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Let $X_n$ denote the n-th skeleton. The CW chain complex of $S^1 \times S^1$ is just $$...0 \rightarrow (X_{-1} = \emptyset) \rightarrow (X_0 = \bullet) \rightarrow (X_1 = S_1 \vee S_1) \rightarrow (X_2 = S^1 \times S^1) \rightarrow 0 ... $$

So attach one $0$-cell, two $1$-cells, and one $2$-cell.

We can generalize this to say that (for example) the chain complex for $S^2 \times S^2$ is $$...0 \rightarrow (X_{-1} = \emptyset) \rightarrow (X_0 = \bullet) \rightarrow (X_1 = \bullet) \rightarrow (X_2 = S^2 \vee S^2) \rightarrow (X_3 = S^2 \times S^2) \rightarrow 0 ... $$

Is that correct? I just wanted to be sure...

Artus
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  • Why do you think that $X_2=S^2\times S^2$? That's not even a $2$-dimensional object. Also, how are these sequences chain complexes? Were you trying to set up the cellular chain complex associated to the CW complex? – Dan Rust May 05 '14 at 11:22
  • @DanielRust Oh I'm sorry...I meant $X_2 = S^1 \times S^1$. I'll edit that. – Artus May 05 '14 at 11:26

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