Let $M$ be a smooth closed simply-connected $4$-manifold with $w\_1 = w\_2 = 0$. Can $TM$ be trivialized in the complement of a point?



Tangent bundle of smooth closed simply-connected $4$-manifold $w_1 = w_2 = 0$ can be trivialized in complement of point?
$M$ is smooth, so there is a cellular decomposition with a single 4-cell. Thus we want $TM$ trivializable over the 3-skeleton, so we look to analyze sections of the frame bundle which has Lie group fiber $SO(4)$ (here $M$ is orientable because $w\_1=0$). Obstruction theory says we can extend over the 2-skeleton, because the class is $w\_2\in H^2(M; \pi\_1SO(4)=\mathbb{Z}\_2)$ and you assumed that to be zero. The next obstruction (extending over the desired 3-skeleton) is automatically vacuous, because $\pi\_2SO(4)=0$.


