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.
For the dimension of the manifold and , then one also speaks of an -framing to mean a trivialization of the “-stabilized tangent bundle” (where the right direct summand denotes the trivial real vector bundle of rank ). These -framed manifolds appear in particular in the construction of the cobordism category of framed -dimensional cobordisms. The cobordism hypothesis asserts essentially that the (∞,n)-category of cobordisms with -framing is the free symmetric monoidal (∞,n)-category with duals.
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.
The following is obvious:
That a 3-manifold has spin structure means that we have a reduction of the structure group of the tangent bundle to the spin group, and hence the tangent bundle is classified by a map . But 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.
But in fact the following stronger statement is also true.
By the argument in the proof of prop. 2, the only possible obstruction is the second Stiefel-Whitney class . By the discussion at Wu class, this vanishes on an oriented manifold precisely if the second Wu class vanishes. This in turn is by definition defined to represent the Steenrod square under cup product, and this vanishes on a 3-manifold by degree reasons.
Prop. 3 has some impact in the context of the first order formulation of gravity, where one is interested in vielbein fields on 4-dimensional spacetime manifolds, and in particular in globally hyperbolic spacetimes, which, as topological spaces, are the Cartesian product of the real line (time) with a 3-manifold (space). Prop. 3 implies that already when that spatial 3-manifold is orientable, then the whole globally hyperbolic spacetime admits a framing. (See also at teleparallel gravity.)
The -spheres that admit a framing are precisely only
the 0-sphere , the unit real numbers
where the algebras appearing are precisely the four normed division algebras.
Rieview in the context of the Kervaire invariant includes
The theorem about the parallizablitiy of spheres is due to
The moduli space of framed surfaces is discussed in
The relation to “2-framings” is duscussed in