= Homology of an equatorial sphere complement
{title2=$S^n\setminus S^m\simeq S^{n-m-1}$}
For the standard coordinate inclusion of <spheres>, write points as $(u,v)\in\mathbb R^{m+1}\times\mathbb R^{n-m}$, with $v\ne0$ in the complement. The map $(u,v)\mapsto(u,v/\lVert v\rVert)$ is a <homeomorphism> to $B^{m+1}_{\mathrm{open}}\times S^{n-m-1}$, with inverse $(u,w)\mapsto(u,\sqrt{1-\lVert u\rVert^2}w)$. Contracting the ball factor proves the displayed <homotopy equivalence>. In codimension one there are two contractible components, so integral $H_0$ is $\mathbb Z^2$; in higher codimension the only nonzero <homology groups> are $H_0=\mathbb Z$ and $H_{n-m-1}=\mathbb Z$.
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