Generators for homology of simplicial complexes

[jhpalmieri]

This is a followup to #6100, and was reported on this ask question. If K is a simplicial complex, the generators argument to K.homology(...) does nothing: K.homology(...) calls the homology method for generic cell complexes, which in turn calls _homology_ for simplicial complexes, and that method ignores the generators keyword.

There is an easy solution: in the generic homology method, if generators is True, then don't call _homology_, just continue in that method. This will give information about chain complex generators. It is possible, although obscure, to extract the simplicial complex information from that.

A better solution: actually do the obscure work, don't force the users to figure it out. That is:

• call K.chain_complex(generators=True), and
• use K._n_cells_sorted() to convert the chain complex generators (which are just vectors in the corresponding free module over the base ring) to actual chains in the simplicial complex.

CC: @tscrim

Component: algebraic topology

Author: John Palmieri

Branch/Commit: a5b41da

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/30838

[jhpalmieri]

Branch: u/jhpalmieri/homology-generators

[jhpalmieri]

Commit: c1eae33

[jhpalmieri]

New commits:

c1eae33

trac 30838: compute generators for homology of simplicial complexes,

[jhpalmieri]

Author: John Palmieri

[sagetrac-git]

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

8ca6c0e

trac 30838: compute generators for homology of simplicial complexes,

[sagetrac-git]

Changed commit from c1eae33 to 8ca6c0e

[sagetrac-git]

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

a5b41da

trac 30838: compute generators for homology of simplicial complexes,

[sagetrac-git]

Changed commit from 8ca6c0e to a5b41da

[jhpalmieri]

comment:5

Before this branch:

sage: simplicial_complexes.Sphere(2).homology(generators=True)                                                        
{0: 0, 1: 0, 2: Z}
sage: cubical_complexes.Sphere(2).homology(generators=True)                                                        
{0: [], 1: [], 2: [(Z, Chain(2:(1, -1, 1, -1, 1, -1)))]}
sage: simplicial_sets.Sphere(2).homology(generators=True)                                 
{0: [], 1: 0, 2: [(Z, Chain(2:(1)))]}

Note that no generators are given at all in the simplicial complex case. In the other cases, the answers are given in terms of the chain groups.

After:

sage: simplicial_complexes.Sphere(2).homology(generators=True)                            
{0: [], 1: [], 2: [(Z, (0, 1, 2) - (0, 1, 3) + (0, 2, 3) - (1, 2, 3))]}
sage: cubical_complexes.Sphere(2).homology(generators=True)                               
{0: [],
 1: [],
 2: [(Z,
   [0,0] x [0,1] x [0,1] - [0,1] x [0,0] x [0,1] + [0,1] x [0,1] x [0,0] - [0,1] x [0,1] x [1,1] + [0,1] x [1,1] x [0,1] - [1,1] x [0,1] x [0,1])]}
sage: simplicial_sets.Sphere(2).homology(generators=True)                                 
{0: [], 1: 0, 2: [(Z, sigma_2)]}

[tscrim]

Reviewer: Travis Scrimshaw

[tscrim]

comment:8

LGTM.

[jhpalmieri]

comment:9

Thanks very much.

[vbraun]

Changed branch from u/jhpalmieri/homology-generators to a5b41da