On the number of coloured triangulations of $d$-manifolds

TL;DR

This paper establishes superexponential bounds on the number of coloured triangulations of d-manifolds, especially in dimension 3, revealing growth rates and properties of random triangulations with implications for manifold topology.

Contribution

It provides the first tight bounds for the number of coloured 3-manifold triangulations and explores related graph properties, advancing understanding of manifold combinatorics.

Findings

Number of coloured 3-manifold triangulations grows as 2^{O(n)} n^{n/6}.

Random coloured 3-manifold triangulations have sublinear vertex counts.

Upper bounds apply to coloured d-spheres, offering the best known estimates in dimensions ≥3.

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 $d\geq 3$ 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 $2^{O(n)} n^{\frac{n}{6}}$ 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. Our upper bounds apply in particular to coloured $d$-spheres for which they seem to be the best known bounds in any dimension $d\geq 3$, 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.