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I need A little bit more clarification when computing the homology of a chain complex. So the problem is:

Compute the simplicial homology of the graph with vertices $$V=\left\{ 1, 2, 3, 4 \right\}$$ and edges $$E=\left \{ (1, 2)\right \}$$

Now, I know $ H_{k}=\frac{ker\partial _{k}}{im\partial _{k+1}} $ and I ‘m supposed to construct a matrix of the boundary maps, and that’s what confuses me.

I have that $C_{0}=\left \{1, 2, 3, 4\right \}$ and $C_{1}=\left \{(1, 2)\right \}$.

And the boundary I got is $\partial _{1}=C_{1}\rightarrow C_{0}=2-1$, is this right? Also this is all done in $F_{2}$

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    $\begingroup$ So, that would be represented by a $1\times 4$ matrix $$\pmatrix{-1&1&0&0}.$$ $\endgroup$ – Lord Shark the Unknown Nov 20 '18 at 4:49

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