  In a previous paper, Sarkar and the third author gave a combinatorial
description of the hat version of Heegaard Floer homology for three-manifolds.
Given a cobordism between two connected three-manifolds, there is an induced
map between their Heegaard Floer homologies. Assume that the first homology
group of each boundary component surjects onto the first homology group of the
cobordism (modulo torsion). Under this assumption, we present a procedure for
finding the rank of the induced Heegaard Floer map combinatorially, in the hat
version.
