nLab Atiyah 2-framing

Definition

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

$$2 T X := T X \oplus T X$$

for the fiberwise direct sum of the tangent bundle with itself. Via the diagonal embedding

$$SO(3) \to SO(3) \times SO(3) \hookrightarrow SO(6)$$

this naturally induces a SO(6)-principal bundle.

Proposition

The underlying $SO(6)$-principal bundle of $2 T X$ always admits a lift to a spin(6)-principal bundle.

Proof

By the sum-rule for Stiefel-Whitney classes (see at SW class – Axiomatic definition) we have that

$$w_2(2 T X) = 2 w_0(T X) \cup w_2(T X) + w_1(T X) w_1(T X) \,.$$

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

Definition

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:

Definition

The moduli stack $At2\mathbf{Frame}$ is the homotopy pullback in

$$\array{ At2\mathbf{Frame} &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}SO(3) &\stackrel{}{\to}& \mathbf{B} Spin(6) }$$

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

$$\array{ && At 2 \mathbf{Frame} \\ & {}^{\mathllap{\hat {\mathbf{o}}}}\nearrow & \downarrow \\ X &\stackrel{\mathbf{o}}{\to}& \mathbf{B}SO(3) } \,.$$

Properties

Relation to bounding 4-manifolds

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).

Relation to String-structures

By (Atiyah 2.1) an Atiyah 2-framing of a 3-manifold $X$ is equivalently a
$p_1$-structure, where $p_1$ is the first Pontryagin class, hence is a homotopy class of a trivialization of

$$p_1(X) \colon X \to B SO(3) \stackrel{p_1}{\to} K(\mathbb{Z},4) \,.$$

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).

Related concepts

• framed manifold

• p1-structure, string structure.

References

The notion of “2-framing” in the sense of framing of the double of the tangent bundle 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 perturbative path integral quantization of 3d Chern-Simons theory in

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

reviewed for instance in

• M. B. Young, section 2 of Chern-Simons theory, knots and moduli spaces of connections (pdf)

(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

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

The relation to $p_1$-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 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).

and more discussion for 3-manifolds with boundary includes

• Thomas Kerler, Volodymyr Lyubashenko, section 1.6.1 of Non-semisimple topological quantum field theories for 3-manifolds with corners, Lecture notes in mathematics 2001

See also

• Greg Kuperberg, MO comment