Spheres are rare

TL;DR

This paper proves that the number of triangulations of homology spheres grows significantly slower than general triangulations, providing a specific growth bound in three dimensions.

Contribution

It establishes a new upper bound on the growth rate of triangulations of homology spheres across all dimensions, especially in 3D.

Findings

Triangulations of homology spheres grow at most like the cube root of general triangulations in 3D.

The growth rate of these special triangulations is much slower than that of general triangulations.

Provides a quantitative bound on the enumeration complexity of homology sphere triangulations.

Abstract

We prove that triangulations of homology spheres in any dimension grow much slower than general triangulations. Our bound states in particular that the number of triangulations of homology spheres in 3 dimensions grows at most like the power 1/3 of the number of general triangulations.