Contents

Definition

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.

Therefore the following definition makes sense

More in detail, we may also remember the groupoid of 2-framings and the smooth structure on collections of them:

In terms of this a 2-frmaing on $X$ with orientation $o:X\to B\mathrm{SO}(3)$ is a lift $\hat{o}$ in

Properties

Relation to bounding 4-manifolds

In (Atiyah) it is shown how a framing on a compact connected oriented 3-manifold $X$ is induced 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).

Relation to String-structures

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, slide 5).

Related concepts

• framed manifold

• string structure.

References

The notion of “2-framing” is due to

• Michael Atiyah, On framings of 3-manifolds , Topology, Vol. 29, No 1, pp. 1-7 (1990) (pdf)

making explicit a structure which slightly implicit in the discussion of the path integral of Chern-Simons theory in

• Edward Witten, Quantum field theory and the Jones Polynomial , Comm. Math. Phys. 121 (1989)

(see Atiyah, page 6). For more on the role of 2-framings in Chern-Simons theory see also

• Dan Freed, Robert Gompf, Computer calculation of Witten’s 3-Manifold invariant, Commun. Math. Phys. 141,79-117 (1991) (pdf)

• Dan Freed, Remarks on Chern-Simons theory (pdf slides)

and for discussion in the context of the M2-brane from p. 7 on in

• Hisham Sati, Geometric and topological structures related to M-branes II: Twisted $\mathrm{String}$ and ${\mathrm{String}}^c$-structures (arXiv:1007.5419).

The relation to string structures is made explicit in section 2.3 of

• Ulrich Bunke, Niko Naumann, Secondary Invariants for String Bordism and tmf (arXiv:0912.4875)

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)

page 9 of

• Stephen Sawin, Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras (arXiv:math/9910106)