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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 thegeometrization theorem implies that there is an algorithm for the homeomorphismproblem for closed, oriented, triangulated 3-manifolds. We give aself-contained proof, with several variations at each stage, that uses only thestatement of the geometrization theorem, basic hyperbolic geometry, and oldresults from combinatorial topology and computer science. For this result, wedo not rely on normal surface theory, methods from geometric group theory, normethods 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 ofexponentials. This result relies on normal surface theory, Mostow rigidity, andbounds on the computational complexity of solving algebraic equations.

Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization