Given a Heegaard splitting of genus $n$, and two distinct orientation preserving homeomorphisms, elements of the mapping class group of the genus $n$ torus, is there a method which shows whether or not these homeomorphisms, when used to identify the boundaries of the pair of handlebodies, will produce the same $3$-manifold?



Heegaard splitting, equivalent homeomorphisms, mapping class group of genus n-torus
The original question was "Is there an algorithm that, given two genus g Heegaard splittings, decides if the resulting two manifolds are homeomorphic?" The answer to this question is "yes" but doesn't have anything to do with Heegaard splittings. 


In theory the solution to the Geometrization Conjecture gives an algorithm to decide if two manifolds are homeomorphic.


In practice one converts the splitting to a triangulations and feeds the result to SnapPy. There are several other programs for three-manifold recognition, such as snap, orb, Regina, and the 3-Manifold Recognizer. However all use SnapPea as the engine for dealing with hyperbolic manifolds. See <http://www.math.uiuc.edu/~nmd/computop/> for links. The Recognizer, for Windows, can be found at <http://www.matlas.math.csu.ru/>.


