I want to show that the $\mathbb{S}^{5}$ does not support two-dimensional foliation. According to the authors in Geometrical Theory of Foliations, it suffices to show that the sphere $\mathbb{S}^{5}$ does not admit a continuous field of 2-tangent plane. What is the obstruction to this?
Thanks in advance.
This is too big a gun, but I believe this works.
The rank $2$ bundles on $S^5$ are classified by $[S^4,SO(2)]$ due to the clutching construction. But $SO(2)\cong S^1$, hence any two plane bundle on $S^5$ must be trivial.
Now Adams showed (much more, but in particular) that $4n+1$ dimensional spheres only admit one linear independent vector field (https://mathoverflow.net/questions/129174/vector-fields-on-4n1-spheres)
Hence the tangent bundle of $S^5$ cannot split of a rank $2$ bundle, as it would then be trivial and we would have two linearly independent vector fields on $S^5$.
You probably can also try to do obstruction theory.