Cup product over integers

[Volker Braun]

It seems that we currently can't compute cup products in cohomology over the integers, only over fields. Is that correct? Is there a ticket to extend it over the integers? Surely that's an interesting question, isn't it? 

[John H Palmieri]

Yes, it is correct that we can only compute cup products over fields. It is possible to do over the integers but hasn't been implemented in Sage yet. See

   P. Pilarczyk and P. Réal, *Computation of cubical homology, cohomology, and (co)homological operations via chain contraction*, Adv. Comput. Math. 41 (2015), pp 253--275.

We have implemented their "Algorithm 1", which works over fields. Their "Algorithm 2" works in general, but will be slower and will require more work to implement. If I remember correctly, implementing it well requires more interaction with linear algebra than Algorithm 1 did: for Algorithm 1, we could just pass all computations off to Sage's pre-existing matrix code. I don't think this will be possible for Algorithm 2.

I would encourage anyone who is interested to work on it. I don't have any immediate plans to do it myself.

[John H Palmieri]

Oh, of course there are probably other algorithms available, but this is the one I knew about off the top of my head...