= Gysin sequence of an embedding
{c}
{title2=$\cdots\to H_j(N\setminus M;\mathbb F_2)\to H_j(N;\mathbb F_2)\to H_{j-c}(M;\mathbb F_2)\to\cdots$}
For a closed smooth submanifold $M\subseteq N$ of codimension $c>0$, a <tubular neighborhood> and the homological <Thom isomorphism theorem> identify
$$
H_j(N,N\setminus M;\mathbb F_2)\cong H_{j-c}(M;\mathbb F_2).
$$
Substitution in the <long exact sequence in relative homology> gives the displayed embedding Gysin sequence. It does not require orientability over $\mathbb F_2$.
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