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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 this better, and have two precise questions.

  1. Using the definition $p_i(E)=(-1)^{i}c_{2i}(E_\mathbb{C})$, the fact that complexification respects direct sums, and the Whitney sum formula for Chern classes, I find for example that $$ p_1(E\oplus F) = -c_2(E_\mathbb{C}\oplus F_\mathbb{C}) = -c_2(E_\mathbb{C}) - c_1(E_\mathbb{C})c_1(F_\mathbb{C}) - c_2(F_\mathbb{C}), $$ and $$p_1(E) + p_1(F)=-c_2(E_\mathbb{C})-c_2(F_\mathbb{C}).$$ I do not see why the difference $c_1(E_\mathbb{C})c_1(F_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing?
  2. edit: I've moved the second question to a separate post Whitney sum formula for Pontryagin classes II
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    $\begingroup$ The reason the classes $c_{2k+1}(E_\mathbb C)$ are 2-torsion is that the complex bundle $E_\mathbb C$ has an anti-linear automorphism (conjugation). $\endgroup$ – Yury Ustinovskiy Nov 4 '15 at 0:25
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    $\begingroup$ @YuryUstinovskiy: Thanks. I see now that I should have gone back and read Milnor and Stasheff more closely before posting this, in particular the first three pages of Chapter 15 (your comment is Lemma 15.1). I still would like a formula for the 2-torsion part though, which I've now asked as a second question. $\endgroup$ – Mark Grant Nov 4 '15 at 8:25
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I do not see why the difference $c_1(E_\mathbb{C})c_1(F_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing?

Real line bundles are classified by $H^1(X, \mathbb{Z}_2)$, which is a 2-torsion group (geometrically, because real line bundles are self-dual, their tensor squares are trivializable). Complex line bundles are classified by $H^2(X, \mathbb{Z})$. Complexification therefore describes a cohomology operation $H^1(X, \mathbb{Z}_2) \to H^2(X, \mathbb{Z})$, so the image of this cohomology operation (the first Chern classes of complexifications of real line bundles) is necessarily also 2-torsion. In fact this cohomology operation is a Bockstein.

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  • $\begingroup$ This is great, thanks. I guess that, more generally, the first Chern class of the complexification of a real bundle of any rank is the Bockstein of its first Stiefel-Whitney class? Probably this follows from the splitting principle? $\endgroup$ – Mark Grant Nov 3 '15 at 21:03
  • $\begingroup$ @Mark: you can reduce to the line bundle case for this without the splitting principle. It follows from the fact that complexification commutes with taking exterior powers (which in turn follows from the fact that it's symmetric monoidal). $\endgroup$ – Qiaochu Yuan Nov 3 '15 at 21:08

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