simplicial complex

hyperbolic manifolds

floer homology

Pacific journal of mathematics

In this paper we prove two results, one semi-historical and the other new.\nThe semi-historical result, which goes back to Thurston and Riley, is that the\ngeometrization theorem implies that there is an algorithm for the homeomorphism\nproblem for closed, oriented, triangulated 3-manifolds. We give a\nself-contained proof, with several variations at each stage, that uses only the\nstatement of the geometrization theorem, basic hyperbolic geometry, and old\nresults from combinatorial topology and computer science. For this result, we\ndo not rely on normal surface theory, methods from geometric group theory, nor\nmethods used to prove geometrization.\n The new result is that the homeomorphism problem is elementary recursive,\ni.e., that the computational complexity is bounded by a bounded tower of\nexponentials. This result relies on normal surface theory, Mostow rigidity, and\nbounds on the computational complexity of solving algebraic equations.\n

Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization