<p>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.</p>
<p>For a more detailed look the authors recommend the books <a href="chapBib.html#biBHudson69PLTop">[Hud69]</a>, <a href="chapBib.html#biBRourke72IntrPLTop">[RS72]</a> on PL-topology and <a href="chapBib.html#biBZiegler95LectPolytopes">[Zie95]</a>, <a href="chapBib.html#biBGruenbaum03ConvPoly">[Gr\03]</a> on the theory of polytopes.</p>
<p>An overview of the more recent developments in the field of combinatorial topology can be found in <a href="chapBib.html#biBLutz05TrigMnfFewVertCombMnf">[Lut05]</a> and <a href="chapBib.html#biBDatta07MinTrigManifolds">[Dat07]</a>.</p>
<h4>2.1 <span class="Heading">Polytopes and polytopal complexes</span></h4>
<p>A convex <em><span class="SimpleMath">d</span>-polytope</em> is the convex hull of <span class="SimpleMath">n</span> points <span class="SimpleMath">p_i ∈ E^d</span> in the <span class="SimpleMath">d</span>-dimensional euclidean space:</p>
<p class="pcenter">P=
conv
\{v_1,\dots,v_n\}\subset E^d,
</p>
<p>where the <span class="SimpleMath">v_1,dots,v_n</span> do not lie in a hyperplane of <span class="SimpleMath">E^d</span>.</p>
<p>From now on when talking about polytopes in this document always convex polytopes are meant unless explicitly stated otherwise.</p>
<p>For any supporting hyperplane <span class="SimpleMath">h ⊂ E^d</span>, <span class="SimpleMath">P∩ h</span> is called a <em><span class="SimpleMath">k</span>-face</em> of <span class="SimpleMath">P</span> if <span class="SimpleMath">dim(P∩ h)=k</span>. The 0-faces are called <em>vertices</em>, the 1-faces <em>edges</em> and the <span class="SimpleMath">(d-1)</span>-faces are called <em>facets</em> of <span class="SimpleMath">P</span>.</p>
<p>A <span class="SimpleMath">d</span>-polytope <span class="SimpleMath">P</span> for which all facets are congruent regular <span class="SimpleMath">(d-1)</span>-polytopes and for which all vertex links are congruent regular <span class="SimpleMath">(d-1)</span>-polytopes is called regular, where the regular <span class="SimpleMath">2</span>-polytopes are regular polygons.</p>
<p>The set of all <span class="SimpleMath">k</span>-faces of <span class="SimpleMath">P</span> is called the <em><span class="SimpleMath">k</span>-skeleton</em> of <span class="SimpleMath">P</span>, written as skel<span class="SimpleMath">_k(P)</span>.</p>
<p>A <em>polytopal complex</em> <span class="SimpleMath">C</span> is a finite collection of polytopes <span class="SimpleMath">P_i</span>, <span class="SimpleMath">1 ≤ i ≤ n</span> for which the intersection of any two polytopes <span class="SimpleMath">P_i ∩ P_j</span> is either empty or a common face of <span class="SimpleMath">P_i</span> and <span class="SimpleMath">P_j</span>. The polytopes of maximal dimension are called the <em>facets</em> of <span class="SimpleMath">C</span>. The <em>dimension</em> of a polytopal complex <span class="SimpleMath">C</span> is defined as the maximum over all dimensions of its facets.</p>
<p>For every <span class="SimpleMath">d</span>-dimensional polytopal complex the <span class="SimpleMath">(d+1)</span>-tuple, containing its number of <span class="SimpleMath">i</span>-faces in the <span class="SimpleMath">i</span>-th entry is called the <em><span class="SimpleMath">f</span>-vector</em> of the polytopal complex.</p>
<p>Every polytope <span class="SimpleMath">P</span> gives rise to a polytopal complex consisting of all the proper faces of <span class="SimpleMath">P</span>. This polytopal complex is called the <em>boundary complex <span class="SimpleMath">C(∂ P)</span> of the polytope <span class="SimpleMath">P</span></em>.</p>
<h4>2.2 <span class="Heading">Simplices and simplicial complexes</span></h4>
<p>A <span class="SimpleMath">d</span>-dimensional <em>simplex</em> or <em><span class="SimpleMath">d</span>-simplex</em> for short is the convex hull of <span class="SimpleMath">d+1</span> points in <span class="SimpleMath">E^d</span> in general position. Thus the <span class="SimpleMath">d</span>-simplex is the smallest (with respect to the number of vertices) possible <span class="SimpleMath">d</span>-polytope. Every face of the <span class="SimpleMath">d</span>-simplex is a <span class="SimpleMath">m</span>-simplex, <span class="SimpleMath">m ≤ d</span>.</p>
<p>A 0-simplex is a point, a <span class="SimpleMath">1</span>-simplex is a line segment, a <span class="SimpleMath">2</span>-simplex is a triangle, a <span class="SimpleMath">3</span>-simplex a tetrahedron, and so on.</p>
<p>A polytopal complex which entirely consists of simplices is called a <em>simplicial complex</em> (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).</p>
<p>The dimension of a simplicial complex is the maximal dimension of a facet. A simplicial complex is said to be <em>pure</em> if all facets are of the same dimension. A pure simplicial complex of dimension <span class="SimpleMath">d</span> satisfies the <em>weak pseudomanifold property</em> if every <span class="SimpleMath">(d-1)</span>-face is part of exactly two facets.</p>
<p>Since simplices are polytopes and, hence, simplicial complexes are polytopal complexes all of the terminology regarding simplicial complexes can be transfered from polytope theory.</p>
<h4>2.3 <span class="Heading">From geometry to combinatorics</span></h4>
<p>Every <span class="SimpleMath">d</span>-simplex has an <em>underlying set</em> in <span class="SimpleMath">E^d</span>, as the set of all points of that simplex. In the same way one can define the <em>underlying set</em> <span class="SimpleMath">|C|</span> of a simplicial complex <span class="SimpleMath">C</span>. If the underlying set of a simplicial complex <span class="SimpleMath">C</span> is a topological manifold, then <span class="SimpleMath">C</span> is called <em>triangulated manifold</em> (or <em>triangulation of <span class="SimpleMath">|C|</span></em>).</p>
<p>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 <em>poset</em>).</p>
<p>Let <span class="SimpleMath">v</span> be a vertex of <span class="SimpleMath">C</span>. The set of all facets that contain <span class="SimpleMath">v</span> is called <em>star of <span class="SimpleMath">v</span> in <span class="SimpleMath">C</span></em> and is denoted by star<span class="SimpleMath">_C(v)</span>. The subcomplex of star<span class="SimpleMath">_C(v)</span> that contains all faces not containing <span class="SimpleMath">v</span> is called <em>link of <span class="SimpleMath">v</span> in <span class="SimpleMath">C</span></em>, written as lk<span class="SimpleMath">_C(v)</span>.</p>
<p>A <em>combinatorial <span class="SimpleMath">d</span>-manifold</em> is a <span class="SimpleMath">d</span>-dimensional simplicial complex whose vertex links are all triangulated <span class="SimpleMath">(d-1)</span>-dimensional spheres with standard PL-structure. A <em>combinatorial pseudomanifold</em> is a simplicial complex whose vertex links are all combinatorial <span class="SimpleMath">(d-1)</span>-manifolds.</p>
<p>Note that every combinatorial manifold is a triangulated manifold. The opposite is wrong: for example, there exists a triangulation of the <span class="SimpleMath">5</span>-sphere that is not combinatorial, the so called <em>Edward's sphere, see [BL00].
<p>A combinatorial manifold carries an induced PL-structure and can be understood in terms of an abstract simplicial complex. If the complex has <span class="SimpleMath">d</span> vertices there exists a natural embedding of <span class="SimpleMath">C</span> into the <span class="SimpleMath">(d-1)</span> simplex and, thus, into <span class="SimpleMath">E^d-1</span>. 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.</p>
<p>Some fundamental properties of an abstract simplicial complex <span class="SimpleMath">C</span> are the following:</p>
<dl>
<dt><strong class="Mark">Dimensionality.</strong></dt>
<dd><p>The dimension of <span class="SimpleMath">C</span>.</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">f</span>, <span class="SimpleMath">g</span> and <span class="SimpleMath">h</span>-vector.</strong></dt>
<dd><p>The <span class="SimpleMath">f</span>-vector (<span class="SimpleMath">f_k</span> equals the number of <span class="SimpleMath">k</span>-faces of a simplicial complex), the <span class="SimpleMath">g</span>- and <span class="SimpleMath">h</span>-vector can be obtained from the <span class="SimpleMath">f</span>-vector via linear transformations.</p>
</dd>
<dt><strong class="Mark">(Co-)Homology.</strong></dt>
<dd><p>The simplicical (co-)homology groups and Betti numbers.</p>
</dd>
<dt><strong class="Mark">Euler characteristic</strong></dt>
<dd><p>The Euler characteristic as the alternating sum over the Betti numbers / the <span class="SimpleMath">f</span>-vector.</p>
</dd>
<dt><strong class="Mark">Connectedness and closedness.</strong></dt>
<dd><p>Whether <span class="SimpleMath">C</span> is strongly connected, path connected, has a boundary or not.</p>
</dd>
<dt><strong class="Mark">Symmetries.</strong></dt>
<dd><p>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.</p>
</dd>
</dl>
<p>All of those properties and many more can be computed on a strictly combinatorial basis.</p>
<h4>2.4 <span class="Heading">Discrete Normal surfaces</span></h4>
<p>The concept of <em>normal surfaces</em> is originally due to Kneser <a href="chapBib.html#biBKneser29ClosedSurfIn3Mflds">[Kne29]</a> and Haken <a href="chapBib.html#biBHaken61TheoNormFl">[Hak61]</a>: A surface <span class="SimpleMath">S</span>, properly embedded into a <span class="SimpleMath">3</span>-manifold <span class="SimpleMath">M</span>, is said to be <em>normal</em>, if it respects a given cell decomposition of <span class="SimpleMath">M</span> in the following sense: It does not intersect any vertex nor touch any <span class="SimpleMath">3</span>-cell of the manifold and does not intersect with any <span class="SimpleMath">2</span>-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 <span class="SimpleMath">M</span> is given as a combinatorial <span class="SimpleMath">3</span>-manifold and we will call the corresponding objects <em>discrete normal surfaces</em>. In order to do this let us first define:<br /> <br /> <strong class="button">Definition</strong><br /> A <em>polytopal manifold</em> is a polytopal complex <span class="SimpleMath">M</span> such that there exists a simplicial subdivision of <span class="SimpleMath">M</span> which is a combinatorial manifold. If <spanclass="SimpleMath">M</span> is a surface we will call it a <em>polytopal map</em>. If, in addition <span class="SimpleMath">M</span> entirely consists of <span class="SimpleMath">m</span>-gons, we call it a <em>polytopal <span class="SimpleMath">m</span>-gon map</em>.<br /> <br /> <strong class="button">Definition</strong> (Discrete Normal surface, <a href="chapBib.html#biBSpreer10NormSurfsCombSlic">[Spr11b]</a>)<br /> Let <span class="SimpleMath">M</span> be a combinatorial <span class="SimpleMath">3</span>-manifold (<span class="SimpleMath">3</span>-pseudomanifold), <span class="SimpleMath">∆ ∈ M</span> one of its tetrahedra and <span class="SimpleMath">P</span> the intersection of <span class="SimpleMath">∆</span> with a plane that does not include any vertex of <span class="SimpleMath">∆</span>. Then <span class="SimpleMath">P</span> is called a <em>normal subset</em> of <span class="SimpleMath">∆</span>. Up to an isotopy that respects the face lattice of <span class="SimpleMath">∆</span>, <span class="SimpleMath">P</span> is equal to one of the triangles <span class="SimpleMath">P_i</span>, <span class="SimpleMath">1 ≤ i ≤ 4</span>, or quadrilaterals <span class="SimpleMath">P_i</span>, <span class="SimpleMath">5 ≤ i ≤ 7</span>, shown in Figure 7. <br /> <br /> A polyhedral map <span class="SimpleMath">S ⊂ M</span> that entirely consists of facets <span class="SimpleMath">P_i</span> such that every tetrahedron contains at most one facet is called <em>discrete normal surface</em> of <span class="SimpleMath">M</span>.<br /> <br /> The second author has recently investigated on the combinatorial theory of discrete normal surfaces, see <a href="chapBib.html#biBSpreer10NormSurfsCombSlic">[Spr11b]</a>.</p>
<h4>2.5 <span class="Heading">Polyhedral Morse theory and slicings</span></h4>
<p>In the field of PL-topology Kühnel developed what one might call a polyhedral Morse theory (compare <a href="chapBib.html#biBKuehnel95TightPolySubm">[K\t95]</a>, 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 binomuwwith 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].
<p>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 <a href="chap2.html#X86275D5979B4B531"><span class="RefLink">2.5</span></a>).<br /> <br /></p>
<h4>2.7 <span class="Heading">Tightness and tight triangulations</span></h4>
<p>Tightness is a notion developed in the field of differential geometry as the equality of the (normalized) <em>total absolute curvature</em> of a submanifold with the lower bound <em>sum of the Betti numbers</em> <a href="chapBib.html#biBKuiper84GeomTotAbsCurvTheo">[Kui84]</a>, <a href="chapBib.html#biBBanchoff97TightSubmSmoothPoly">[BK97]</a>. It was first studied by Alexandrov, Milnor, Chern and Lashof and Kuiper and later extended to the polyhedral case by Banchoff <a href="chapBib.html#biBBanchoff65TightEmb3DimPolyMnf">[Ban65]</a>, Kuiper <a href="chapBib.html#biBKuiper84GeomTotAbsCurvTheo">[Kui84]</a> and Kühnel <a href="chapBib.html#biBKuehnel95TightPolySubm">[K\t95]</a>. 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:<br /> <br /> <strong class="button">Definition</strong> (Tightness, <a href="chapBib.html#biBKuehnel95TightPolySubm">[K\t95]</a>)<br /> Let <span class="SimpleMath">F</span> be a field. An embedding <span class="SimpleMath">M → E^N</span> of a compact manifold is called <em><span class="SimpleMath">k</span>-tight with respect to <span class="SimpleMath">F</span></em> if for any open or closed halfspace <span class="SimpleMath">h⊂ E^N</span> the induced homomorphism</p>
<p>is injective for all <span class="SimpleMath">i≤ k</span>. <span class="SimpleMath">M</span> is called <span class="SimpleMath">F</span><em>-tight</em> if it is <span class="SimpleMath">k</span>-tight for all <span class="SimpleMath">k</span>. The standard choice for the field of coefficients is <span class="SimpleMath">F_2</span> and an <span class="SimpleMath">F_2</span>-tight embedding is called <em>tight</em>.<br /> <br /> With regard to PL embeddings of PL manifolds tightness of <em>combinatorial manifolds</em> can also be defined via a purely combinatorial condition as follows. For an introduction to PL topology see <a href="chapBib.html#biBRourke72IntrPLTop">[RS72]</a>.<br /> <br /> <strong class="button">Definition</strong> (Tight triangulation <a href="chapBib.html#biBKuehnel95TightPolySubm">[K\t95]</a>)<br /> Let <span class="SimpleMath">F</span> be a field. A combinatorial manifold <span class="SimpleMath">K</span> on <span class="SimpleMath">n</span> vertices is called <em>(<span class="SimpleMath">k</span>-) tight w.r.t. <span class="SimpleMath">F</span></em> if its canonical embedding <span class="SimpleMath">K⊂ ∆^n-1⊂ E^n-1</span> is (<span class="SimpleMath">k</span>-)tight w.r.t. <span class="SimpleMath">F</span>, where <span class="SimpleMath">∆^n-1</span> denotes the <span class="SimpleMath">(n-1)</span>-dimensional simplex.<br /> <br /> In dimension <span class="SimpleMath">d=2</span> the following are equivalent for a triangulated surface <span class="SimpleMath">S</span> on <span class="SimpleMath">n</span> vertices: (i) <span class="SimpleMath">S</span> has a complete edge graph <span class="SimpleMath">K_n</span>, (ii) <span class="SimpleMath">S</span> appears as a so called <em>regular case</em> in Heawood's Map Color Theorem [Rin74], compare [K\t95] 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\t95].
Kühnel investigated the tightness of combinatorial triangulations of manifolds also in higher dimensions and codimensions, see [K\t94]. It turned out that the tightness of a combinatorial triangulation is closely related to the concept of Hamiltonicity of a polyhedral complexes (see [K\t95]): A subcomplex Aof 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\t95] 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\t95], [K\t86].
<p>The <em>blowing up process</em> or <em>Hopf <span class="SimpleMath">σ</span>-process</em> can be described as the resolution of nodes or ordinary double points of a complex algebraic variety. This was described by H.~Hopf in <a href="chapBib.html#biBHopf1951">[Hop51]</a>, compare <a href="chapBib.html#biBHirzebruch1953">[Hir53]</a> and <a href="chapBib.html#biBHauser2000">[Hau00]</a>. 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 <span class="SimpleMath">CP^1 ≅ S^2</span> of all complex lines through that point. This is often called <em>blowing up</em> of the point or just <em>blowup</em>. In general the process can be applied to non-singular 4-manifolds and yields a transformation of a manifold <em>M</em> to <span class="SimpleMath">M # (+CP^2)</span> or <span class="SimpleMath">M # (-CP^2)</span>, 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 <a href="chapBib.html#biBSpanier56HomKummerMnf">[Spa56]</a> as a so-called <em>dilatation process</em>.<br /> <br /> 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 <span class="SimpleMath">RP^3</span>. The boundary of the resulting space is this triangulated <span class="SimpleMath">RP^3</span>. Now we glue back in a triangulated version <span class="SimpleMath">mathbfC</span> of a complex projective plane with a <span class="SimpleMath">4</span>-ball removed where antipodal points of the boundary are identified. <span class="SimpleMath">mathbfC</span> is called a triangulated mapping cylinder and by construction its boundary is PL homeomorphic to <span class="SimpleMath">RP^3</span>.<br /> <br /> 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:<br /> <br /> <strong class="button">Definition</strong> (Simplicial blowup, <a href="chapBib.html#biBSpreer09CombPorpsOfK3">[SK11]</a>)<br /> Let <span class="SimpleMath">v</span> be a vertex of a combinatorial <span class="SimpleMath">4</span>-pseudomanifold <span class="SimpleMath">M</span> whose link is isomorphic with the particular <span class="SimpleMath">11</span>-vertex triangulation of <span class="SimpleMath">RP^3</span> which is given by the boundary complex of the triangulated <span class="SimpleMath">mathbfC</span> given in <a href="chapBib.html#biBSpreer09CombPorpsOfK3">[SK11]</a>. Let <span class="SimpleMath">ψ : operatornamelk(v) → ∂mathbfC</span> denote such an isomorphism. A simplicial resolution of the singularity <span class="SimpleMath">v</span> is given by the following construction <span class="SimpleMath">M ↦ widetildeM := (M ∖ operatornamestar(v)^∘) ∪_ψ mathbfC.</span><br /> <br /> The process is described in more detail in <a href="chapBib.html#biBSpreer09CombPorpsOfK3">[SK11]</a>. In particular it is used to transform a <span class="SimpleMath">4</span>-dimensional Kummer variety into a K3 surface.</p>
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