Hello everyone!
I would like to compute the cup-product of two chains in the cohomology of
a simplicial complex.
What I have so far, is that I have the simplicial complex realized as a
SimplicialComplex in sage and I can compute its cohomology groups. They
are: {0: 0, 1: Z x Z, 2: Z^28, 3: Z^9}. What I would like to do, is to take
two generators of the cohomology group in degree 1 and compute that their
cup-product is non-trivial. What is the best way to do this?
As I gather from Trac issue
#6102<http://trac.sagemath.org/sage_trac/ticket/6102>,
I can't simply compute the cohomology ring, as this functionality is not
yet implemented. So I have to do this by hand, but that's okay. AFAICT,
there are two things I need to figure out.
1) How can I get my hands on two representatives of the generators of the
first cohomology group? I am not sure, whether I can have sage compute
these for me (see ticket
#6100<http://trac.sagemath.org/sage_trac/ticket/6100>).
If yes, how? If no, I could maybe construct these two by hand (as I "know"
how my simplicial complex "looks"). But in what format do I have to
construct these chains so that I can use them in step 2?
2) How can I compute coboundaries of chains in the cochain complex? I know
that sage will give me the cochain complex induced by my simplicial complex
including the (co)boundary maps. But how do I apply these coboundary maps
to chains? What format do these chains need to have?
Sorry, if these questions are somewhat fuzzy, but I am new to sage. Thank
you for any help you can give!
Best,
Felix
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