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May 22, 2020 at 17:29 comment added skd Ah, yes. Sorry, didn't read carefully enough.
May 22, 2020 at 14:20 vote accept Igor Belegradek
May 22, 2020 at 12:40 comment added Igor Belegradek @sdk: Brown's paper is about vector bundles, while my question is about locally trivial bundles with fiber homeomorphic to $\mathbb R^n$.
May 22, 2020 at 7:05 answer added Oscar Randal-Williams timeline score: 8
May 22, 2020 at 4:23 comment added skd Let p(E) denote the total (integral) Pontryagin class of a real bundle E. It's not too hard to show that p(E (+) F) = p(E) * p(F) mod 2-torsion. The 2-torsion term is stated in Theorem 1.6 of Brown's "The Cohomology of BSO_n and BO_n with Integer Coefficients".
May 22, 2020 at 2:07 history asked Igor Belegradek CC BY-SA 4.0