Computational Software in Algebraic Topology – Tools and Applications

at.algebraic-topologycomputational geometrycomputational-topologymathematical-software

I was wondering if there is any good software out there that allows you to do specific computations in algebraic topology. For example:

  • Create a simplicial complex/set and ask questions about its homology, cohomology;
  • Build manifolds using handle decompositions;
  • Calculate homotopy limits, colimits.

Something quite flexible and robust in the vein of MAGMA

Thank you.

Best Answer

There are several programs that answer to your first demand whilst the others, as Ryan says, are a bit more vague. There are books written on computational homology (and its applications) for instance, see http://chomp.rutgers.edu/ and the computational homology project. For simplicial complexes, the Plex routines written for Matlab are at http://comptop.stanford.edu/u/programs/plex/ and that leads to a lot of other interesting programs for which see http://comptop.stanford.edu/ and follow links. The main problems are always speed of computing with large simplicial complexes. (Work by Edelsbrunner and collaborators is good for some of this.)

For homotopy colimits, it seems likely that the only programs that might go some way are related to Kenzo project: see http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/

but that is more difficult to use.

There are programs for detecting (small) handles used in computer graphics, but I cannot say anything about them.

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