Whitney sum formula and Euler class

Question

About the Whitney sum formula and Euler class: If the fiber $F \to$ the base $X$ is an oriented, real vector bundle, then the Euler class of their direct sum is given by $e(E\oplus F)=e(E)\smile e(F).$ I do not quite understand the precise definition Lhs and Rhs formula : $$e(E\oplus F)=e(E)\smile e(F).$$

• Can one fill in more detail why this is true? (is that a cup product? how to define it precisely in this case?)

• Do we have similar statements about the Whitney sum formula for other characteristic classes? (e.g. see the link Whitney sum formula for Pontryagin classes are rather different? $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion

Answer 1

Yes, it is cup product. This property of the Euler class is proved in Milnor and Stasheff's Characteristic Classes on page $100$.

If $w$ denotes the total Stiefel-Whitney class, then $w(E\oplus F) = w(E)w(F)$; this is proved on pages $92$ and $93$.

If $p$ denotes the total Pontryagin class, then $p(E\oplus F) = p(E)p(F)$ modulo two-torsion (i.e. $2(p(E\oplus F) - p(E)p(F)) = 0$); this is proved on page $175$.

If $E$ and $F$ are complex vector bundles and $c$ denotes the total Chern class, then $c(E\oplus F) = c(E)c(F)$; this is proved on pages $164 - 166$.