  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.
