On the first page of Milnor-Kervaire's paper "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", they assert without proof or reference that if $M$ is a compact connected oriented differentiable $4$-manifold such that $w\_2(M)=0$, then $M$ is **almost parallelizable**, that is, for all $x\_0 \in M$ the tangent bundle of $M \setminus x\_0$ is trivial. Try as I might, I cannot figure out how to prove this. Can someone help me?



Almost parallelizable 4-manifolds
You want to trivialise the restriction of the tangent bundle to the 3-skeleton of $M$. Since $\pi\_0 O(4) = \pi\_1 O(4) = Z/2$, there are obstructions $w\_1(E) \in H^1(X; Z/2)$ and $w\_2(E) \in H^2(X;Z/2)$ to trivialising a rank 4 bundle over the 1- and 2-skeleta of a cell complex $X$. Because $\pi\_2 O(4)$ is trivial, there is no further obstruction to extending a trivialisation from the 2-skeleton to the 3-skeleton. This is outlined in a nice way at the beginning of chaper 3 in [Hatcher's book on vector bundles](http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html).


