
  [22X1 ≤ i ≤ 4[122X, or quadrilaterals [22XP_i[122X, [22X5 ≤ i ≤ 7[122X, shown in Figure 7.[133X
  [33X[0;0YA  polyhedral map [22XS ⊂ M[122X that entirely consists of facets [22XP_i[122X such that every
  tetrahedron  contains at most one facet is called [13Xdiscrete normal surface[113X of
  [22XM[122X.[133X
  [33X[0;0YThe  second  author has recently investigated on the combinatorial theory of
  discrete normal surfaces, see [Spr11b].[133X
  
  
  [1X2.5 [33X[0;0YPolyhedral Morse theory and slicings[133X[101X
  
  [33X[0;0YIn  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 [14X2.6[114X):[133X
  [33X[0;0YLet  [22XM[122X  be a combinatorial [22Xd[122X-manifold. A function [22Xf:M -> R[122X is called [13Xregular
  simplexwise  linear (rsl)[113X if [22Xf(v) ≠ f(w)[122X for any two vertices [22Xw ≠ v[122X and if [22Xf[122X
  is linear when restricted to an arbitrary simplex of the triangulation.[133X
  [33X[0;0YA  vertex  [22Xx ∈ M[122X is said to be [13Xcritical[113X for an rsl-function [22Xf:M -> R[122X, if [22XH_⋆
  (M_x , M_x backslash { x } , F) ≠ 0[122X where [22XM_x := { y ∈ M | f(y) ≤ f(x) }[122X and
  [22XF[122X is a field.[133X
  [33X[0;0YIt  follows that no point of [22XM[122X can be critical except possibly the vertices.
  In arbitrary dimensions we define:[133X
  [33X[0;0Y[12XDefinition[112X (Slicing, [Spr11b])[133X
  [33X[0;0YLet  [22XM[122X  be  a  combinatorial  pseudomanifold  of dimension [22Xd[122X and [22Xf:M -> R[122X an
  rsl-function.  Then we call the pre-image [22Xf^-1 (α)[122X a [13Xslicing[113X of [22XM[122X whenever [22Xα
  ≠ f(v)[122X for any vertex [22Xv ∈ M[122X.[133X
  [33X[0;0YBy construction, a slicing is a polytopal [22X(d-1)[122X-manifold and for any ordered
  pair  [22Xα  ≤  β[122X  we  have [22Xf^-1 (α) ≅ f^-1 (β)[122X whenever [22Xf^-1([α,β])[122X contains no
  vertex of [22XM[122X. In particular, a slicing [22XS[122X of a closed combinatorial [22X3[122X-manifold
  [22XM[122X is a discrete normal surface: It follows from the simplexwise linearity of
  [22Xf[122X  that  the  intersection of the pre-image with any tetrahedron of [22XM[122X either
  forms  a  single  triangle  or  a  single quadrilateral. In addition, if two
  facets  of  [22XS[122X  lie  in adjacent tetrahedra they either are disjoint or glued
  together  along  the  intersection  line  of  the  pre-image  and the common
  triangle.[133X
  [33X[0;0YAny  partition  of  the  set  of  vertices  [22XV  =  V_1  dot∪ V_2[122X of [22XM[122X already
  determines a slicing: Just define an rsl-function [22Xf: M -> R[122X with [22Xf(v) ≤ f(w)[122X
  for  all  [22Xv  ∈  V_1[122X  and  [22Xw  ∈  V_2[122X and look at a suitable pre-image. In the
  following  we  will  write [22XS_(V_1,V_2)[122X for the slicing defined by the vertex
  partition [22XV = V_1 dot∪ V_2[122X.[133X
  [33X[0;0YEvery  vertex  of  a  slicing  is  given  as  an  intersection  point of the
  corresponding  pre-image with an edge [22X⟨ u,w ⟩[122X 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 [22Xbinomuw[122X with [22Xu ∈ V_1[122X and [22Xw ∈ V_2[122X.[133X
  [33X[0;0YEvery  slicing  decomposes  the surrounding combinatorial manifold [22XM[122X into at
  least  [22X2[122X  pieces  (an  upper part [22XM^+[122X and a lower part [22XM^-[122X). This is not the
  case  for  discrete  normal  surfaces (see [14X2.4[114X) 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.[133X
  [33X[0;0YSince  every combinatorial pseudomanifold [22XM[122X has a finite number of vertices,
  there  exist  only  a  finite number of slicings of [22XM[122X. Hence, if [22Xf[122X is chosen
  carefully,  the  induced  slicings  admit  a useful visualization of [22XM[122X, c.f.
  [SK11].[133X
  
  
  [1X2.6 [33X[0;0YDiscrete Morse theory[133X[101X
  
  [33X[0;0YFor  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 [14X2.5[114X).[133X
  
  
  [1X2.7 [33X[0;0YTightness and tight triangulations[133X[101X
  
  [33X[0;0YTightness is a notion developed in the field of differential geometry as the
  equality  of the (normalized) [13Xtotal absolute curvature[113X of a submanifold with
  the  lower  bound  [13Xsum  of  the  Betti numbers[113X [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:[133X
  [33X[0;0Y[12XDefinition[112X (Tightness, [K{\95])[133X
  [33X[0;0YLet  [22XF[122X  be  a  field.  An  embedding [22XM → E^N[122X of a compact manifold is called
  [13X[22Xk[122X-tight  with  respect  to  [22XF[122X[113X if for any open or closed halfspace [22Xh⊂ E^N[122X the
  induced homomorphism[133X
  
  
        [33X[1;6Y[24X[33X[0;0YH_i(M\cap h;\mathbb{F})\longrightarrow H_i(M;\mathbb{F})[133X [124X[133X
  
  
  [33X[0;0Yis  injective  for all [22Xi≤ k[122X. [22XM[122X is called [22XF[122X[13X-tight[113X if it is [22Xk[122X-tight for all [22Xk[122X.
  The  standard  choice  for the field of coefficients is [22XF_2[122X and an [22XF_2[122X-tight
  embedding is called [13Xtight[113X.[133X
  [33X[0;0YWith  regard  to  PL  embeddings  of PL manifolds tightness of [13Xcombinatorial
  manifolds[113X  can  also  be  defined  via  a  purely combinatorial condition as
  follows. For an introduction to PL topology see [RS72].[133X
  [33X[0;0Y[12XDefinition[112X (Tight triangulation [K{\95])[133X
  [33X[0;0YLet  [22XF[122X  be  a field. A combinatorial manifold [22XK[122X on [22Xn[122X vertices is called [13X([22Xk[122X-)
  tight  w.r.t.  [22XF[122X[113X  if  its  canonical  embedding [22XK⊂ ∆^n-1⊂ E^n-1[122X is ([22Xk[122X-)tight
  w.r.t. [22XF[122X, where [22X∆^n-1[122X denotes the [22X(n-1)[122X-dimensional simplex.[133X
  [33X[0;0YIn  dimension  [22Xd=2[122X the following are equivalent for a triangulated surface [22XS[122X
  on  [22Xn[122X  vertices: (i) [22XS[122X has a complete edge graph [22XK_n[122X, (ii) [22XS[122X appears as a so
  called  [13Xregular case[113X in Heawood's Map Color Theorem [Rin74], compare [K{\95]
  and  (iii)  the  induced  piecewise  linear  embedding  of  [22XS[122X into Euclidean
  [22X(n-1)[122X-space has the two-piece property [Ban74], and it is tight [K{\95].[133X
  [33X[0;0YKühnel   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 [13XHamiltonicity[113X of a polyhedral complexes (see [K{\95]): A
  subcomplex [22XA[122X of a polyhedral complex [22XK[122X is called [13X[22Xk[122X-Hamiltonian[113X if [22XA[122X contains
  the  full [22Xk[122X-dimensional skeleton of [22XK[122X (not to be confused with the notion of
  a  [22Xk[122X-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 [22X0[122X-Hamiltonian subcomplex of a
  [22X1[122X-dimensional graph or of a higher-dimensional complex.[133X
  [33X[0;0YA  triangulated  [22X2k[122X-manifold  that  is  a  [22Xk[122X-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
  [13X[22X(k+1)[122X-neighborly  triangulation[113X  since  any  [22Xk+1[122X vertices in a [22Xk[122X-dimensional
  simplex  are  common neighbors. Moreover, [22X(k+1)[122X-neighborly triangulations of
  [22X2k[122X-manifolds  are  also referred to as [13Xsuper-neighborly[113X triangulations -- in
  analogy  with neighborly polytopes the boundary complex of a [22X(2k+1)[122X-polytope
  can  be  at  most  [22Xk[122X-neighborly  unless  it  is  a simplex. Notice here that
  combinatorial  [22X2k[122X-manifolds  can  go  beyond  [22Xk[122X-neighborliness, depending on
  their topology.[133X
  [33X[0;0YWhereas  in the [22X2[122X-dimensional case all tight triangulations of surfaces were
  classified  by  Ringel  and  Jungerman  and Ringel, in dimensions [22Xd≥ 3[122X 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].[133X
  
  
  [1X2.8 [33X[0;0YSimplicial blowups[133X[101X
  
  [33X[0;0YThe  [13Xblowing up process[113X or [13XHopf [22Xσ[122X-process[113X 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 [22XCP^1 ≅ S^2[122X of all complex lines through
  that  point. This is often called [13Xblowing up[113X of the point or just [13Xblowup[113X. In
  general  the process can be applied to non-singular 4-manifolds and yields a
  transformation  of  a manifold [13XM[113X to [22XM # (+CP^2)[122X or [22XM # (-CP^2)[122X, 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 [13Xdilatation process[113X.[133X
  [33X[0;0YA 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 [22XRP^3[122X. The boundary of the
  resulting   space  is  this  triangulated  [22XRP^3[122X.  Now  we  glue  back  in  a
  triangulated  version  [22XmathbfC[122X  of  a complex projective plane with a [22X4[122X-ball
  removed  where  antipodal  points of the boundary are identified. [22XmathbfC[122X is
  called  a  triangulated mapping cylinder and by construction its boundary is
  PL homeomorphic to [22XRP^3[122X.[133X
  [33X[0;0YFor  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:[133X
  [33X[0;0Y[12XDefinition[112X (Simplicial blowup, [SK11])[133X
  [33X[0;0YLet  [22Xv[122X  be  a  vertex  of  a  combinatorial [22X4[122X-pseudomanifold [22XM[122X whose link is
  isomorphic  with  the  particular  [22X11[122X-vertex  triangulation of [22XRP^3[122X which is
  given  by  the boundary complex of the triangulated [22XmathbfC[122X given in [SK11].
  Let  [22Xψ  :  operatornamelk(v)  →  ∂mathbfC[122X  denote  such  an  isomorphism.  A
  simplicial  resolution  of  the  singularity  [22Xv[122X  is  given  by the following
  construction [22XM ↦ widetildeM := (M ∖ operatornamestar(v)^∘) ∪_ψ mathbfC.[122X[133X
  [33X[0;0YThe  process is described in more detail in [SK11]. In particular it is used
  to transform a [22X4[122X-dimensional Kummer variety into a K3 surface.[133X
  
