A manifold that admits a framing is also called a parallelizable manifold. A manifold equipped with a framing is also called a parallelized manifold.
More generally, one means by a framing not a trivialization of the tangent bundle itself, but
of the stable tangent bundle.
Accordingly, a framed cobordism is a cobordism equipped with a framing on the underlying manifold.
Every Lie group is a parallelizable manifold.
Every non-zero invariant vector field on the Lie group provides an everywhere non-vanishing section of the tangent bundle.
That a 3-manifold is orientable means that we have a reduction of the structure group of the tangent bundle to the special orthogonal group, and hence the tangent bundle is classified by a map . But since has vanishing homotopy groups in degree . Therefore its delooping classifying space has vanishing homotopy groups below degree 4 and hence every morphism out of a 3-dimensional manifold into it is homotopically constant.
The -spheres that admit a framing are precisely only