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I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand the 2-torsion part better.

Is there a reference which describes the difference between $p(E\oplus F)$ and $p(E)\cdot p(F)$, perhaps in terms of Bocksteins of Stiefel-Whitney classes of $E$ and $F$?

This question was previously part of Whitney sum formula for Pontryagin classes I; Qiaochu Yuan's answer to that question might be helpful.

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Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.

Theorem 1.6, last sentence:

Under Whitney sum, $p_q\mapsto \sum_j r_{2q-j}\otimes r_j$, where $r_{2s} = p_s$ and $r_{2s+1} = (\delta w_{2s})^2+ p_s\delta w_1$.

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    $\begingroup$ I don't suppose anyone wants to provide that sentence here to have the question and answer in one place? $\endgroup$
    – Greg Friedman
    Commented Nov 5, 2015 at 7:08
  • $\begingroup$ @GregFriedman: I do. $\endgroup$
    – user78588
    Commented Nov 5, 2015 at 8:38

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