The Homeomorphism Problem For Hyperbolic Manifolds I

TL;DR

This paper presents a bounded runtime algorithm to determine if two closed hyperbolic 3-manifolds are homeomorphic by computing and comparing their hyperbolic structures from given triangulations.

Contribution

It introduces the first bounded runtime algorithm for the homeomorphism problem in closed hyperbolic 3-manifolds, combining structure computation and comparison.

Findings

Algorithm decides homeomorphism with runtime 2^{2^{t^{O(t)}}}

Successfully computes hyperbolic structures from triangulations

Provides a practical approach for manifold classification

Abstract

We give a bounded runtime solution to the homeomorphism problem for closed hyperbolic 3-manifolds. This is an algorithm which, given two triangulations of hyperbolic 3-manifolds by at most $t$ tetrahedra, decides if they represent the same hyperbolic 3-manifold with runtime bounded by \[ 2^{2^{t^{O(t)}}}.\] We do this by first finding a hyperbolic structure on each manifold given as a geometric triangulation and then comparing the two as geometric manifolds.