Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials $$ d_4^{p,m}:E_4^{p,3} \to E^{p+4,0} $$ for $m \in \{2,4,\ldots,2n\}$?
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7$\begingroup$ Doesn't the serre sseq for a sphere bundle reduce to the Gysin sequence? So, using cohomology instead of homology, the differential should be multiplication by the Euler class, which in this case is like... $klc^2$?where c is the standard generator in H^2 of CP^n $\endgroup$– Dylan WilsonCommented Aug 12, 2017 at 21:27
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$\begingroup$ @DylanWilson Oh, apparently this is the case. Thanks for pointing this out! $\endgroup$– 54321userCommented Aug 12, 2017 at 22:06
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