15-vertex triangulations of an 8-manifold

Abstract

Mathematics Subject ClassO~cation (1991): 57Q15, 52B70, 53C42 Introduction and main theoremIt is well known that any simplicial decomposition of the real projective plane RP 2 must have at least 6 vertices, and that the 2-fold quotient of the icosahedron provides the unique 6-vertex triangulation RP62. By construction it is invariant under the action of the icosahedral group As. Moreover it has the following properties:(1) It has the smallest number of vertices among all simplicial 2-manifolds which are not PL homeomorphic to the sphere,(2) any two vertices are contained in a common edge. This property is also called 2-neighborliness.Regarding NP62 as a subcomplex of the 5-dimensional simplex A 5 we have Skl(A 5) C RP~ C Sk2(Z~5),(3) NP 2 C z55 C E 5 is a tight polyhedral embedding with the maximal possible essential codimension. Recall that there is a tight algebraic embedding ~.p2 C S 4 C E 5 known as the Veronese surface (and these two embeddings are essentially the only tight embeddings RP 2 --+ E 5, see [KP]). A submanifold M C E n is called tight if for any half space h c E n the induced homomorphism H,(M N h) ~ H,(M) is injective where H. denotes the ordinary simplicial or singular homology with coefficients in Z2. A d-dimensional combinatorial manifold M d is a simplicial decomposition of a topological manifold such that the link of each vertex is a combinatorial (d-1)-sphere. M is called k -neighborly if any k-tuple of vertices spans a (k -1)-dimensional simplex of M, or equivalently, if M contains the full (k -1)-dimensional skeleton of the (n -1)-simplex A '~-I where n is the number of vertices. 168 U. Brehm and W. KtihnelThe corresponding result for the complex projective plane CP 2 says that any combinatorial triangulation of CP 2 must have at least 9 vertices, and that there is exactly one such triangulation CP~ (up to relabeling). It is invariant under the action of a 2-fold extension of the Heisenberg group over Z3, see [KB, KL]. Moreover it has the following analogous properties:(1) It has the smallest number of vertices among all combinatorial 4-manifolds which are not PL homeomorphic to the sphere (see [BG, BK2]),(2) any three vertices are contained in a common triangle (3-neighborliness), thus Sk2(A s) C CP~ C Sk4(As),(3) CP92 C A 8 C E 8 is a tight polyhedral embedding with the maximal possible essential codimension. It corresponds to the tight algebraic embedding CP 2 c S 7 c E s, see [KB, Kui2] (compare [Kuil, Theorem 15] for the uniqueness of the tight algebraic embedding).The analogy between RP 2 and CP~ and the correspondence with the algebraic standard embeddings RP 2 ---, E 5, CP 2 --+ E s suggests to consider also the cases of the quaternionic projective plane HP 2 and of the Cayley plane CaP 2. The tight standard embeddings HP 2 --* E 14, CaP 2 --~ E 26 (see [T, TK]) suggest n = 15 or n = 27 for the number n of vertices of such a combinatorial triangulation. Similarly the tightness suggests the 5-neighborliness for a possible HP25 and the 9

Heuristics for Sphere Recognition

“…Brehm and Kühnel [6] introduced a basic version of (4) to show that some 8-dimensional simplicial complex with 15 vertices is a combinatorial 8-manifold. A necessary condition for K to be homeomorphic to the d-sphere is that the homology, say with integer coefficients, is trivial.…”

“…Enumeration. When enumerating triangulations of manifolds of a given dimension with a fixed number of vertices or facets, we want to ensure that the objects produced are indeed manifolds and discard all others [6,7,31]. 2.…”

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

“…See [5] for 5-neighborly triangulations of an 8-manifold with χ = 3 and with 15 vertices. This leads to a new design S 168 (2, 9, 15) without repeated blocks, invariant under the vertex transitive action of the icosahedral group A 5 .…”

Manifolds associated to simple games

“…Example 7 corresponds to the 6-vertex triangulation of projective 2-plane, and can be generalized as vertex-minimal triangulation of projective space only in dimensions 4, 8, 16, see [2,3].…”

“…It appeared as "simple game with constant sum" in game theory, see [12,4] and also as "strongly complementary simplicial complex", see [2,3].…”