cohomology ring of simplicial complexes #6102

antieau · 2009-05-21T03:47:14Z

Add functionality in sage to compute the cohomology ring of a simplicial complex.

This relies on #6099, #6100, and #5882.

These will be examples of graded alebras, finite as modules over their bases, that are graded-commutative.

Depends on #19179

CC: @jhpalmieri @sagetrac-fbreuer @tscrim @fchapoton

Component: algebraic topology

Author: John Palmieri, Travis Scrimshaw

Branch/Commit: 9bfc2d2

Reviewer: Travis Scrimshaw, John Palmieri

Issue created by migration from https://trac.sagemath.org/ticket/6102

The text was updated successfully, but these errors were encountered:

jhpalmieri · 2012-01-04T23:59:42Z

comment:1

The following code was posted by Felix Breuer on sage-support

def cup_product(X,c1,dim1,c2,dim2):
    d = dim1 + dim2 
    faces1 = list(X.n_faces(dim1))
    faces2 = list(X.n_faces(dim2))
    faces = list(X.n_faces(d))
    res = []
    for sigma in faces:
        sigma1 = Simplex(sigma[0:dim1+1])
        sigma2 = Simplex(sigma[dim1:d+1])
        index1 = faces1.index(sigma1)
        index2 = faces2.index(sigma2)
        coeff1 = c1[index1]
        coeff2 = c2[index2]
        coeff = coeff1 * coeff2
        res.append(coeff)
    return vector(tuple(res))

To use it on the Torus, for example, you can do this:

X = simplicial_complexes.Torus()
C = X.chain_complex(cochain=True)
H = C.homology(generators=True)
gen1 = H[1][1][0]
gen2 = H[1][1][1]
d1 = C.differential()[1]
q = cup_product(X,gen1,1,gen1,1)
print q
print d1.solve_right(q)
p = cup_product(X,gen1,1,gen2,1)
print p
print d1.solve_right(p) #error

jhpalmieri · 2015-09-10T02:14:07Z

Dependencies: #19179

jhpalmieri · 2015-09-10T02:29:22Z

Branch: u/jhpalmieri/AT-model

jhpalmieri · 2015-09-10T02:30:15Z

Author: John Palmieri

jhpalmieri · 2015-09-10T02:30:15Z

Commit: 763a8a8

jhpalmieri · 2015-09-10T02:30:15Z

comment:9

Here is an initial attempt.

New commits:

`dece275`	`trac 19179: chain homotopies, chain contractions, and duals of chain maps`
`763a8a8`	`trac 6102: cup products, cohomology rings, mod 2 cohomology operations`

sagetrac-git · 2015-09-10T15:24:14Z

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

`d7b9ed4`	`change AssertionError to other errors`
`d63774b`	`Merge branch 'chains' into AT-model`
`c0918b8`	`change AttributeErrors to other errors`

sagetrac-git · 2015-09-10T15:24:14Z

Changed commit from 763a8a8 to c0918b8

sagetrac-git · 2015-09-11T18:24:33Z

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

`d65ba8d`	trac 19179: change `_repr_` for chain maps, chain homotopies
`c38751b`	`Merge branch 'chains' into AT-model`
`f70e3a2`	trac 19179: doctest fixes for `_repr_` changes
`92b17e8`	`Merge branch 'chains' into AT-model`
`6cd5b7c`	trac 6102: doctest fixes for `_repr_` changes

sagetrac-git · 2015-09-11T18:24:33Z

Changed commit from c0918b8 to 6cd5b7c

sagetrac-git · 2015-09-12T16:24:36Z

Changed commit from 6cd5b7c to 869636b

sagetrac-git · 2015-09-12T16:24:36Z

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

`4e50776`	`trac 19179: hashing of chain maps, chain homotopies`
`869636b`	`Merge branch 'chains' into AT-model`

jhpalmieri · 2015-09-12T16:30:55Z

comment:15

#18246 broke the default hashing of chain homotopies, so I've added a __hash__ method, and also one for chain maps. This is necessary so that we can cache the methods algebraic_topological_model, homology_basis, and cohomology_ring in cell_complex.

jhpalmieri · 2015-10-12T19:05:38Z

comment:37

Turns out that cup products for Delta complexes weren't too hard to implement, so I did that.

New commits:

`d00d31b`	`trac 6102: cup products for Delta complexes`

jhpalmieri · 2015-10-12T19:05:38Z

Changed commit from 716469a to d00d31b

sagetrac-git · 2015-10-12T20:40:05Z

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

`12d4cc7`	`trac 6102: fix typo "left" -> "right"`

sagetrac-git · 2015-10-12T20:40:05Z

Changed commit from d00d31b to 12d4cc7

tscrim · 2015-10-13T00:50:40Z

comment:39

I'm still working on my refactoring, but I did #19397 for getting the degree d components.

tscrim · 2015-10-13T20:33:07Z

Changed branch from u/jhpalmieri/AT-model to u/tscrim/AT-model

tscrim · 2015-10-13T20:33:07Z

Changed commit from 12d4cc7 to 3751753

tscrim · 2015-10-13T20:33:07Z

comment:40

Done. I spent so much time trying to get the cup_product to iterate over cohomology, but I realized that it was support to be over homology... Anyways, it works now. With the category framework, I was able to remove __pow__ (at a small cost of a not correct error for negative powers, at least for now I didn't want to muck with the AlgebrasWithBasis code). So if you're happy with my changes, then you can set a positive review.

New commits:

`8b59ed3`	`Making (co)homology into a graded module (algebra).`
`3751753`	`Making Sq work for inhomogeneous elements.`

jhpalmieri · 2015-10-14T23:05:57Z

comment:41

I will have some reviewer's changes on top of your changes soon. Meanwhile, I noticed that you removed the code related to the FiniteDimensionalAlgebra class. I don't know much about that class, and I don't mind the removal of that code. We could also reinstate it as a method for the class CohomologyRing ("exporting" it as a FiniteDimensionalAlgebra). Is that worth doing?

tscrim · 2015-10-15T02:52:59Z

comment:42

No, the CohomologyRing class takes the place of the FiniteDimensionalAlgebra. Was there something in that class that you were using that this version can't do? If there was, it is probably something we should generalize (on a followup ticket).

jhpalmieri · 2015-10-15T04:30:35Z

comment:43

There isn't anything that I was using, but it has some methods (cardinality, is_unitary, is_commutative) that I suppose some people might want.

tscrim · 2015-10-15T13:08:39Z

comment:44

The cardinality should work, but it is not there and is something we should implement in generality. A default is_unitary that returns True could perhaps go in UnitalAlgebras, but the cohomology ring is unital as the sum of the 0-th degree components, correct (you had this in your __pow__ method too)? There should be a generic is_commutative test for finite dimensional algebras with basis. I will open up a separate ticket when I get to my office.

tscrim · 2015-10-15T14:29:18Z

comment:45

This is now #19416.

jhpalmieri · 2015-10-15T18:50:06Z

Changed branch from u/tscrim/AT-model to u/jhpalmieri/AT-model

jhpalmieri · 2015-10-15T18:57:15Z

Changed commit from 3751753 to 9bfc2d2

jhpalmieri · 2015-10-15T18:57:15Z

comment:47

Okay, your turn again. If you're happy with these changes, set it to positive review. A summary of my changes:

various documentation fixes: some docstrings didn't get changed in your refactoring, some cross-references didn't work, etc.
I moved several instances of if base_ring is None to the methods in cell_complex.py. Before, some were there but some were in algebraic_topological_model.py.
I removed the explicit check about immutability for simplicial complexes because it wasn't being used: once we cache the method, immutability is checked as soon as the method is called, so checking again in the method is redundant. (And if we ever decide that the method should not be cached, there is no reason to check immutability.)

jhpalmieri · 2015-10-15T18:57:38Z

Changed author from John Palmieri to John Palmieri, Travis Scrimshaw

jhpalmieri · 2015-10-15T18:57:38Z

Changed reviewer from Travis Scrimshaw to Travis Scrimshaw, John Palmieri

tscrim · 2015-10-15T19:19:26Z

comment:49

Then it is a positive review. This is a very nice addition to Sage which I'm hoping to get some good use from (especially once #18175 and more of SageManifolds gets merged in). Thanks for all your work.

jhpalmieri · 2015-10-15T19:50:06Z

comment:51

Great! Thanks very much.

vbraun · 2015-10-18T12:06:14Z

Changed branch from u/jhpalmieri/AT-model to 9bfc2d2

antieau mannequin added this to the sage-5.11 milestone May 21, 2009

antieau mannequin added c: algebraic topology labels May 21, 2009

antieau mannequin self-assigned this May 21, 2009

jdemeyer modified the milestones: sage-5.11, sage-5.12 Aug 13, 2013

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.1, sage-6.2 Jan 30, 2014

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.2, sage-6.3 May 6, 2014

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.3, sage-6.4 Aug 10, 2014

jhpalmieri added the s: needs review label Sep 10, 2015

jhpalmieri added p: major / 3 and removed p: minor / 4 labels Sep 10, 2015

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tscrim added s: positive review and removed s: needs review labels Oct 15, 2015

vbraun removed the s: positive review label Oct 18, 2015

vbraun closed this as completed in 8ebbe9a Oct 18, 2015

This was referenced Oct 28, 2015

computation of induced morphism on homology and cohomology of simplicial complex morphisms #6101

Closed

cohomology of simplicial complexes with finite coefficients as modules over steenrod algebras #6103

Closed

This was referenced Apr 7, 2011

more examples of simplicial complexes: RP^n, CP^2, etc. #9125

Closed

Chain homotopies #19179

Closed

Some clean up in simplicial complexes #20103

Closed

cohomology ring of simplicial complexes #6102

cohomology ring of simplicial complexes #6102

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