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