When is the frame bundle of a manifold trivial?

Question

My question is what are the topological restrictions on a manifold $M$ such that its frame bundle is trivial?

http://math.stackexchange.com/questions/46297/which-manifolds-are-parallelizable, http://math.stackexchange.com/questions/50127/what-kind-of-manifold-is-one-with-a-trivial-tangent-bundle?rq=1 — Moishe Kohan, Apr 20 '16 at 05:35

score 1 · Accepted Answer · answered Apr 20 '16 at 05:37

A manifold with this property is called parallelizable or frameable. A necessary condition is that all of its characteristic classes (e.g. Stiefel-Whitney, Pontryagin) vanish; in particular, $M$ must be orientable and, if closed, must have Euler characteristic $0$ by the Poincare-Hopf theorem. In dimensions divisible by $4$ and for closed $M$ we also get that the signature of $M$ vanishes by the signature theorem.

For example, among the closed orientable surfaces $\Sigma_g$, only $\Sigma_1 = T^2$, the torus, is frameable, because it's the only one with Euler characteristic $0$.

When is the frame bundle of a manifold trivial?

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