  We define integral measures of complexity for Heegaard splittings based on
the graph dual to the curve complex and on the pants complex defined by Hatcher
and Thurston. As the Heegaard splitting is stabilized, the sequence of
complexities turns out to converge to a non-trivial limit depending only on the
manifold. We then use a similar method to compare different manifolds, defining
a distance which converges under stabilization to an integer related to Dehn
surgeries between the two manifolds.
