A Cheeger-type exponential bound for the number of triangulated manifolds

TL;DR

This paper extends known exponential bounds on the number of triangulations from surfaces to higher-dimensional space forms with bounded geometry, establishing a combinatorial analogue of Cheeger's finiteness theorem.

Contribution

It provides the first geometric extension of exponential triangulation bounds to higher dimensions, linking geometric constraints to combinatorial enumeration.

Findings

Exponential bounds on the number of geometric triangulations of space forms.

Exponential bounds on the number of geometric triangulations of spheres.

Exponential bounds on convex triangulations of the d-ball.

Abstract

In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric extension of this result to higher dimensions. We show that in terms of the number of facets, there are only exponentially many geometric triangulations of space forms with bounded geometry in the sense of Cheeger (curvature and volume bounded below, and diameter bounded above). This establishes a combinatorial version of Cheeger's finiteness theorem. Further consequences of our work are: (1) There are exponentially many geometric triangulations of $S^d$. (2) There are exponentially many convex triangulations of the d-ball.