2018-06-20 11:00:45 +0200 | asked a question | Problem in Relative Homology Computation ? I am trying to understand how to compute relative homologies between cubical complexes and a given subcomplex. Consider the cubical complex of the elementary cube [0,1]x[0,1],defined via: I further refer to the edges ofthe complex as: First Edge: [0,0] x [0,1] Second Edge: [0,1] x [1,1] Third Edge: [1,1] x [0,1] Fourth Edge: [0,1] x [0,0] Imagine labeling the edges of a square in a clockwise fashion, with the vertical leftmost one being the first edge. When i try to compute the relative homology of Square in relation to the subcomplex generated by the First, Second and Third edges, i do: Then, the calculation of the homology and the result is: {0: 0, 1: Z, 2: Z} (which I suspect is wrong). In order to calculate the homology in relation to the subcomplex generated by the First ,Third and Fourth edges, i first define: To calculate the relative homology: And the result is: {0: 0, 1: 0, 2: 0}. I am not experienced with homology calculations, but I believe the two results should be the same, since the latter configuration is just a rotation of the first one by 180 degrees.I also believe that the right result should given by {0: 0, 1: 0, 2: 0} in both cases, which is the same as considering the relative homology of Square and a single arbitrary edge. Are these calculations correct ? Is my intuition wrong about these two relative homologies groups? If anyone could point out some mistake, I would very much appreciate :) |

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