# nLab 2-framing

cohomology

### Theorems

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Definition

For $X$ a compact, connected, oriented 3-dimensional manifold, write

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

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

$\mathrm{SO}\left(3\right)\to \mathrm{SO}\left(3\right)×\mathrm{SO}\left(3\right)↪\mathrm{SO}\left(6\right)$SO(3) \to SO(3) \times SO(3) \hookrightarrow SO(6)

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

###### Proposition

The underlying $\mathrm{SO}\left(6\right)$-principal bundle of $2TX$ 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

${w}_{2}\left(2TX\right)=2{w}_{0}\left(TX\right)\cup {w}_{2}\left(TX\right)+{w}_{1}\left(TX\right){w}_{1}\left(TX\right)\phantom{\rule{thinmathspace}{0ex}}.$w_2(2 T X) = 2 w_0(T X) \cup w_2(T X) + w_1(T X) w_1(T X) \,.

Since $TX$ is assumed oriented, ${w}_{1}\left(TX\right)=0$ (since this is the obstruction to having an orientation). So ${w}_{2}\left(2TX\right)=0\in {H}^{2}\left(X,{ℤ}_{2}\right)$ 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 on acompact, 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 $2\mathrm{Frame}$ is the homotopy pullback in

$\begin{array}{ccc}2\mathrm{Frame}& \to & *\\ ↓& & ↓\\ B\mathrm{SO}\left(3\right)& \stackrel{}{\to }& B\mathrm{Spin}\left(6\right)\end{array}$\array{ 2\mathbf{Frame} &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}SO(3) &\stackrel{}{\to}& \mathbf{B} Spin(6) }

in Smooth∞Grpd.

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

$\begin{array}{ccc}& & 2\mathrm{Frame}\\ & {}^{\stackrel{^}{o}}↗& ↓\\ X& \stackrel{o}{\to }& B\mathrm{SO}\left(3\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && 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 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

${p}_{1}\left(X\right):X\to B\mathrm{SO}\left(3\right)\stackrel{{p}_{1}}{\to }K\left(ℤ,4\right)\phantom{\rule{thinmathspace}{0ex}}.$p_1(X) \colon X \to B SO(3) \stackrel{p_1}{\to} K(\mathbb{Z},4) \,.

This perspective on framings is made explicit in (Bunke-Naumann, section 2.3). It is mentioned for instance also in (Freed, slide 5).

## 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

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)

page 9 of

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

Revised on November 12, 2012 22:58:21 by Urs Schreiber (82.113.98.69)