## Spheres and homology groups

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

Thanks.