= Mod-two homology of a submanifold complement
{title2=$\widetilde H_i(S^n\setminus M;\mathbb F_2)$}
For a closed connected $k$-manifold smoothly embedded in $S^n$, with $k<n$ and $n\geq1$, its complement has
$$
\widetilde H_i(S^n\setminus M;\mathbb F_2)\cong H_{i+k+1-n}(M;\mathbb F_2)\quad(0\leq i\leq n-2),
$$
and zero reduced homology for $i\geq n-1$, where negative-index groups are zero. The <Thom isomorphism theorem> for the normal bundle and the pair's exact sequence give this formula. The top ambient <fundamental class> maps isomorphically to the fundamental class of $M$, canceling the exceptional top term.
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