TITLE: Parallelizability of the Milnor's exotic spheres in dimension 7 
QUESTION [79 upvotes]: Are the Milnor's seven dimensional exotic spheres parallelizable?

REPLY [40 votes]: Here's another way to answer the original question. There is a theorem of Bredon and Kosinski (Annals, 1966) which says that if a manifold $M^n$ is stably parallelizable, then either $M^n$ is parallelizable or the maximum number of linearly independent vector fields on $M^n$ is the same as on $S^n$. Since $S^7$ is parallelizable, this implies that exotic 7-spheres are parallelizable (since they are stably parallelizable).