On the number of coloured triangulations of d-manifolds

Abstract

We give superexponential lower and upper bounds on the number of coloured d-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and d=3 is fixed. In the special case of dimension 3, the lower and upper bounds match up to exponential factors, and we show that there are 2O(n)nn6 coloured triangulations of 3-manifolds with n tetrahedra. Our results also imply that random coloured triangulations of 3-manifolds have a sublinear number of vertices. The upper bounds apply in particular to coloured d-spheres for which they seem to be the best known bounds in any dimension d=3, even though it is often conjectured that exponential bounds hold in this case. We also ask a related question on regular edge-coloured graphs having the property that each 3-coloured component is planar, which is of independent interest.