If you're willing to work with field coefficients, there is the method
"homology_with_basis":
sage: T = simplicial_complexes.Torus()
sage: H = T.homology_with_basis()
sage: H
Homology module of Minimal triangulation of the torus over Rational Field
sage: H.basis()
Finite family {(2, 0): h_{2,0}, (1, 0): h_{1,0}, (0, 0): h_{0,0}, (1, 1):
h_{1,1}}
sage: h10 = H.basis()[1,0]; h10
h_{1,0}
sage: h11 = H.basis()[1,1]
sage: x = h10 + 3/2 * h11
sage: x.to_cycle() # a representative of x as a linear combination of
chains
(0, 1) + 3/2*(0, 2) - (0, 3) - 3/2*(0, 5) + (1, 3) + 3/2*(2, 5)
--
John
On Friday, September 9, 2016 at 8:28:31 AM UTC-7, Albert Haase wrote:
>
> For an abstract finite simplicial complex C, the method
> homology(algorithm='auto',generators=True) uses CHomP to compute the
> homology groups and their generators. The generators are conveniently
> expressed as sums of simplices from C rather than as coefficient vectors of
> chains from the chain complex. Assume we manipulate the generators, for
> instance by letting a group act on them.
>
> (1) Are there sage functions, or a "setting", that allows us to calculate
> with homology classes, where representatives of the classes are expressed
> as sums of simplices from the simplicial complex C?
>
> (2) Is there a sage function that takes an element of a homology group
> (represented by a sum of simplices from C) and returns its coordinates
> w.r.t. a basis of the homology group?
>
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