nLab
framed manifold

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.

For dim(X)dim(X) the dimension of the manifold and ndim(X)n \geq dim(X), then one also speaks of an nn-framing to mean a trivialization of the “nn-stabilized tangent bundle” TX ndim(X)T X \oplus \mathbb{R}^{n-dim(X)} (where the right direct summand denotes the trivial real vector bundle of rank ndim(X)n - dim(X)). These nn-framed manifolds appear in particular in the construction of the cobordism category of framed nn-dimensional cobordisms. The cobordism hypothesis asserts essentially that the (∞,n)-category of cobordisms with nn-framing is the free symmetric monoidal (∞,n)-category with duals.

Beware that there is also the term “2-framing” due to (Atiyah), which is related but different.

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 XX 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 XBSpin(3)X \to B Spin(3). But Spin(3)Spin(3) has vanishing homotopy groups in degree 0k20 \leq k \leq 2. Therefore its delooping classifying space BSpin(3)B Spin(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 2w_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 nn-spheres that admit a framing are precisely only

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

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

Properties

Relation to intersection pairing and Kervaire invariant

On a framed manifold there is a canonical quadratic refinement of the intersection pairing. The associated invariant is the Kervaire invariant.

Properties

Moduli of framings

The homotopy type of the moduli space of framings on a fixed manifold is a disjoint union of subgroups of the oriented mapping class group which fix a given isotopy type of framings.

References

General

Rieview in the context of the Kervaire invariant includes

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.

Relation to existence of flat connections on the tangent bundle is discussed in

For 2-manifolds

The moduli space of framed surfaces is discussed in

For 3-manifolds

  • Rob Kirby, Paul Melvin, Canonical framings for 3-manifolds, Turkish Journal of Mathematics, volume 23, number 1,1999 (pdf)

The relation to “2-framings” is duscussed in

Revised on September 15, 2014 07:45:13 by Urs Schreiber (185.26.182.31)