The context is integral simplicial homology. Consider the following triangulation of the Klein bottle: link to picture. I want to write down a basis for the 1-cycles explicitly, without resorting to linear algebra (i.e., without writing down a huge matrix and reducing).

The chain complex looks like this: 0 → Z¹⁸ → Z²⁷ → Z⁹ → 0. Helpful facts: since the bottle is non-orientable, the second homology is trivial, i.e., there are no non-trivial 2-cycles. This implies that all 18 triangles that generate Z¹⁸ embed into the 1-cycles. Moreover, since the bottle is connected, the zeroth homology has rank 1, which implies that the rank of 0-boundaries is 8, which in turn implies that the 1-cycles have rank 27 - 8 = 19. Therefore, our basis should consist of 19 1-cycles, and we already have 18 of them (the boundaries of the 18 triangles). This means that any other 1-cycle that is linearly independent with the 18 triangle cycles, such as ag gd da, should complete the picture. But this doesn't feel right as I can't figure out a way to write down an arbitrary 1-cycle, such as bc cf fi ib, in terms of these 19.