Gysin sequence of a sphere bundle
= Gysin sequence of a sphere bundle
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= Gysin sequence
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A <Gysin sequence of a sphere bundle> is the <long exact sequence> induced by the <Gysin homomorphism> and the <Euler class of a vector bundle>.
For the unit sphere bundle $S(E)\to B$ of an oriented rank-$d$ vector bundle, the Gysin sequence contains
$$
\cdots\to H^{q-d}(B)\xrightarrow{\smile e(E)}H^q(B)\to H^q(S(E))\to H^{q-d+1}(B)\xrightarrow{\smile e(E)}H^{q+1}(B)\to\cdots.
$$