group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
In the general terminology of -framing, a 2-framing of a manifold of dimension is a trivialization of .
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 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 -principal bundle of 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 is assumed oriented, (since this is the obstruction to having an orientation). So 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 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:
In terms of this a 2-framing on with orientation is a lift in
In (Atiyah) it is shown how a framing on a compact connected oriented 3-manifold is induced by 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 2004.
By (Atiyah 2.1) an Atiyah 2-framing of a 3-manifold is equivalently a
-structure, where is the first Pontryagin class, hence is a homotopy class of a trivialization of
This perspective on Atiyah 2-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 -structure is made explicit in
Ulrich Bunke, Niko Naumann, section 2.3 of Secondary Invariants for String Bordism and tmf, Bull. Sci. Math. 138 (2014), no. 8, 912–970 (arXiv:0912.4875)
C. Blanchet, N. Habegger, Gregor Masbaum, Pierre Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology Vol 34, No. 4, pp. 883-927 (1995) (pdf)
More discussion in terms of bounding 4-manifolds:
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, Journal of Knot Theory and its Ramifications 13 7 (2004) 947-996 [pdf, doi:10.1142/S0218216504003536]
Stephen F. Sawin, page 9 of: Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras, Advances in Mathematics 165 1 (2002) 35-70 [arXiv:math/9910106, doi:10.1006/aima.2000.1935]
(cf. also at spin Chern-Simons theory)
and more discussion for 3-manifolds with boundary:
See also
Last revised on March 26, 2025 at 08:48:41. See the history of this page for a list of all contributions to it.