Opened 20 months ago
Closed 18 months ago
#31691 closed enhancement (fixed)
Turn mixed form algebra into de Rham complex
Reported by:  Michael Jung  Owned by:  

Priority:  major  Milestone:  sage9.4 
Component:  manifolds  Keywords:  chain_complex 
Cc:  Eric Gourgoulhon, Travis Scrimshaw, Matthias Köppe, John Palmieri  Merged in:  
Authors:  Michael Jung  Reviewers:  Travis Scrimshaw 
Report Upstream:  N/A  Work issues:  
Branch:  566176a (Commits, GitHub, GitLab)  Commit:  566176a74a480456b7836c13a173e217b721b70a 
Dependencies:  Stopgaps: 
Description (last modified by )
We turn the algebra of mixed differential forms into a de Rham complex and add it to the category of ChainComplexes
, see #31669.
Furthermore, we add de Rham cohomology to SageManifolds? with limited functionality. For now, the implementation will only consist of abstract elements that are given by representatives of mixed forms, i.e. we take closed mixed forms, put a bracket around it and do all computations in the algebra of mixed forms.
Change History (37)
comment:1 Changed 20 months ago by
Summary:  Make mixed forms to de Rham complex → Turn mixed forms into de Rham complex 

comment:2 Changed 20 months ago by
Description:  modified (diff) 

comment:3 Changed 20 months ago by
Branch:  → public/31691_de_rham_complex 

comment:4 Changed 20 months ago by
Commit:  → eeeadf5aa539af8eb994705093f9972d12281c82 

comment:5 followup: 8 Changed 20 months ago by
Here is a first draft. Unfortunately I get an error with the current test:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: C = M.de_rham_complex() sage: d0 = C.differential(0); d0 Exception raised: ... ValueError: Algebra of differentiable scalar fields on the 2dimensional differentiable manifold M is not in Category of modules over Algebra of differentiable scalar fields on the 2dimensional differentiable manifold M
This is somewhat peculiar. Each algebra should automatically be a module over itself... Should I simply add the category to the scalar field algebra or does that need a broader fix?
comment:6 Changed 20 months ago by
Commit:  eeeadf5aa539af8eb994705093f9972d12281c82 → 252cb8f157e8df943f2666ae1683397adb61ff79 

Branch pushed to git repo; I updated commit sha1. New commits:
252cb8f  Trac #31691: invoke derivative instead of differential

comment:7 Changed 20 months ago by
Cc:  Matthias Köppe added 

comment:8 Changed 20 months ago by
Replying to ghmjungmath:
Here is a first draft. Unfortunately I get an error with the current test:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: C = M.de_rham_complex() sage: d0 = C.differential(0); d0 Exception raised: ... ValueError: Algebra of differentiable scalar fields on the 2dimensional differentiable manifold M is not in Category of modules over Algebra of differentiable scalar fields on the 2dimensional differentiable manifold MThis is somewhat peculiar. Each algebra should automatically be a module over itself... Should I simply add the category to the scalar field algebra or does that need a broader fix?
See #31713.
comment:9 Changed 20 months ago by
Commit:  252cb8f157e8df943f2666ae1683397adb61ff79 → ab3d601704d47453641f4e1a225a596784507a4b 

Branch pushed to git repo; I updated commit sha1. New commits:
ab3d601  Trac #31691: de Rham cohomology ring

comment:10 Changed 20 months ago by
This draft should already reflect what I have in mind. Comments are welcome.
comment:11 Changed 20 months ago by
I must apologize. I was extremely sloppy here: differentials are no morphisms in the category of modules over scalar fields, they are morphisms in the category of modules over K
!
I'll fix this.
comment:12 Changed 19 months ago by
That was expected. Same problem, but now with differential form modules not considered as vector spaces.
There should be some way to define these morphisms, no?
comment:13 Changed 19 months ago by
Commit:  ab3d601704d47453641f4e1a225a596784507a4b → 4746ddb61e6d2da8e7d7b5e76d39772b286f56a2 

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:
4746ddb  Trac #31691: turn mixed form algebra into de rham complex

comment:15 Changed 19 months ago by
This is just a first step.
My idea (not for this ticket): Optimally, we want to cover our manifold by contractible chart domains. Therefore it would be helpful if we could determine whether an open subset is contractible or not. Perhaps this needs some input from the user. But then, we can probably check for exactness (I still have to think about it) and, other than that, construct Cech cohomology.
comment:16 Changed 19 months ago by
P.S. I said "simply connected" in the first place, but of course I meant contractible.
comment:17 Changed 19 months ago by
Authors:  → Michael Jung 

comment:18 Changed 19 months ago by
Summary:  Turn mixed forms into de Rham complex → Turn mixed form algebra into de Rham complex 

comment:19 Changed 19 months ago by
Description:  modified (diff) 

comment:20 Changed 19 months ago by
Description:  modified (diff) 

comment:21 Changed 19 months ago by
Commit:  4746ddb61e6d2da8e7d7b5e76d39772b286f56a2 → fb8331abc00f6886b511819bff1e4116bb2e6d30 

Branch pushed to git repo; I updated commit sha1. New commits:
fb8331a  Trac #31691: fix doctest in manifold.py

comment:23 Changed 19 months ago by
Cc:  John Palmieri added 

comment:24 followup: 25 Changed 19 months ago by
I think it would be best to split off the cohomology ring into a separate file.
Do you want to give a special (latex) name to zero
and one
?
For the classlevel docstring of DeRahmComologyRing
:
Define the de Rham cohomology ring of a de Rham complex.
The de Rham cohomology ring of a de Rham complex.
I don't think you should assume that the parents are the same for cup
(unlike what you can do for _mul_
).
In your cohomology()
method, you have both \left.
and \right.
. I don't understand the point of this.
The input block to differential()
is overindented. Also, remove the fullstop/period.
comment:25 followup: 26 Changed 19 months ago by
Replying to tscrim:
I think it would be best to split off the cohomology ring into a separate file.
Sounds reasonable.
Do you want to give a special (latex) name to
zero
andone
?
I don't think so. That would make more sense if we can distinguish cohomology classes properly.
For the classlevel docstring of
DeRahmComologyRing
:Define the de Rham cohomology ring of a de Rham complex. The de Rham cohomology ring of a de Rham complex.
Thanks, done.
I don't think you should assume that the parents are the same for
cup
(unlike what you can do for_mul_
).
What do you mean?
In your
cohomology()
method, you have both\left.
and\right.
. I don't understand the point of this.
The point is that \middle/
has the appropriate size.
The input block to
differential()
is overindented. Also, remove the fullstop/period.
Thanks, done.
comment:26 followup: 27 Changed 19 months ago by
Replying to ghmjungmath:
Replying to tscrim:
I don't think you should assume that the parents are the same for
cup
(unlike what you can do for_mul_
).What do you mean?
If you take the cup
with something the coerces in (perhaps 2
?) but is not actually an element of the same parent, then it will either fail or could end up with an element that is not actually in the ring. However, for _mul_
, this can never happen because the coercion framework would make sure both arguments belong to the same parent.
comment:27 followup: 28 Changed 19 months ago by
Replying to tscrim:
Replying to ghmjungmath:
Replying to tscrim:
I don't think you should assume that the parents are the same for
cup
(unlike what you can do for_mul_
).What do you mean?
If you take the
cup
with something the coerces in (perhaps2
?) but is not actually an element of the same parent, then it will either fail or could end up with an element that is not actually in the ring. However, for_mul_
, this can never happen because the coercion framework would make sure both arguments belong to the same parent.
Currently, there is not even a natural coercion implemented. The only coercion that would be admissible is the one from closed differential forms (and everything that is contained in it) into cohomology, but we don't have that subspace implemented. Therefore, it wouldn't even work with _mul_
.
comment:28 followup: 29 Changed 19 months ago by
Replying to ghmjungmath:
Replying to tscrim:
Replying to ghmjungmath:
Replying to tscrim:
I don't think you should assume that the parents are the same for
cup
(unlike what you can do for_mul_
).What do you mean?
If you take the
cup
with something the coerces in (perhaps2
?) but is not actually an element of the same parent, then it will either fail or could end up with an element that is not actually in the ring. However, for_mul_
, this can never happen because the coercion framework would make sure both arguments belong to the same parent.Currently, there is not even a natural coercion implemented. The only coercion that would be admissible is the one from closed differential forms (and everything that is contained in it) into cohomology, but we don't have that subspace implemented. Therefore, it wouldn't even work with
_mul_
.
However, it will fail with a much more reasonable message and it will be all setup for when there is a coercion implemented. Most importantly, it will handle the natural coercion from the scalar field into the ring, but cup
will not. You should have cup
simply redirect to *
. This will also make it easier if this ever gets subclassed too.
comment:29 Changed 19 months ago by
Replying to tscrim:
However, it will fail with a much more reasonable message and it will be all setup for when there is a coercion implemented. Most importantly, it will handle the natural coercion from the scalar field into the ring, but
cup
will not. You should havecup
simply redirect to*
. This will also make it easier if this ever gets subclassed too.
It will get subclassed. My idea is that a characteristic class becomes a subclass of DeRhamCohomologyClass
with a dedicated representative
method.
comment:31 Changed 19 months ago by
Commit:  fb8331abc00f6886b511819bff1e4116bb2e6d30 → 566176a74a480456b7836c13a173e217b721b70a 

Branch pushed to git repo; I updated commit sha1. New commits:
566176a  Trac #31691: new file + improved doc + cup product delegates to *

comment:35 Changed 19 months ago by
Status:  needs_review → positive_review 

comment:37 Changed 18 months ago by
Branch:  public/31691_de_rham_complex → 566176a74a480456b7836c13a173e217b721b70a 

Resolution:  → fixed 
Status:  positive_review → closed 
Branch pushed to git repo; I updated commit sha1. New commits:
Trac #31691: add category and differential
Trac #31691: more concrete examples