Problem in Relative Homology Computation ?

asked 2018-06-19 18:22:06 -0500

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:

Square = CubicalComplex([([0,1],[0,1])])

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:

FirstandSecondandThirdEdges = CubicalComplex([([0,0],[0,1]),([0,1],[1,1]),([1,1],[0,1])])

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:

FirstandThirdandFourthEdges = CubicalComplex([([0,0],[0,1]),([0,1],[0,0]),([1,1],[0,1])])

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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