Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization

TL;DR

This paper proves that the homeomorphism problem for closed, oriented 3-manifolds is algorithmically solvable and establishes that its computational complexity is elementary recursive, relying on geometrization and normal surface theory.

Contribution

It provides a new elementary recursive complexity bound for the 3-manifold homeomorphism problem, building on geometrization and normal surface theory.

Findings

Algorithm for the homeomorphism problem derived from geometrization.

Homeomorphism problem is elementary recursive in complexity.

Self-contained proof avoiding advanced methods like normal surface theory.

Abstract

In this paper 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.