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?

## Advertising

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? >> > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.