Remarks on the entropy of 3-manifolds

TL;DR

This paper provides a combinatorial proof of an exponential bound on the number of 3-manifolds constructed via boundary triangle identification and discusses the potential for all 3-spheres to be obtained through this method.

Contribution

It introduces a simple combinatorial proof for the exponential upper bound and explores the construction of all 3-spheres through boundary identifications.

Findings

Exponential upper bound on the number of constructible 3-manifolds.

All closed simplicial manifolds constructed by this method are homeomorphic to S^3.

Example of a 3-ball boundary identification leading to a simply connected 3-manifold.

Abstract

We give a simple combinatoric proof of an exponential upper bound on the number of distinct 3-manifolds that can be constructed by successively identifying nearest neighbour pairs of triangles in the boundary of a simplicial 3-ball and show that all closed simplicial manifolds that can be constructed in this manner are homeomorphic to $S^3$. We discuss the problem of proving that all 3-dimensional simplicial spheres can be obtained by this construction and give an example of a simplicial 3-ball whose boundary triangles can be identified pairwise such that no triangle is identified with any of its neighbours and the resulting 3-dimensional simplicial complex is a simply connected 3-manifold.