Canonical framings for 3-manifolds

A framing of an oriented trivial bundle is a homotopy class of sections of the associated oriented frame bundle. This paper is a study of the framings of the tangent bundle τM of a smooth closed oriented 3-manifold M , often referred to simply as framings of M . We shall also discuss stable framings and 2-framings of M , that is framings of ε ⊕ τM (where ε is an oriented line bundle) and 2τM = τM ⊕ τM . The notion of a canonical 2-framing of M was introduced by Atiyah [1]. Motivated by Witten’s paper [27] generalizing the Jones polynomial to links in M , Atiyah observed that Witten’s invariant contained a phase factor specified by the choice of a 2-framing on M , and thus was an invariant of links in a 2-framed 3-manifold. Independent calculations by Reshetikhin and Turaev [23] for a related invariant, defined from a framed link description of M , did not appear to depend on a 2-framing. As Atiyah noted, however, the framed link description naturally gave a 2-framing of M , explained further by Freed and Gompf in [6, §2], and so Reshetikhin and Turaev were in fact calculating Witten’s invariant forM with this framing times a phase factor depending on the difference between this framing and the canonical 2-framing, i.e. Witten’s invariant for M with its canonical 2-framing. In this paper we give a leisurely exposition of framings, stable framings and 2-framings of M , including some of the material in [1] and [6]. Our principal objective is to define the notion of a canonical (stable) framing within each spin structure on M . This is the content of §2. The set of possible framings φ for a given spin structure form an affine space Z, corresponding to π3(SO3), and we choose a canonical framing in this space by minimizing the absolute value of the “Hirzebruch defect” h(φ) (defined in §1). More generally, there are Z ⊕ Z = π3(SO4) possible stable framings φ, and here we must also minimize a certain “degree” d(φ) associated with φ. The typical application may be to calculate the difference between a naturally occurring framing and the canonical one, and this is carried out in a number of instances. In §3 we consider framings on quotients of S by finite subgroups, where the calculations use signature defects and the G-signature theorem, and on certain circle bundles over surfaces. Natural framings also arise by restriction from framings on 4-manifolds bounded by M , and this situation is taken up in §4. In particular we discuss surgery on an even framed link L in S , which connects with the work of Freed and Gompf.