TITLE: Computer-aided homology computations
QUESTION [16 upvotes]: Background
I am currently working on the homology of some moduli space and there exists a much simpler chain complex with the same homology.
It is a quotient of a bisimplicial complex by a subcomplex. One could say that some faces are degenerate, thus are zero.
Unfortunately, it's too hard to compute the homology groups by hand.

Given a free chain complex of finite type over the coefficient ring $\mathbb{Z}$ or $\mathbb{Z}/{p^k}$, with $p$ prime, I cannot compute the homology groups by hand. Some matrices have approximately $18.000.000 \times 15.000.000$ entries.
Question: Is there a C/C++ library or an external program that does the following (in descending order of importance)?

computes the homology groups of the given chain complex
describes the image of one or multiple cycles as a linear combination of homology classes
is efficient (e.g. it is capable of using multiple CPUs)

REPLY [10 votes]: I realize that this is an old question and it is likely that the OP has moved on, but let me summarize the state-of-the-art as it exists now.

The given sizes (18 million by 15 million) are pretty hopeless if you are planning to resort to matrix methods. To put things in perspective, latest version of Linbox has been chugging away at a 12,000 x 11,000 dense integer matrix on my cluster for over three months with no end in sight. So, you would almost certainly need to resort to a reduction method and the most effective ones are based on discrete Morse theory.
Since discrete Morse theory exploits the presence of units in the boundary matrix, it is important to keep in mind that there is a world of difference computationally between computing with $\mathbb{Z}$ coefficients versus computing over a field. Over a field, there is no trouble since everything nonzero is a unit, and you are guaranteed to be able to Morse-reduce very effectively.
If you are working over the integers, then everything depends on whether your matrix has many units (i.e., 1's and -1's) in it. If you are unit-rich, and if the actual homology of your complex is light on torsion, then there is a pretty good chance that the Morse-reductions will be drastic even over the integers; and they might produce much smaller boundary matrices which can then be fed into Linbox.
I've written some C++ software to implement these discrete Morse theoretic reductions before computing homology (over the integers!): see here. Although it doesn't mention this explicitly on the website, you can easily feed the software various matrix representations of boundary operators (rather than simplicial complexes) and produce smaller matrices which fit into a quasi-isomorphic chain complex.