Special and general types
Manifolds and cobordisms
For 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:
The moduli stack is the homotopy pullback in
In terms of this a 2-framing on with orientation is a lift in
Relation to bounding 4-manifolds
In (Atiyah) it is shown how a framing on a compact connected oriented 3-manifold is induced a 4-manifold with boundary . 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 is equivalently a
-structure, where 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).
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
The relation to string structures is made explicit in section 2.3 of
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
- Stephen Sawin, Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras (arXiv:math/9910106).