manifolds and cobordisms
cobordism theory, Introduction
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 $\mathbb{R}^d$
of the stable tangent bundle.
Accordingly, a framed cobordism is a cobordism equipped with a framing on the underlying manifold (see also at MFr).
For $dim(X)$ the dimension of the manifold and $n \geq dim(X)$, then one also speaks of an $n$-framing to mean a trivialization of the “$n$-stabilized tangent bundle” $T X \oplus \mathbb{R}^{n-dim(X)}$ (where the right direct summand denotes the trivial real vector bundle of rank $n - dim(X)$). These $n$-framed manifolds appear in particular in the construction of the cobordism category of framed $n$-dimensional cobordisms. The cobordism hypothesis asserts essentially that the (∞,n)-category of cobordisms with $n$-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.
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:
Every 3-dimensional manifold with spin structure admits a framing.
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 $B Spin(3)$ 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.
Every orientable 3-dimensional manifold admits a framing.
By the argument in the proof of prop. , 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.
Prop. 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. 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 $n$-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.
Left invariant framings $\mathcal{L}$ on compact connected Lie groups $G$ with $dim(G)=k$ give rise to elements $[G,\mathcal{L}]$ in the stable homotopy group $\pi_k^s$ of spheres. One can restrict attention to semisimple Lie groups $G$ since this construction behaves well with respect to products, $G \to T\times G/T$ gives a framed diffeomorphism and $[S^1,\mathcal{L}] \in \pi_1^s$ is the generator. The following facts are assembled from (Ossa 1982) and (Minami 2016).
For any semisimple compact connected Lie group $G$, Ossa proved that $72[G,\mathcal{L}]=0$. In particular the only possible torsion is at the primes 2 and 3.
The left invariant framings on $SO(n),Spin(n)$ and $SU(n)$ give the zero element for $n\geq 7$, those on $Sp(n)$ for $n\geq 4$, as do those on $F_4,E_6,E_7,E_8, SO(4),SO(6),SU(5),SU(6)$. (this means that the undetermined cases of rank 4 in Ossa’s Table 1 are in fact all 0)
The left invariant framings on the remaining groups ($SU(2)=Spin(3)=Sp(1),SU(3),SU(4)=Spin(6),Sp(2)=Spin(5),Sp(3),SO(3),SO(5)$ and $G_2$) represent known nonzero classes in $\pi_\ast^s$.
On a framed manifold there is a canonical quadratic refinement of the intersection pairing. The associated invariant is the Kervaire invariant.
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.
Formalization in differential cohesion is discussed there.
Rieview in the context of the Kervaire invariant includes
The theorem about the parallizablitiy of spheres is due to
Relation to existence of flat connections on the tangent bundle is discussed in
The moduli space of framed surfaces is discussed in
The relation to “2-framings” is discussed in
Erich Ossa, Lie groups as framed manifolds, Topology, 21 (1982), 315–323, (doi)
Haruo Minami, On framed simple Lie groups, (pdf)
For an extension to p-compact groups see
Last revised on March 3, 2021 at 08:44:26. See the history of this page for a list of all contributions to it.