Computational software in Algebraic Topology? 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.
 A: Although it might not be exactly what you are looking for (e.g. lack of homotopy-theoretic constructions), 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."
JavaPlex allows straightforward construction of chain complexes and things like the Mayer-Vietoris sequences, as well as computational techniques for persistent homology.
Link: http://git.appliedtopology.org/javaplex/
A: 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.  
A: Sage allows you to play with simplicial complexes and their (co)homology.
http://www.sagemath.org/doc/reference/sage/homology/simplicial_complex.html
http://www.sagemath.org/doc/reference/sage/homology/examples.html
A: (1) The Computational Homology Project offers free software CHomP that will compute homology of simplicial complexes, at least with finite field coefficients.
(2) Dionysus, from the computational topology group at Stanford, is good for computing persistent homology of Rips and Cech complexes, etc.  This might be especially useful, for example, if you had points sampled from a manifold.
(3) Afra Zomorodian has apparently recently written some code for computing homology of clique (i.e. flag) complexes very quickly and with small memory requirement by going through calculations involving simplicial sets, but I don't know if the code can compute homology of arbitrary simplicial sets, and (more importantly) I don't know if the code is publicly available.
