Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization

Greg Kuperberg

Vol. 301 (2019), No. 1, 189–241

DOI: 10.2140/pjm.2019.301.189

Abstract

We prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof, with several variations at each stage, that uses only the statement of the geometrization theorem, basic hyperbolic geometry, and old results from combinatorial topology and computer science. For this result, we do not rely on normal surface theory, methods from geometric group theory, nor methods used to prove geometrization.

The new result is that the homeomorphism problem is elementary recursive, i.e., that the computational complexity is bounded by a bounded tower of exponentials. This result relies on normal surface theory, Mostow rigidity, and bounds on the computational complexity of solving algebraic equations.

Keywords

geometrization, computational complexity, 3-manifold recognition

Mathematical Subject Classification 2010

Primary: 57M27, 57M50

Secondary: 68Q15

Milestones

Received: 16 June 2016

Revised: 23 August 2017

Accepted: 4 July 2018

Published: 16 September 2019