#8302: cubical complexes, delta complexes, and more
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Reporter: jhpalmieri | Owner: jhpalmieri
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.3.3
Component: algebraic topology | Keywords:
Author: John Palmieri | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Description changed by jhpalmieri:
Old description:
> The attached patch adds lots of functionality to Sage's algebraic
> topology capabilities:
>
> - it implements cubical complexes: complexes constructed from cubes of
> various dimensions, glued together in prescribed ways.
>
> - it implements Delta complexes: this is a variant on a simplicial
> complex which allows for more efficient construction of many spaces. For
> example, the minimal triangulation of the torus as a simplicial complex
> uses 14 triangles, while there is a Delta complex version with only two
> triangles. Allen Hatcher uses these in his popular algebraic topology
> book.
>
> - it "implements" generic cell complexes, as a parent class to the
> previous two, and also to simplicial complexes. This is not intended for
> use by casual Sage users, but instead for developers who want to add
> another kind of complex (CW complexes? Prodsimplicial complexes?) Many
> methods in this class are not implemented, but instead provide a template
> of what should be implemented in any derived class.
>
> - it modifies simplicial complexes a bit, allowing them to be defined
> without specifying a vertex set: just list the maximal simplices and it
> will deduce what the vertex set is. It also defines {{{connected_sum}}}
> for arbitrary simplicial complexes, not just simplicial surfaces, with a
> warning that it's not well-defined if you don't call it on manifolds. It
> renames {{{ProjectivePlane}}} to {{{RealProjectivePlane}}} (keeping the
> old name as an alias for backward compatibility).
>
> - it provides an interface to CHomP, which is now an experimental spkg
> for Sage. CHomP provides programs to compute homology which are faster
> than anything Sage can do. See [http://chomp.rutgers.edu/] for more
> information.
>
> - it changes how the {{{homology}}} and {{{chain_complex}}} methods
> work: these now pass keywords to each other, so it's easy to implement
> new keywords: just implement it for {{{ChainComplex.homology}}}, for
> instance, and when you compute the homology of any simplicial complex,
> you can give it the keyword and it will get passed on to this method.
New description:
The attached patch adds lots of functionality to Sage's algebraic topology
capabilities:
- it implements cubical complexes: complexes constructed from cubes of
various dimensions, glued together in prescribed ways.
- it implements Delta complexes: this is a variant on a simplicial
complex which allows for more efficient construction of many spaces. For
example, the minimal triangulation of the torus as a simplicial complex
uses 14 triangles, while there is a Delta complex version with only two
triangles. Allen Hatcher uses these in his popular algebraic topology
book.
- it "implements" generic cell complexes, as a parent class to the
previous two, and also to simplicial complexes. This is not intended for
use by casual Sage users, but instead for developers who want to add
another kind of complex (CW complexes? Prodsimplicial complexes?) Many
methods in this class are not implemented, but instead provide a template
of what should be implemented in any derived class.
- it modifies simplicial complexes a bit, allowing them to be defined
without specifying a vertex set: just list the maximal simplices and it
will deduce what the vertex set is. It also defines {{{connected_sum}}}
for arbitrary simplicial complexes, not just simplicial surfaces, with a
warning that it's not well-defined if you don't call it on manifolds. It
renames {{{ProjectivePlane}}} to {{{RealProjectivePlane}}} (keeping the
old name as an alias for backward compatibility).
- it provides an interface to CHomP, which is now an experimental spkg
for Sage. CHomP provides programs to compute homology which are faster
than anything Sage can do. See [http://chomp.rutgers.edu/] for more
information.
- it changes how the {{{homology}}} and {{{chain_complex}}} methods work:
these now pass keywords to each other, so it's easy to implement new
keywords: just implement it for {{{ChainComplex.homology}}}, for instance,
and when you compute the homology of any simplicial complex, you can give
it the keyword and it will get passed on to this method.
- it adds a file {{{tests.py}}} which compares the results of CHomP with
the built-in Sage homology computations, raising an error if they ever
disagree.
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8302#comment:3>
Sage <http://www.sagemath.org>
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