Prove that $\displaystyle S^{m+n+1}=X\cup Y$ with $\displaystyle X\cong S^{m}\times B^{n+1}$, $\displaystyle Y\cong B^{m+1}\times S^{n}$, and $\displaystyle X\cap Y\cong S^{m} \times S^{n}$. Use this to compute the homology groups of $\displaystyle S^m \times S^n$.

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