Cup product for cochains

[vbraun]

It would be nice to have the cup product available for arbitrary cochains, without involving homology_vector_space_with_basis. Of course its not well-defined on the level of cochains, but at least one could check that products are non-zero if there is torsion over ZZ etc.

There are potentially two places where to put it:

• Method of Chain.Element, then chains would need to know about their cell (sub)complex. Right now they don't reference their cell complex as far as I can see.

• Method GenericCellComplex.cup_product(self, left_cochain, right_cochain). Then chains need to know their degree (dimension of cells). Right now they don't store their degree, although the documentation says that they must be of homogeneous degree.

Thoughts?

CC: @jhpalmieri @tscrim

Component: algebraic topology

Author: Volker Braun

Branch/Commit: e2b36a3

Reviewer: Frédéric Chapoton, John Palmieri

Issue created by migration from https://trac.sagemath.org/ticket/21081

[jhpalmieri]

comment:1

Using Chain.Element seems good to me. It doesn't look like the Chains class is used much, so I think we could add extra arguments to the __init__ method specifying the degree and the associated cell complex.

(The Chains class could use other improvements, too, like a map to the associated chain complex.)

[vbraun]

comment:2

What about splitting Chains and Cochains into separate parents? Of course its somehow the same thing but notation is different and people will probably be confused if there are cochain.boundary() and chain.cup_product() methods...

[jhpalmieri]

comment:3

That sounds like a good idea.

[vbraun]

Branch: u/vbraun/cup_product_for_cochains

[vbraun]

Commit: 294aa8e

[vbraun]

New commits:

294aa8e

Split chains/cochains as separate parents and expose cup product

[vbraun]

Author: Volker Braun

[fchapoton]

comment:6

There is something wrong here (cochains should be chains in the OUTPUT):

+    def dual(self):
+        """
+        Return the chains
+
+        OUTPUT:
+
+        The cochains of the same cells with the same base ring.

and also here (two blank lines is too much):

+        Integer. The dimension of the cells.
+
+        EXAMPLES::
+
+
+            sage: X = simplicial_complexes.Simplex(2)

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

54c27b4

Fix typo

[sagetrac-git]

Changed commit from 294aa8e to 54c27b4

[jhpalmieri]

Changed branch from u/vbraun/cup_product_for_cochains to u/jhpalmieri/cup_product_for_cochains

[jhpalmieri]

Changed commit from 54c27b4 to e2b36a3

[jhpalmieri]

comment:9

I made some trivial changes: fixed a few typos (like :class:`Cohains`), added chains.py to the reference manual, wrapped some long lines, and deleted some trailing whitespace. I'm happy with this, so positive review if my changes are okay.

---

New commits:

e2b36a3

trac 21081: fix a few typos, add chains.py to reference manual,

[jhpalmieri]

Reviewer: Frédéric Chapoton, John Palmieri

[vbraun]

comment:10

Thanks!

[vbraun]

Changed branch from u/jhpalmieri/cup_product_for_cochains to e2b36a3