= Cohomological Gysin map of an embedding
{title2=$i_!:H^j(N;R)\to H^{j+r}(M;R)$}
For a closed <smooth embedding> $i:N\hookrightarrow M$ of codimension $r$ whose <normal bundle> is oriented over a coefficient ring $R$, a <tubular neighborhood>, <excision> and the <Thom isomorphism theorem> identify
$$
H^{q-r}(N;R)\cong H^q(M,M\setminus N;R).
$$
The relative-to-absolute map defines $i_!:H^j(N;R)\to H^{j+r}(M;R)$. For closed oriented $M,N$, it is characterized by
$$
\langle i_!b\smile c,[M]\rangle=\langle b\smile i^*c,[N]\rangle
$$
with compatible orientations. It obeys $i_!(b\smile i^*c)=i_!b\smile c$. If $j:M\setminus N\hookrightarrow M$ and $\delta:H^q(M\setminus N;R)\to H^{q+1-r}(N;R)$ is the connecting map followed by the inverse Thom isomorphism, then, over $\mathbb F_2$,
$$
\delta(j^*c\smile u)=i^*c\smile\delta u.
$$
These identities follow from naturality of the relative <cup product> and show how the exact sequence determines multiplication in complements. Over $\mathbb F_2$ every real <normal bundle> has the required orientation.
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