nLab
Atiyah 2-framing

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Manifolds and cobordisms

Contents

Definition

In the general terminology of nn-framing then a 2-framing of a manifold Σ\Sigma of dimension d2d \leq 2 is a trivialization of TΣ 2dT \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 XX a compact, connected, oriented 3-dimensional manifold, write

2TX:=TXTX 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)SO(3)×SO(3)SO(6) SO(3) \to SO(3) \times SO(3) \hookrightarrow SO(6)

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

Proposition

The underlying SO(6)SO(6)-principal bundle of 2TX2 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(2TX)=2w 0(TX)w 2(TX)+w 1(TX)w 1(TX). w_2(2 T X) = 2 w_0(T X) \cup w_2(T X) + w_1(T X) w_1(T X) \,.

Since TXT X is assumed oriented, w 1(TX)=0w_1(T X) = 0 (since this is the obstruction to having an orientation). So w 2(2TX)=0H 2(X, 2)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 XX 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 At2FrameAt2\mathbf{Frame} is the homotopy pullback in

At2Frame * BSO(3) BSpin(6) \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 XX with orientation o:XBSO(3)\mathbf{o} \colon X \to \mathbf{B}SO(3) is a lift o^\hat {\mathbf{o}} in

At2Frame o^ X o BSO(3). \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 XX is induced by a 4-manifold ZZ with boundary ZX\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 XX is equivalently a
p 1p_1-structure, where p 1p_1 is the first Pontryagin class, hence a homotopy class of a trivialization of

p 1(X):XBSO(3)p 1K(,4). 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, page 6, slide 5).

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

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

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

Revised on October 28, 2014 09:16:09 by Urs Schreiber (141.0.9.62)