nLab
framed manifold

Idea

In one sense of the term, a framing of a manifold is a choice of trivialization of its tangent bundle, hence a choice of section of the corresponding frame bundle.

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 normal bundle if the manifold is understood embedded in some Cartesian space ${ℝ}^d$

• of the stable tangent bundle.

Accordingly, a framed cobordism is a cobordism equipped with a framing on the underlying manifold.

Examples

Proposition

Every Lie group is a parallelizable manifold.

Proof

Every non-zero invariant vector field on the Lie group provides an everywhere non-vanishing section of the tangent bundle.

Proposition

Every orientable 3-dimensional manifold admits a framing.

Proof

That a 3-manifold $X$ 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 $X\to B\mathrm{SO}(3)$. But since $\mathrm{SO}(3)$ has vanishing homotopy groups in degree $0\le k\le 2$. Therefore its delooping classifying space $B\mathrm{SO}(3)$ has vanishing homotopy groups below degree 4 and hence every morphism out of a 3-dimensional manifold into it is homotopically constant.

Theorem

The $n$-spheres that admit a framing are precisely only

• the 1-sphere $S^1$, the circle underlying the circle group (the unit complex numbers);

• the 3-sphere $S^3$, underlying the special unitary group $\mathrm{SU}(2)$, is isomorphic to the unit quaternions;

• the 7-sphere $S^7$, which underlies a Moufang loop internal to Diff, namely the unit octonions.

Related concepts

• 2-framing