group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
In the general terminology of $n$-framing then a 2-framing of a manifold $\Sigma$ of dimension $d \leq 2$ is a trivialization of $T \Sigma \oplus \mathbb{R}^{2-d}$.
In (Atiyah 90) the term “2-framing” is instead used for a trivialization of the double of the tangent bundle of a 3-manifold. So this is a different concept, but it turns out to be closely related to the 3-framing (in the previous sense) of surfaces.
For $X$ a compact, connected, oriented 3-dimensional manifold, write
for the fiberwise direct sum of the tangent bundle with itself. Via the diagonal embedding
this naturally induces a SO(6)-principal bundle.
The underlying $SO(6)$-principal bundle of $2 T X$ always admits a lift to a spin(6)-principal bundle.
By the sum-rule for Stiefel-Whitney classes (see at SW class – Axiomatic definition) we have that
Since $T X$ is assumed oriented, $w_1(T X) = 0$ (since this is the obstruction to having an orientation). So $w_2(2 T X) = 0 \in H^2(X,\mathbb{Z}_2)$ and since this in turn is the further obstruction to having a spin structure, this does exist.
Therefore the following definition makes sense
A 2-framing in the sense of (Atiyah 90) on a compact, connected, oriented 3-dimensional manifold $X$ is the homotopy class of a trivializations of the spin-group-principal bundle underlying twice its tangent bundle.
More in detail, we may also remember the groupoid of 2-framings and the smooth structure on collections of them:
The moduli stack $At2\mathbf{Frame}$ is the homotopy pullback in
in Smooth∞Grpd.
In terms of this a 2-framing on $X$ with orientation $\mathbf{o} \colon X \to \mathbf{B}SO(3)$ is a lift $\hat {\mathbf{o}}$ in
In (Atiyah) it is shown how a framing on a compact connected oriented 3-manifold $X$ is induced by a 4-manifold $Z$ with boundary $\partial Z \simeq X$. In fact, a framing is equivalently a choice of cobordism class of bounding 4-manifolds (Kerler).
Discussion of 2-framing entirely in terms of bounding 4-manifolds is for instance in (Sawin).
By (Atiyah 2.1) a 2-framing of a 3-manifold $X$ is equivalently a
$p_1$-structure, where $p_1$ is the first Pontryagin class, hence a homotopy class of a trivialization of
This perspective on framings is made explicit in (Bunke-Naumann, section 2.3). It is mentioned for instance also in (Freed, page 6, slide 5).
The notion of “2-framing” in the sense of framing of the double of the tangent bundle is due to
making explicit a structure which slightly implicit in the discussion of the perturbative path integral quantization of 3d Chern-Simons theory in
reviewed for instance in
(see Atiyah, page 6). For more on the role of 2-framings in Chern-Simons theory see also
Daniel Freed, Robert Gompf, Computer calculation of Witten’s 3-Manifold invariant, Commun. Math. Phys. 141,79-117 (1991) (pdf)
Gregor Masbaum, section 2 of Spin TQFT and the Birman-Craggs Homomorphism, Tr. J. of Mathematics 19 (1995) pdf
Daniel Freed, Remarks on Chern-Simons theory (arXiv:0808.2507, pdf slides)
and for discussion in the context of the M2-brane from p. 7 on in
The relation to string structures is made explicit in section 2.3 of
More discussion in terms of bounding 4-manifolds is in
Thomas Kerler, Bridged links and tangle presentations of cobordism categories. Adv. Math., 141(2):207–281, (1999) (arXiv:math/9806114)
Stephen F. Sawin, Three-dimensional 2-framed TQFTS and surgery (2004) (pdf)
and page 9 of
and more discussion for 3-manifolds with boundary includes
See also