Images of continuous maps and embedded submanifolds as manifold subsets

[mkoeppe]

We define a subclass of ManifoldSubset, instances of which represent the image of a continuous map.

In particular, this will allow an embedded submanifold to be considered as a subset of the ambient manifold.

[mkoeppe]

There are a number of related existing methods in various components of Sage. FreeModuleMorphism.pushforward can compute the image of a submodule,

and RingHomomorphism.pushforward can compute the image of an ideal.

[mkoeppe]

Also there are various image methods of various morphism classes, not all with the same signature:

• without extra argument: category FiniteDimensionalModulesWithBasis parent method, CrystalMorphismByGenerators, SimplicialComplexMorphism, Matrix, MatrixMorphism_..., SchemeMorphism_...
• with an extra argument (substructure to compute the image of): AbelianGroupMorphism, PermutationGroupMorphism, SimplicialSetMorphism,
• incompatible: WordMorphism.image

[egourgoulhon]

Thanks for opening this ticket; I've added it to the metaticket #30525.

[mkoeppe]

Likewise, there is some diversity regarding inverse_image and preimage methods.

[git]

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

​d757adb

ImageManifoldSubset.__contains__: New

[git]

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

​4b59194

ImageManifoldSubset._an_element_: New

[git]

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

​18c1bd9

ImageManifoldSubset, ContinuousMap.image: Add optional 'inverse' argument, use it in __contains__

[git]

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

​645aa2f

src/sage/manifolds/continuous_map_image.py: Update doctests

[git]

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

​5170c15

TopologicalSubmanifold.as_subset: New

[mkoeppe]

Early stage. Comments on this design are very welcome

[gh-mjungmath]

Yes, I have two comments:

• I advocate f-strings:

-        return "Image of the {}".format(self._map)
+        return f"Image of the {self._map}"

More readable and even (slightly) faster.

• I don't quite get the purpose of

  return self._map(preimage) == point
  

Isn't that always true if preimage in self._map.domain() is already met (and the inverse is correctly defined)?

[mkoeppe]

This notion of inverse is taken from TopologicalSubmanifold. This inverse maps the whole ambient space to the domain of the map - so one needs to check

[mkoeppe]

Replying to gh-mjungmath:

• I advocate f-strings:

Yes, I like them too. Unfortunate clash of these braces with LaTeX notation though

[gh-mjungmath]

Replying to mkoeppe:

This notion of inverse is taken from TopologicalSubmanifold. This inverse maps the whole ambient space to the domain of the map - so one needs to check

Alright, I see it now. It's just a very unusual definition of inverse...

The rest looks pretty nice already. :)

[egourgoulhon]

Replying to gh-mjungmath:

The rest looks pretty nice already. :)

Yes, it looks nice!

[egourgoulhon]

Replying to mkoeppe:

Early stage. Comments on this design are very welcome

I like it.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​cd8d6af	sage.manifolds.continuous_map_image, ContinuousMap.image: New
​ae0b7f9	ImageManifoldSubset.__contains__: New
​0135892	ImageManifoldSubset._an_element_: New
​459758e	ImageManifoldSubset, ContinuousMap.image: Add optional 'inverse' argument, use it in __contains__
​26c7e56	src/sage/manifolds/continuous_map_image.py: Update doctests
​3e273bb	TopologicalSubmanifold.as_subset: New
​9726d36	Docstring work

[mkoeppe]

I am considering possible extensions of the current code:

• handle a domain subset directly in ImageManifoldSubset - which would allow us to extend ContinuousMap.image to take a subset that is not open
• if the map is an isomorphism (and the domain subset open), we could create a submanifold -- but should this use extra classes ImageTopologicalManifold, ImageDifferentiableManifold, which remember the map, or just compute the result? (in either case - follow-up ticket)

[git]

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

​19762ae

ImageManifoldSubset: New parameter domain_subset, use it in ContinuousMap.image

[git]

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

​964f9f7

src/sage/manifolds/continuous_map.py: Update copyright

[egourgoulhon]

Some TESTS:: fields are actually empty.

[egourgoulhon]

Replying to mkoeppe:

I am considering possible extensions of the current code:

• handle a domain subset directly in ImageManifoldSubset - which would allow us to extend ContinuousMap.image to take a subset that is not open

Sounds legitimate.

• if the map is an isomorphism (and the domain subset open), we could create a submanifold -- but should this use extra classes ImageTopologicalManifold, ImageDifferentiableManifold, which remember the map, or just compute the result? (in either case - follow-up ticket)

Maybe just compute the result?

[mkoeppe]

Replying to egourgoulhon:

Replying to mkoeppe:

• if the map is an isomorphism (and the domain subset open), we could create a submanifold -- but should this use extra classes ImageTopologicalManifold, ImageDifferentiableManifold, which remember the map, or just compute the result? (in either case - follow-up ticket)

Maybe just compute the result?

OK, I have opened #31685 for this.

[git]

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

​e711215

src/sage/manifolds/continuous_map_image.py: Add tests

[mkoeppe]

Ready for review

[egourgoulhon]

To generate the documentation regarding the new class ImageManifoldSubset, the following line

   sage/manifolds/continuous_map_image

should added at the end of src/doc/en/reference/manifolds/continous_map.rst. Also, it would be nice to modify the head of src/sage/manifolds/subsets.py so that it refers to it, e.g. something like:

The class :class:`ManifoldSubset` implements generic subsets of a
topological manifold. Open subsets are implemented by the class
:class:`~sage.manifolds.manifold.TopologicalManifold` (since an
open subset of a manifold is a manifold by itself), which inherits
- from :class:`ManifoldSubset`.
+ from :class:`ManifoldSubset`. Besides, subsets that are images of
+ a manifold subset under a continuous map are implemented by the
+ subclass :class:`~sage.manifolds.continuous_map_image.ImageManifoldSubset`. 

[git]

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

​0f3e36d

Link in documentation of sage.manifolds.continuous_map_image

[egourgoulhon]

Thanks!

[mkoeppe]

Thanks for reviewing!