# Whitney sum formula for Pontryagin classes I

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
• 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). – Yury Ustinovskiy Nov 4 '15 at 0:25
• @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. – Mark Grant Nov 4 '15 at 8:25

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.