Canonical framings for 3-manifolds. (English) Zbl 0947.57020

Kirby, Rob; Melvin, Paul

Turk. J. Math. 23, No. 1, 89-115 (1999).

This is a relatively self-contained exposition of framings, stable framings and 2-framings for a smooth, closed oriented 3-manifold \(M\). The authors state that their principal objective is to define the notion of a canonical (stable) framing within each spin structure on \(M\). This is done by minimizing a “Hirzebruch defect” and/or “degree” associated with the appropriate class of framings. They calculate the difference between a naturally occuring framing and the canonical one for quotients of \(S^3\) by finite subgroups and for manifolds occuring as the boundary of a simply connected spin 4-manifold. See also [M. Atiyah, Topology 29, No. 1, 1-7 (1990; Zbl 0716.57011); D. S. Freed and R. E. Gompf, Commun. Math. Phys. 141, No. 1, 79-117 (1991; Zbl 0739.53065)].

Reviewer: J.Hebda (St.Louis)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57R15	Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M27	Invariants of knots and \(3\)-manifolds (MSC2010)
57R22	Topology of vector bundles and fiber bundles

Keywords:

stable framings; 2-framings

Citations:

Zbl 0739.53065; Zbl 0716.57011