Partitioning the triangles of the cross polytope into surfaces
PPartitioning the triangles of the cross polytope intosurfaces
Jonathan Spreer
Abstract
We present a constructive proof that there exists a decomposition of the -skeleton of the k -dimensional cross polytope β k into closed surfaces of genus g ď , each with a transitiveautomorphism group given by the vertex transitive Z k -action on β k . Furthermore weshow that for each k ” , p q the -skeleton of the p k ´ q -simplex is a union of highlysymmetric tori and Möbius strips. MSC 2010: 52B12 ; 52B70; 57Q15; 57M20; 05C10;
Keywords: cross polytope, simplicial complexes, triangulated surfaces, difference cycles.
It is an obvious fact that the p k ´ q -simplex ∆ k ´ contains all triangulations on k vertices(possibly after a relabeling of the vertices). Hence, its i -skeleton skel i p ∆ k ´ q , i. e. the setof all i -dimensional faces of ∆ k ´ , i ă k , can be seen as the space of all triangulated i -manifolds with m ď k vertices.One way of analyzing this space is to create partitions of skel i p ∆ k ´ q where every partfulfills certain conditions.Of course, one could divide skel i p ∆ k ´ q into ` ki ` ˘ parts, each containing one i -simplexwhich is a topological ball and hence fulfills the constraint to be a bounded manifold.However, if we want to partition skel i p ∆ k ´ q into fewer parts, with the additional condi-tion that, for example, each part is a bounded manifold, more restrictions apply. As aconsequence, the partition becomes more meaningful regarding the structure of ∆ k ´ .In the centrally symmetric case, the k -dimensional cross polytope , i. e. the convex hullof k points x ˘ i “ p , . . . , , ˘ , , . . . , q P R k , ď i ď k , is the space of all centrallysymmetric triangulations, i. e. all triangulations having a symmetry of order withoutfixed points. Thus, partitions of the i -skeleton of β k which follow suitable constrainscould give new insights into the class of centrally symmetric triangulations which forms aninteresting family of triangulations, previously investigated by Grünbaum [7], Jockush [9],Kühnel [10], Effenberger and Kühnel [4], Lassmann and Sparla [12] and many others.1 a r X i v : . [ m a t h . C O ] J u l owever, a more general and systematic approach to this method of analyzing a poly-tope is to consider partitions of the i -skeleton of arbitrary (simplicial) polytopes, orderedby the size of the partition, say l . In the special case l “ , i. e. where the i -skeleton skel i p P q , ď i ď p k ´ q , of a k -polytope P can be partitioned into two (possibly bounded)PL i -manifolds M and M with M Y M “ skel i p P q such that M X M Ă skel i ´ p P q , thepolytope P is called decomposable .For i “ , finding decomposable polytopes P means decomposing the edge graph of P into two (connected) graphs of degree at most . This problem was solved by Grünbaumand Malkevitch [8] as well as Martin [14].In the case i “ , the question was investigated by Betke, Schulz and Wills (see [3])with the result that there exist only five polytopes which allow a decomposition of theset of triangles into two surfaces with boundary, namely the -simplex, the -simplex, the -dimensional cross polytope, the -vertex cyclic -polytope and the double pyramid overthe -simplex.It is fairly easy to agree that in general decompositions of a polytope P with l small arevery restrictive towards the local combinatorial structure of P : Just counting the numberof triangles sharing one common edge of β k leads to the conclusion that the -skeleton of β k cannot be partitioned into less than l “ k ´ surfaces.On the other hand, a decomposition with not too many parts could be enlighteningtowards a deeper understanding of how skeletons of high co-dimension in a simplicialpolytope are structured locally. For surfaces in the k -dimensional cross polytope this canbe demonstrated by our main result. Theorem 1.1.
The -skeleton of the k -dimensional cross polytope β k can be decomposedinto triangulated vertex transitive closed surfaces.More precisely, if k ” , p q , skel p β k q decomposes into p k ´ qp k ´ q triangulated vertextransitive closed surfaces of Euler characteristic on k vertices and, if k ” p q , into k disjoint copies of B β (on vertices each) and k p k ´ q triangulated vertex transitive closedsurfaces of Euler characteristic on k vertices. The case k “ equals one of the five decompositions already described by Betke, Schulzand Wills in [3]. It is worthwhile mentioning that Theorem 1.1 defines a decompositionof skel p β k q not only into triangulated surfaces but into -Hamiltonian closed surfaceswith transitive automorphism group, i. e. each part of the partition contains k verticesand has an automorphism group of order at least k (a cyclic part with k vertices plusthe centrally symmetric element). Hence, the space of all centrally symmetric triangulatedsurfaces - which is a highly symmetric space - can be partitioned into relatively large piecesof such surfaces while these surfaces in some cases consist of surprisingly many differentcombinatorial types (see Table 3.1). In fact, it follows from the construction of the partitionthat each centrally symmetric triangulated torus or Klein bottle with cyclic Z k symmetryoccurs in this partition as a proper part.Under this point of view it is surprising that a similar partition of the -skeleton of ∆ k ´ fails to exist. Instead, it seems that only a partition into partly bounded triangulated2urfaces is possible by virtue of the following theorem. Theorem 1.2.
Let k ą , k ” , p q . Then the -skeleton of ∆ k ´ decomposes into k ´ collections of Möbius strips M l,k : “ tp l : l : k ´ l qu , ď l ď k ´ each with n : “ gcd p l, k q isomorphic connected components on kn vertices and k ´ k ` collections of tori S l,j,k : “ tp l : j : k ´ l ´ j q , p l : k ´ l ´ j : j qu , ď l ă j ă k ´ l ´ j , with m : “ gcd p l, j, k q connected components on km vertices each. In order to proof our main results, we will first introduce some terms and methodsto work with triangulations with transitive cyclic symmetry (cf. Section 2) before we willestablish some useful lemma and a proof for Theorem 1.1 (cf. Section 3) and Theorem 1.2(cf. Section 4).
Highly symmetric simplicial complexes C allow a very efficient description by the generatorsof its automorphism group Aut p C q together with a system of orbit representatives of thecomplex under the action of Aut p C q . In this way, all kinds of statements involving highlysymmetric simplicial complexes often have elegant and easy to handle proofs.In particular, this is true for complexes with transitive automorphism group, sometimesjust called transitive complexes : As the automorphism group acts transitive on the set ofvertices, all vertex links of a simplicial complex are combinatorially isomorphic , i. e. they areequal up to a relabeling of their vertices. Hence, a neighborhood of each vertex determinesthe complex globally and, as a consequence, global properties can be calculated locally.An important class of transitive complexes are the ones which contain a cyclic auto-morphism acting regular on the set of vertices.By a relabeling of the vertices to an integer labeling , . . . , n , the cyclic group actioncan be represented by the vertex shift v ÞÑ v ` n . In this way, the simplices ofa complex can be divided into equivalence classes by simply calculating the differencesbetween the vertex labels of each simplex and each of these equivalence classes defines acyclic orbit. More precisely we have the following definition. Definition 2.1 (Difference cycle) . Let a i P N zt u , ď i ď d , n : “ ř di “ a i and Z n “xp , , . . . , n ´ qy . The simplicial complex p a : . . . : a d q : “ Z n ¨ t , a , . . . , Σ d ´ i “ a i u , where ¨ is the induced cyclic Z n -action on subsets of Z n , is called difference cycle of di-mension d on n vertices . The number of its elements is referred to as the length of the3ifference cycle. If a complex C is a union of difference cycles of dimension d on n verticesand λ is a unit of Z n such that the complex λC (obtained by multiplying all vertex labelsmodulo n by λ ) equals C , then λ is called a multiplier of C .Note that for any unit λ P Z ˆ n , the complex λC is combinatorially isomorphic to C . Inparticular, all λ P Z ˆ n are multipliers of the complex Ť λ P Z ˆ n λC by construction.The definition of a difference cycle above is similar to the one given in [11]. For a morethorough introduction into the field of the more general difference sets and their multiplierssee Chapter VI and VII in [2].Throughout this article we will look at difference cycles as simplicial complexes with atransitive automorphism group given by the cyclic Z n -action on its elements: Every p d ` q -tuple t x , . . . , x d u is interpreted as a d -simplex ∆ d “ x x , . . . , x d y . A simplicial complex C is called transitive , if its group of automorphisms acts transitively on the set of vertices.In particular, any union of difference cycles is a transitive simplicial complex. Remark . It follows from Definition 2.1 that the set of difference cycles of dimension d on k vertices defines a partition of the d -skeleton of the p k ´ q -simplex. Two p d ` q -tuples p a , . . . , a d q and p b , . . . , b d q with Σ di “ a i “ Σ di “ b i “ k define the same difference cycle ifand only if for a fixed j P Z we have a p i ` j q mod p d ` q “ b i for all ď i ď d . Proposition 2.3.
Let p a : . . . : a d q be a difference cycle of dimension d on n vertices and ď k ď d ` the smallest integer such that k | p d ` q and a i “ a i ` k , ď i ď d ´ k . Then p a : . . . : a d q is of length ř k ´ i “ a i “ nkd ` . Proof.
We set m : “ nkd ` and compute A ` m, a ` m, . . . , p Σ d ´ i “ a i q ` m E “ A Σ k ´ i “ a i , Σ ki “ a i , . . . , Σ d ´ i “ a i , , a , . . . , Σ k ´ i “ a i E “ A , a , . . . , Σ d ´ i “ a i E (all entries are computed modulo n ). Hence, for the length l of p a : . . . : a d q we have l ď nkd ` and since k is minimal with k | p d ` q and a i “ a i ` k , the upper bound isattained. skel p β k q into closed cyclic sur-faces In this section we will prove our main result Theorem 1.1.First of all let us mention that it will turn out to be very convenient to look at theboundary of the k -dimensional cross polytope in terms of the abstract simplicial complex B β k “ tx a , . . . , a k y | a i P t , . . . , k ´ u , t i, k ` i u Ę t a , . . . , a k u , @ ď i ď k ´ u . (3.1)4n particular, the diagonals of β k are precisely the edges x i, k ` i y , ď i ď k , and thuscoincide with the difference cycle p k : k q .The proof of Theorem 1.1 itself will consist of an explicit construction of pairwisedisjoint cyclic closed surfaces in the -skeleton of the cross polytope β k as well as a proofof their topological types for any given integer k ě .However, let us first state a number of lemmata which will be helpful in the following. Lemma 3.1.
The -skeleton of β k can be written as the following set of difference cycles: p l : j : 2 k ´ l ´ j q , p l : 2 k ´ l ´ j : j q for ă l ă j ă k ´ l ´ j , k R t l, j, l ` j u , and p j : j : 2 p k ´ j qq for ă j ă k with j ‰ k . If k ı p q all of them are of length k , if k ” p q thedifference cycle p k : k : k q has length k .Proof. Let β k be the k -dimensional cross polytope with vertices t , . . . , k ´ u and diag-onals t j, k ` j u , ď j ď k ´ . It follows from the recursive construction of β k as thedouble pyramid over β k ´ that it contains all -tuples of vertices as triangles except theones including a diagonal. Thus, a difference cycle of the form p a : b : c q lies in skel p β k q ifand only if k R t a, b, a ` b u . In particular, skel p β k q is a union of difference cycles.Note that each ordered -tuple ă l ă j ă k ´ l ´ j defines exactly two distinctdifference cycles on the set of k vertices, namely p l : j : 2 k ´ l ´ j q and p l : 2 k ´ l ´ j : j q and it follows immediately that there is no other difference cycle p a : b : c q , k R t a, b, a ` b u on k vertices with a, b, c pairwise distinct.For any positive integer ă j ă k with j ‰ k there is exactly one difference cycle p j : j : 2 k ´ j qq , and since j must fulfill ă j ă k , there are no further difference cycles without diagonalswith at most two different entries.The length of the difference cycles follows directly from Proposition 2.3 with d “ and n “ k . Lemma 3.2.
A closed -dimensional pseudomanifold S defined by m difference cycles offull length on the set of n vertices has Euler characteristic χ p S q “ p ´ m q n .Proof. Since all difference cycles are of full length, S consists of n vertices and m ¨ n triangles. Additionally, the pseudo manifold property asserts that S has m ¨ n edges andthus χ p S q “ n ´ m ¨ n ` m ¨ n “ n p ´ m q . emma 3.3. Let ă l ă j ă k ´ l ´ j , k R t l, j, l ` j u and m : “ gcd p l, j, k q . Then S l,j, k : “ tp l : j : 2 k ´ l ´ j q , p l : 2 k ´ l ´ j : j qu – t , . . . , m u ˆ T , where all connected components of S l,j, k are combinatorially isomorphic to each other.Proof. The link of vertex in S l,j, k is equal to the cycle lk S l,j, k (0) = l l + jj k − l k − j − l k − j Since ă l ă j ă k ´ l ´ j and k R t l, j, l ` j u , all vertices are distinct and lk S l,j, k p q is the boundary of a hexagon. By the vertex transitivity all other links are also hexagonsand S l,j, k is a surface.Since l , j and k ´ l ´ j are pairwise distinct, both p l : j : 2 k ´ l ´ j q and p l : 2 k ´ l ´ j : j q have full length and by Lemma 3.2 the surface has Euler characteristic .In order to see that S l,j, k is oriented, we look at the (oriented) boundary of the trianglesin S l,j, k in terms of -dimensional difference cycles: Bp l : j : 2 k ´ l ´ j q “ p j : 2 k ´ j q ´ p l ` j : 2 k ´ l ´ j q ` p l : 2 k ´ l qBp l : 2 k ´ l ´ j : j q “ p k ´ l ´ j : l ` j q ´ p k ´ j : j q ` p l : 2 k ´ l q“ p j : 2 k ´ j q ´ p l ` j : 2 k ´ l ´ j q ` p l : 2 k ´ l q and thus Bp l : j : 2 k ´ l ´ j q ´ Bp l : 2 k ´ l ´ j : j q “ and S l,j, k is oriented.Now consider p l : j : 2 k ´ l ´ j q “ Z k ¨ x , l, l ` j y Clearly, xp ` i q mod 2 k, p l ` i q mod 2 k, p l ` j ` i q mod 2 k y share at least one vertex if i P t , l, k ´ l, j, k ´ j, k ´ l ´ j, l ` j u . For any other value of i ă k , the intersectionof the triangles is empty. By iteration it follows that p l : j : 2 k ´ l ´ j q has exactly gcd p , l, k ´ l, j, k ´ j, k ´ l ´ j, l ` j q “ gcd p l, j, k q “ m connected components. Thesame holds for p l : 2 k ´ l ´ j : j q . The complex p , . . . , p k ´ qq i ¨ x , l, l ` j y is disjointto x , l, k ´ j y for i R t , l, k ´ l, j, k ´ j, k ´ l ´ j, l ` j u . Together with the fact that star S l,j, k p q consists of triangles of both p l : j : 2 k ´ l ´ j q and p l : 2 k ´ l ´ j : j q it follows that S l,j, k has m connected components and by a shift of the indices one can see that all of themmust be combinatorially isomorphic. Altogether it follows that S l,j, k – t , . . . , m uˆ T . Remark . Some of the connected components of the surfaces presented above are com-binatorially isomorphic to the so-called Altshuler tori tp n ´ q , p n ´ qu with n “ km ě vertices (cf. proof of Theorem in [1]). However, other triangulationsof transitive tori are part of the decomposition as well: in the case k “ , there are four6 comb. types k comb. types k comb. types k comb. types Figure 3.1: Number of combinatorially distinct types of centrally symmetric transitive toriin β k , k ď .different combinatorial types of tori. This is in fact the total number of combinatorialtypes of transitive tori on vertices (cf. Table 1). The number of distinct combinatorialtypes of centrally symmetric transitive tori for k ď is listed in Table 3.1. Remark . All centrally symmetric transitive surfaces ( cst-surfaces for short) S l,j, k fromLemma 3.3 can be constructed using the function SCSeriesCSTSurface(l,j,2k) from the
GAP -package simpcomp [6, 5], maintained by Effenberger and the author. If the secondparameter is not provided (
SCSeriesCSTSurface(l,2k) ), the surface S l, k from Lemma3.7 is generated. Lemma 3.6.
Let M : “ " p j : j : 2 p k ´ j qq | ă j ă k ; 2 j ‰ k * ,M : “ " p l : l : 2 p k ´ l qq | ď l ď Z k ´ ^* and M : “ " p k ´ l : k ´ l : 2 l qq | ď l ď Z k ´ ^* . For all k ě the triple p M, M , M q defines a partition M “ M Y M into two sets of equal size. In particular, we have | M |” p q .Proof. From ď l ď t k ´ u it follows that k ´ l ą l and l ă k ă p k ´ l q . Thus, M X M “ H and M Y M Ď M .On the other hand let t k ´ u ă j ă k ´ t k ´ u . If k is odd, then k ´ ă j ă k ` whichis impossible for j P N . If k is even, then k ´ ă j ă k ` , hence it follows that j “ k which is excluded in the definition of M . Altogether M Y M “ M holds and | M | “ Z k ´ ^ “ " k ´ if k is odd k ´ else.7 emma 3.7. The complex S l, k : “ tp l : l : 2 p k ´ l qq , p k ´ l : k ´ l : 2 l qu , ď l ď t k ´ u , is a disjoint union of k copies of B β if | k and l “ k and a surface ofEuler characteristic otherwise.Proof. We prove that S l, k is a surface by looking at the link of vertex : lk S j, k (0) = l lk + lk − l k − l k − l where l “ k ´ l and k ´ l “ k ` l if and only if l “ k . Thus, lk S l,j, k p q is either theboundary of a hexagon or, in the case l “ k , the boundary of a quadrilateral and S l, k is asurface.Furthermore, if l ‰ k the surface S l, k is a union of two difference cycles of full lengthand by Lemma 3.2 we have χ p S l, k q “ . If l “ k , p k : k : k q is of length k and it followsthat χ p S k , k q “ k ´ k ` k “ k. By a calculation analogue to the one in the proof of Lemma 3.3, one obtains that S k , k consists of gcd p l, k q “ k isomorphic connected components of type t , u . Hence, S k , k isa disjoint union of k copies of B β .Subsection 3.1 of [13] by Lutz contains a series of transitive tori and Klein bottles with n “ ` m vertices, m ě . The series is given by A p n q : “ tp p n ´ qq , p p n ´ q : p n ´ qu .A p n q is a torus for m even and a Klein bottle for m odd. Lemma 3.8.
Let k ě , ď l ď t k ´ u , l ‰ k and n : “ gcd p l, k q . Then S l, k is isomorphicto n copies of A p kn q .Proof. Since n “ gcd p l, k q “ gcd p l, k ´ l q , we have n “ min t gcd p l, p k ´ l qq , gcd p l, k ´ l qu and either ln P Z ˆ kn or k ´ ln P Z ˆ kn holds. It follows by mulitplying with l or k ´ l that A p kn q is isomorphic to S ln , kn “ S k ´ ln , kn . The monomorphism Z kn Ñ Z k j ÞÑ p lj mod 2 k q represents a relabeling of S ln , kn and a small computation shows that the relabeledcomplex is equal to the connected component of S l, k containing . By a shift of thevertex labels we see that all other connected components of S l, k are isomoprhic to the onecontaining what states the result. 8et us now come to the proof of Theorem 1.1. Proof.
Lemma 3.1 and Lemma 3.6 describe skel p β k q in terms of series of pairs of differ-ence cycles tp l : j : 2 k ´ l ´ j q , p l : 2 k ´ l ´ j : j qu and tp l : l : 2 p k ´ l qq , p k ´ l : k ´ l : 2 l qu for certain parameters j and l . Lemma 3.3 determines the topological type of the first andTheorem 3 together with Lemma 3.7 and 3.8 determines the type of the second series.Since | skel p β k q| “ ` k ˘ ´ k p k ´ q and for k ı p q all surfaces have exactly k triangles, we get an overall number of p k ´ qp k ´ q surfaces. If k ” p q , all surfaces butone have k triangles, the last one has k triangles. Altogether this implies that there are k p k ´ q surfaces of Euler characteristic and k copies of B β .Table 1 shows the decomposition of skel p β k q for ď k ď . The table was computedusing the GAP package simpcomp [6]. For a complete list of the decomposition for k ď see [15]. skel p ∆ k ´ q First, note that skel p ∆ k ´ q , k ě , equals the set of all triangles on k vertices. By lookingat its vertex links we can see that in the case that k is an even number the complex tp l : k ´ l : k qu cannot be part of a triangulated surface for any ă l ă k . Thus, thedecomposition of skel p β k q cannot be extended to a decomposition of skel p ∆ k ´ q in anobvious manner. However, following Theorem 1.2, in the case that k is neither even nordivisible by the situation is different.Again, we will first prove some lemma before we will come to the actual proof of thetheorem. Lemma 4.1.
The complex M l,k with k ě , k ı p q and k ı p q is a triangulation of n : “ gcd p l, k q cylinders r , s ˆ B ∆ if kn is even and of n Möbius strips if kn is odd.Proof. We first look at M ,k “ tx , , y , x , , y , . . . , x k ´ , k ´ , y , x k ´ , , yu for k ě (see Figure 4.1). Every triangle has exactly two neighbors. Thus, the alternatingsum `x , , y ´ x , , y ` . . . ´ `p´ q k ´ x k ´ , , y induces an orientation if and only if k is even and for any l P Z ˆ k the complex M l,k is acylinder if k is even and a Möbius strip if k is odd. Now suppose that n “ gcd p l, k q ą .Since k ı p q and k ı p q we have kn ě and by a relabeling we see that the connectedcomponents of M l,k are combinatorially isomorphic to M ln , kn – M , kn .9 l − l . . .. . .
01 2 3 4 2 l − l − l − . . .. . . CylinderM¨obius strip
Figure 4.1: The cylinder p l ´ q and the Möbius strip p l ´ q . The verticalboundary components ( x , y )are identified. Remark . If k ” p q the connected components of M k ,k “ tp k : k : k qu are equivalentto tp qu , the boundary of ∆ . If k ” p q , then M k ,k is a collection of disjointtriangles. Lemma 4.3.
The complex S l,j,k , ă l ă j ă k , k ı p q , is a triangulation of m : “ gcd p l, j, k q connected components of isomorphic tori on km vertices.Proof. The link of vertex equals lk S l,j,k (0) = l l + jjk − lk − j − lk − j (cf. proof of Lemma 3.3). Since ă l ă j ă k ´ l ´ j and k ı p q the link is theboundary of a hexagon, km ě and S l,j,k is a surface. By Lemma 3.2, the complex S l,j,k is of Euler characteristic . The proof of the orientability and the number of connectedcomponents is analogue to the one given in the proof of Lemma 3.3. It follows that S l,j,k – t , . . . , m u ˆ T . Together with Lemma 4.1 and Lemma 4.3 in order to proof Theorem 1.2, it suffices toshow that the two series presented above contain all triangles of ∆ k ´ . Proof.
Let x a, b, c y P skel p ∆ k ´ q , a ă b ă c . Then x a, b, c y P p b ´ a : c ´ b : k ´ p c ´ a qq .Now if b ´ a , c ´ b and k ´ p c ´ a q are pairwise distinct, we have • x a, b, c y P S b ´ a,c ´ b,k “ S b ´ a,k ´p c ´ a q ,k if b ´ a ă c ´ b, k ´ p c ´ a q , x a, b, c y P S c ´ b,b ´ a,k “ S c ´ b,k ´p c ´ a q ,k if c ´ b ă b ´ a, k ´ p c ´ a q or • x a, b, c y P S k ´p c ´ a q ,b ´ a,k “ S k ´p c ´ a q ,c ´ b,k if k ´ p c ´ a q ă c ´ b, b ´ a . If on the other hand at least two of the entries are equal, then p b ´ a : c ´ b : k ´ p c ´ a qq “p l : l : k ´ l q for ď l ď k ´ . Thus, the union of all Möbius strips M l,k and collections oftori S l,j,k equals the full -skeleton of the k -simplex skel p ∆ k ´ q .Table 2 shows the decomposition of skel p ∆ k ´ q for k P t , , , , u . For a completelist of the decomposition of skel p β k q and skel p ∆ k ´ q for k ď see [15].Table 1: The decomposition of the -skeleton of β k ( k ď ) into transitive surfaces. k f p β k q topological type difference cycles S tp q , p qu T tp q , p qu , tp q , p qu T tp q , p qu , tp q , p qu K tp q , p qu , tp q , p qu t , u ˆ S tp q , p qu T tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu T tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qut , u ˆ T tp q , p qu K tp q , p qu , tp q , p qu , tp q , p qu T tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qut , u ˆ T tp q , p qu , tp q , p qu t , , u ˆ S tp q , p qu T tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qut , u ˆ T tp q , p qu , tp q , p qu , tp q , p qu K tp q , p qu , tp q , p qu , tp q , p qu
10 960 T tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qut , u ˆ T tp q , p qu , tp q , p qut , u ˆ K tp q , p qu , tp q , p qu -skeleton of ∆ k ´ ( k P t , , , , u ) by topologicaltypes. k topological type difference cycles M tp qu , tp qu M tp qu , tp qu , tp qu T tp q , p qu M tp qu , tp qu , tp qu , tp qu , tp qu T tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu M tp qu , tp qu , tp qu , tp qu , tp qu , tp qu T tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu M tp qu , tp qu , tp qu , tp qu , tp qu , tp qu , tp qu , tp qu , tp qu , tp qu , tp qu , tp qut , . . . , u ˆ M tp qu , tp qu , tp qut , . . . , u ˆ M tp qu , tp qu T tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qu , tp q , p qut , . . . , u ˆ T tp q , p qu eferences [1] A. Altshuler. Polyhedral realization in R of triangulations of the torus and -manifolds in cyclic -polytopes. Discrete Math. , 1(3):211–238, 1971/1972.[2] T. Beth, D. Jungnickel, and H. Lenz.
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