# nLab framed manifold

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## 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

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.

The following is obvious:

###### Proposition

Every 3-dimensional manifold with spin structure admits a framing.

###### Proof

That a 3-manifold $X$ 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 $X \to B Spin(3)$. But $Spin(3)$ has vanishing homotopy groups in degree $0 \leq k \leq 2$. Therefore its delooping classifying space $B SO(3)$ 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.

###### Proposition

Every orientable 3-dimensional manifold admits a framing.

###### Proof

By the argument in the proof of prop. 2, the only possible obstruction is the second Stiefel-Whitney class $w_2$. 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.

###### Theorem

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

• the 0-sphere $S^0 = \ast \coprod \ast$, the unit real numbers

• 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 $SU(2)$, is isomorphic to the unit quaternions;

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

where the algebras appearing are precisely the four normed division algebras.

This is due to (Adams 58), proven with the Adams spectral sequence.

## References

The theorem about the parallizablitiy of spheres is due to

• John Adams, On the Non-Existence of Elements of Hopf Invariant One Bull. Amer. Math. Soc. 64, 279-282, 1958, Ann. Math. 72, 20-104, 1960.