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Cohomology classes associated to vector bundles. Includes Stiefel-Whitney classes, Chern classes, Pontryagin classes, and the Euler class.
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Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles
I can hardly give an exact meaning to ii). The characteristic class $c$ has to be stable and hence a natural transformation from (say) $K^*$ to $ h^*$. The best one can hope is that If $\pi \colon …