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](https://mathoverflow.net/questions/222616/whitney-sum-formula-for-pontryagin-classes-ii)



Whitney sum formula for Pontryagin classes I

> 
> 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. 


