Hey Marek,
>
> I have two questions regarding cohomology ring of (delta/simplicial)
> complexes.
>
> First:
> How to change the way the generators are displayed?
> (currently it's h^{i,j}, where i is the degree/gradation and j is the
> index of generator in its gradation)
> It would be much nicer to be able to name them (actually: display) with
> more human readable names.
>
> e.g. for the cohomology ring You could do
>
> >S_2 = delta_complexes.SurfaceOfGenus(2)
>> >H = S_2.cohomology_ring(QQ); H
>> Cohomology ring of Delta complex with 3 vertices and 29 simplices over
>> Rational Field
>> >H.__custom_name= 'H*(S_2)'; H
>> H*(S_2)
>>
>
> But I was not able to change how x_1 = H.basis(1)[0] is displayed.
> This is probably trivial, it's just me not beeing able to find an option
> to do this. I tried things around Indexed generators
>
IndexedGenerators has some control over the print representation when
allowed access. However, it is just coming along for the ride with
CombinatorialFreeModule, which the class HomologyVectorSpaceWithBasis is
based off of. HomologyVectorSpaceWithBasis explicitly sets its
representation behavior with _repr_term.
It's not too hard to add options; in particular, there is the class
GlobalOptions (see, e.g,, Partitions or Tableaux for examples) that we can
add to include options.
>
> Second: Consider cartesian product:
>
> >SxS = S_2.product(S_2); SxS
>>
>> Delta complex with 9 vertices and 3635 simplices
>>
>>
> This doesn't seem prohibitively large, yet computation of
> SxS.cohomology_ring(QQ) takes a lot of time.
> So I decided to play smart and set:
>
> H2 = tensor([H,H])
>>
>
> However the product in the tensor product is twisted (see e.g. A. Hatcher,
> Algebraic Topology, Section 3.2 on the cup product, p. 219):
>
> (a \otimes b) \cup (c \otimes d) = (-1)^{|b||c|} (a \cup c) \otimes (b
>> \cup d)
>>
>
> yet I don't see a way to ask sage to respect that? maybe tensor([H,H],
> twisted=True) could be introduced
>
> I'd be willing to address both issues in code if someone would be willing
> to guide me.
>
> Great! I would be happy to help. Please cc me on the ticket (tscrim), and
we can discuss details there. You should all cc John Palmieri (jhpalmieri),
who should also be interested in this.
Best,
Travis
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