Block DesignsS2n−8(2,5,n) and Triangulated Eulerian 4-manifolds

Abstract

Three-neighborly triangulations of eulerian 4-manifolds with n vertices can be interpreted as block designs S 2n−8 (2, 5, n). We discuss this correspondence and present a new cyclic example with 14 vertices. c 1998 Academic Press LimitedIt is well known that any 2-neighborly triangulation of a compact 2-manifold M with n vertices can be regarded as a block design S 2 (2, 3, n) or twofold triple system if one interprets the triangles as abstract triples (or blocks). This is possible only if n(7 − n) = 6χ(M) or, equivalent,, the so-called regular cases in Heawood's map color problem [18, 20]. As usual, χ denotes the Euler characteristic. In general, k-neighborliness means that the number of (k − 1)-dimensional simplices is n k where n is the number of vertices. Vice versa, any twofold triple system S 2 (2, 3, n) can be regarded as a triangular (and 2-neighborly) decomposition of a pseudo-surface. A pseudo-surface is a surface except at isolated points, see [1,10, 21]. In the literature, these regular cases are also known as triangular embeddings of the complete graph K n into surfaces or pseudo-surfaces.For compact triangulated 4-manifolds M with n vertices there are similar regular cases, characterized by the equation. This coincides with the case of 3-neighborly triangulations, meaning that any triple of vertices determines a triangle of the triangulation [12,13]. The 3-neighborliness implies that the link of every edge contains all the remaining n −2 vertices. If it is a 2-sphere then the number of triangles is 2(n −4), the number of edges is 3(n −4), in accordance with the Euler equation χ = n −2−3(n −4)+2(n −4) = 2. The same holds for any 4-dimensional simplicial complex such that the link of each simplex has the same Euler characteristic as one would expect in a 4-manifold. Such complexes are called eulerian manifolds in [12]. In this paper we study the induced block designs S 2n−8 (2, 5, n).

Generalized Heawood Numbers

“…It has been checked that there is no 14-vertex 3-neighborly triangulation of any 4-manifold with χ = 14 admitting a vertex transitive automorphism group Z 14 . However, there is a 4-dimensional pseudomanifold with the same properties otherwise and with singularities precisely along an embedded Klein bottle [39]. Besides the trivial case of the simplex itself, so far we know only one example with a twofold transitive group: the 16-vertex triangulation of the K3 surface [14].…”