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I'm investigating the cohomology of complex manifolds, and need to construct chain complexes, Mayer-Vietoris sequences etc. I use Mathematica for numerically tracing manifolds with geodesics, but haven't implemented any representation of cohomological classes, differentials and complexes. Has anyone used Mathematica in this area, or anything related to this (eg multidimensional topology)?

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I am doing similar computations and I hope that you will get a better answer from someone else. My current method is to use Macaulay 2 to compute representatives for my cohomology classes (people.math.gatech.edu/~aleykin3/Dmodules/index.html ) and then compute with them using Mathematica's differential forms package (library.wolfram.com/infocenter/MathSource/482 ). –  David Speyer Feb 15 '13 at 14:36
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@DavidSpeyer I confess I don't understand a word of either question or comment, but I'm pretty sure your comment would rate as an answer if you expanded on it a little more. –  Verbeia Feb 21 '13 at 10:20

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It's not a completely satisfying answer, but there is a nice computational package called javaPlex that "implements persistent homology and related techniques from computational and applied topology, in a library designed for ease of use, ease of access from Matlab and java-based systems, and ease of extensions for further research projects and approaches." It is written in Java, and so should, in principle, be accessible in Mathematica using Jlink. It has a well-worked-out connection to Matlab, so that might be an easier approach. One could use javaPlex via Matlink, which gives "seamless two-way communication" between Matlab and Mathematica.

JavaPlex allows straightforward construction of chain complexes and things like the Mayer-Vietoris sequences, as well as solid computational techniques for persistent homology.

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second time you mention it...are you involved with that package? –  Stefan Jul 9 '13 at 18:06
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@Stefan -- I've used it, if that constitutes "involvement". It functions well. –  bill s Jul 10 '13 at 0:57

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