Mod-two homology of a submanifold complement (${\tilde{H}}_i(S^n\setminus M;{\mathbb{F}}_2)$)

For a closed connected $k$-manifold smoothly embedded in $S^n$, with $k<n$ and $n\ge 1$, its complement hasand zero reduced homology for $i\ge n-1$, 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.