Thanks for your speedy reply, John! I have a follow-up question. For some
cycle representing a homology class, is there a function that expresses it
as a linear combination of the basis in the CohomologyRing?
The reason I'm interested in this is that I have a group acting on my space
and want to determine what the group action does to the basis elements: I
want to take a basis element, act on it, and express it w.r.t. the basis.
So far, the most sensible thing in my eyes is to take the basis element,
write it as a cycle, act on the cycle, and convert the result back to a
linear combination of the basis.
On Friday, September 9, 2016 at 6:19:52 PM UTC+2, John H Palmieri wrote:
>
> 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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