Implement twisted homology of simplicial sets.

[miguelmarco]

Right now we can compute universal covers of simplicial sets if the fundamental group is finite. This PR implements
twisted homology with respect to the abelianization character.

In particular, if the fundamental group is abelian, this gives modules whose underlying additive group are isomorphic to the homology groups of the universal cover.

Addresses #36690

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

[jhpalmieri]

Perhaps delete lines 600 and 601, since they're covered in 603-604.

[miguelmarco]

Before the changes in universal_cover_dict, these two examples behaved differently (the hashes of the simplices in the n-skeleton of a wedge didn't necessarily match the ones in the total space).

I think it is good to have a test that makes sure that that problem is solved.

[jhpalmieri]

I'm talking about (a) a line that evaluates Y._canonical_twisting_operator() as compared to (b) a line that does d2 = Y._canonical_twisting_operator() and then evaluates d2. Are you saying that those behaved differently? If so, it would be good to add a comment explaining that, or else someone may come along later, not understand their purpose, and delete those tests.

[miguelmarco]

Oh sorry, you are right. What worked differently was the example with X. The lines you mention are indeed unnecessarily duplicated.

[jhpalmieri]

My first impression is that it adds important functionality to Sage, and it looks good. Should K.twisted_homology(n) (etc.) exit gracefully if K is simply connected? Right now I just get an error; should it just return the ordinary homology? Should _canonical_twisting_operator return a trivial operator, assigning 1 to each edge?

[jhpalmieri]

I have a design question: when I wrote the initial simplicial set code, I set it up so that fundamental_group was only available to pointed simplicial sets. Mathematically this is certainly the right thing to do, but should Sage be sloppier and (for example) if the simplicial set is path connected, choose an arbitrary base point?

This would allow moving some methods around between files, and it might make it possible to unify twisted_chain_complex with chain_complex — chain_complex could just take an optional argument for the twisting operator.

None of this would necessarily be done on this PR, but is it worth thinking about, or should we keep it as is?

[enriqueartal]

There are some small comments.

[vbraun]

merge conflict

[github-actions]

Documentation preview for this PR (built with commit 82bea65; changes) is ready! 🎉

[miguelmarco]

merge conflict

Fixed.