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Thanks! I totally understand the answer. Moreover, if we change $S^m$ to other manifolds, for example, projective spaces, Grassmannians, then how to compute the corresponding Stiefel-Whitney class?
Thanks! Why the exact sequence of vector bundles can imply that the Stiefel-Whitney class satisfy the product formula? I am only clear with the special case when the short exact sequence split, then we have the Whitney sum.
Thanks, Prof. I have another question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind of $I$ can we choose? How is the exterior algebra $H^*(\Omega^{n+1}\Omega^{n+k+1};\mathbb{Z}_2)$ related to $n$?
Dear Prof., you mentioned that "since the fundamental class is primitive, so all the classes obtained by Dyer-Lashof operations are primitive as well". But if this is true for the homology, the cup product of cohomology would be trivial.
