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With the example you started with:
K = SimplicialComplex([[0, 1], [1, 2], [2, 0]])
C1 = K.n_chains(1)
Then C1 is
Free module generated by {(0, 1), (0, 2), (1, 2)} over Integer Ring
so we can do
c01, c02, c12 = C1.gens()
z = c01 - c02 + c12
Then
z.is_cycle()
should return True, while
z.is_boundary()
should return False. So z represents a nonzero class in homology.
