Combinatorial manifolds with complementarity

Combinatorial manifolds with complementarity

Abstract. A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a face of the complex. We show that if a d-dimensional combinatorial manifold M with n vertices satisfies complementarity then d = 0, 2, 4, 8 or 16 with n = 3d/2 + 3 and IMI is a "manifold like a projective plane". Arnoux and Marin had earlier proved th...

Triangulated Manifolds with Few Vertices: Combinatorial Manifolds

Combinatorial Embeddings of Manifolds

The following results on embedding manifolds resemble in their form Dehn's Lemma, the Sphere Theorem, and, especially, embedding theorems obtained for differentiable manifolds by A. Haefliger [i]. Let M, Q be finite combinatorial manifolds of dimensions m and q, respectively. Let M, Q be their boundaries (possibly empty), and let ƒ : M—>Q be a piecewise linear map. We define sing (ƒ) to be the ...

Combinatorial 3-Manifolds with 10 Vertices

We give a complete enumeration of combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product S×S and 615 triangulations of the twisted sphere product S×S. All the 3-spheres with up to 10 vertices are shellable, but there are 29 vertexminimal non-shellable 3-balls with 9 vertices.

Neighborly combinatorial 3-manifolds with 9 vertices