Remarks on the entropy of 3-manifolds

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.

12 pages, 5 figures available from authors

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• Genus dependence of the number of (non-)orientable surface triangulations
• 2-LC triangulated manifolds are exponentially many
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• Shellable tilings on relative simplicial complexes and their h-vectors
• Many Triangulated 3-Spheres
• Five Dimensional Dynamical Triangulations
• A conditional lower bound for the Turán number of spheres
• The structure of spatial slices of three-dimensional causal triangulations