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cell complexes vs simplicial complexes

asked 2021-02-09 06:41:25 +0100

Ingrid gravatar image

I have spent several days googling, but am not able to find any basic information for working with cell complexes in sage. I am using windows 10 , sage 9.1 where the help function doesn't work.

What I need to be able to do is to enter a cell complex, add cells, delete cells and subdivide. I already have functions defined that could be used to specify the gluing maps. There seems to be slightly more information available for working with simplicial complexes. If I were to subdivide my cell complex, I would get a simplicial complex; there are no exotic gluing maps. However, because the cells are high dimensional, the number of simplicies is huge after subdivision, and I am not sure I could correctly enter all that information by hand. Any suggestions would be very appreciated! If anyone has a file to share in which they have worked with cell complexes or simplicial complexes, that would also be really helpful to me to learn the syntax.

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answered 2021-02-09 11:53:56 +0100

jipilab gravatar image

I would recommend you to have a look at the documentation:

For general cell complexes For simplicial complexes

Note that they have different methods available and that they might not offer the state-of-the-art methods for what you mean, but it is worth trying!

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Thanks, both of you, for confirming what I was beginning to be afraid of. I don't know how to describe continuous functions in sage, but for all the examples I can think of, the maps can be described as being piecewise linear.

Ingrid gravatar imageIngrid ( 2021-02-10 12:37:36 +0100 )edit
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answered 2021-02-09 17:34:19 +0100

From the documentation for generic cell complexes cited by @jpilab:

This module defines a class of abstract finite cell complexes. This is meant as a base class from which other classes (like SimplicialComplex, CubicalComplex, and DeltaComplex) should derive. As such, most of its properties are not implemented. It is meant for use by developers producing new classes, not casual users.

In particular, there is no way for a Sage user to use a "cell complex"; you can instead work with a simplicial complex, a Delta complex, or a simplicial set. There are plenty of examples of all of those in the documentation.

Simplicial sets can be very efficient; to define a 10-dimensional sphere, for example, you only have to specify a 10-dimensional simplex and a 0-dimensional simplex. Delta complexes are somewhat more efficient, and simplicial complexes are the least efficient, in terms of numbers of simplices required.

(Here's a question: if Sage were to implement CW complexes, for example, how should it encode the gluing maps? Is there a good way to describe a continuous function in a computer algebra system like Sage? With a good answer to this, we could think about how to handle the rest of the implementation.)

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Asked: 2021-02-09 06:41:25 +0100

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Last updated: Feb 09