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The cellular chain complex $C_{\ast}(X)$ of an $n$-sphere $X=S^{n}$ (with any CW-complex structure), gives rise to an exact sequence $$ 0 \rightarrow \mathbb{Z} \rightarrow C_{n}(X) \rightarrow C_{n-1}(X) \rightarrow \ldots \rightarrow C_{0}(X) \rightarrow \mathbb{Z} \rightarrow 0. $$By glueing two copies of this sequence together, one gets the exact sequence $$ 0 \rightarrow \mathbb{Z} \rightarrow C_{n}(X) \rightarrow C_{n-1}(X) \rightarrow \ldots \rightarrow C_{0}(X)\rightarrow C_{n}(X) \rightarrow C_{n-1}(X) \rightarrow \ldots \rightarrow C_{0}(X) \rightarrow \mathbb{Z} \rightarrow 0. $$ What (if any) topological construction can one do with two copies of the $n$-sphere, to obtain this complex as cellular chain complex of the resulting space?


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