TITLE: Computing homology of very large posets
QUESTION [12 upvotes]: 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.

REPLY [3 votes]: Short note for similar problems.
We solved the problem using RedHom software in 2012, 3 days of computations, ~400GB of RAM.
However, now we have better methods and we can check Cohen-Macaulay property of the huge complex in 30 minutes, 340GB of RAM, using 64-cores Intel E7-8837@2.6GHz (assuming hdf5 input file format). It is so fast, because in parallel we can build a data structure for the complex and compute an optimal discrete Morse vector field.