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Let $j : SO(2n) \to SO(2n+1)$ be the standard subgroup embedding and let $Bj : BSO(2n) \to BSO(2n+1)$ be the induced fibration obtained by factoring the universal $SO(2n+1)$-bundle by the subgroup $SO(2n)$. I am looking for an explicit formula for the connecting homomorphism of the corresponding Gysin sequence in integer cohomology. I think the answer is $(Bj^\ast \alpha) \cup e_{2n} = 2 \alpha$, for $\alpha$ in the cohomology of $BSO(2n+1)$, where $e_{2n}$ is the Euler class of the fibration $BSO(2n-1) \to BSO(2n)$, but I don't really have a proof for that.

Many thanks.

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