I'm studying the homology of a couple of very large posets (one has over 4 million vertices, though the dimension is only 3). I want to show the posets are spherical (homology vanishes except in top dimension) or better Cohen-Macaulay. Because of the size, a direct computation of homology seems impossible, even with a clever program like Simplicial Homology for GAP.

Does anyone know of an algorithm for this sort of thing which could make life easier? I have tried to find an explicit recursive atom ordering already, but haven't found one.