I have been working on improving Sage's capabilities with simplicial complexes. With the changes at #19179, #6101, and #6102 (all ready for review), we now have
- chain homotopies
- cup products (with field coefficients)
- cohomology as a FiniteDimensionalAlgebra (with field coefficients)
- Steenrod squares acting on cohomology (mod 2 coefficients)
- maps on homology and cohomology induced by maps of simplicial complexes (with field coefficients)
There is one speed bottleneck, the computation of an "algebraic topological model" for the simplicial complex. This should probably be cythonized, but I am not the person to do that. This could also be done with integer coefficients, but that requires a different algorithm. I can provide a reference if anyone wants to work on it. That will be even slower than with field coefficients, of course.
If you are interested in this sort of mathematics, please take a look at the relevant tickets.http://trac.sagemath.org/ticket/19179
(which is also incorporated into 6101)http://trac.sagemath.org/ticket/6101http://trac.sagemath.org/ticket/6102
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