On neighbourly triangulations

Abstract

A simplicial complex is called d d -neighbourly if any d + 1 d + 1 vertices determine a d d -simplex. We give methods for constructing 1 1 -neighbourly triangulations of 3 3 - and 4 4 -manifolds; further we discuss some relationships between d d -neighbourly triangulations, chromatic numbers and the problem of finding upper and lower bounds on the number of simplices and locating the zeros of the characteristic polynomial of a triangulation. A triangulation of an orientable manifold is called order-orientable if there exists some ordering of the vertices which orients the manifold. We give necessary conditions for their existence; also we construct such triangulations on 3 3 -dimensional handlebodies and discuss the problem of recognising finite monotone subsets of an affine space by using these ideas.