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Suppose we have a space $X$ consisting of a 1-cell $S^1$ and two 2-cells attached via maps of degree 2 and 3 respectively. I want to be able to compute the homology groups of such a space using cellular homology. The cell complex of $X$ is clearly given by one 0-cell, one 1-cell and two 2-cells, so we have the cellular chain complex $$0 \xrightarrow{ \ \ \ \ } \mathbb{Z} \oplus \mathbb{Z} \xrightarrow{ \ \ d_2 \ \ } \mathbb{Z} \xrightarrow{ \ \ d_1 \ \ } \mathbb{Z} \longrightarrow 0.$$

Since the attaching maps are of degree 2 and 3, $d_2$ is defined by mapping $(1,1) \longmapsto 2 \cdot 3 = 6$, so the image of $d_2 = 6\mathbb{Z}$ and the kernel is trivial. I'm unsure of how to proceed.

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For a connected CW complex $X$ with a single $0$-cell, the boundary map $d_1$ is always $0$. I believe that $d_2$ should be the map given by $(1,0)\mapsto 2$ and $(0,1)\mapsto 3$ (in particular, surjective). Now use exactness of the sequence.

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