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Fibered knots and involutions on homotopy spheres. (English) Zbl 0567.57015

Four-manifold theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Durham/N.H. 1982, Contemp. Math. 35, 1-74 (1984).
[For the entire collection see Zbl 0549.00018.]
S. E. Cappell and J. L. Shaneson showed that by performing (generalized) surgery on the mapping tori of diffeomorphisms of the 3- torus one could obtain on the one hand fibred 2-knots in smooth homotopy 4-spheres which were not determined by their exterior [Ann. Math., II. Ser. 103, 349-353 (1976; Zbl 0338.57008)] and on the other hand exotic 4- dimensional real projective spaces [ibid. 104, 61-72 (1976; Zbl 0345.57003)]. The latter are double covered by homotopy 4-spheres bearing interesting involutions. Although since M. H. Freedman’s work [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)] all homotopy 4-spheres are now known to be homeomorphic to \(S^ 4\), the question of their differentiable structure remains of interest. This article makes a detailed study of the handle body structure of these mapping tori and of related spaces. The first half of the paper (34 pages) is geometric, a principal tool being the Reidemeister-Singer stabilization theorem for Heegaard decompositions of 3-manifolds. There is a long appendix (13 pages) on conjugacy in SL(3;\({\mathbb{Z}})\), the orientation preserving diffeotopy group of the 3-torus. (The question mark in Table 1 of the appendix can be replaced by ”YES”; if \(a=13\) or -8 the ring R is again integrally closed.) Finally there are 26 pages of figures illustrating the geometry.
Reviewer: J.Hillman

MSC:

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57R50 Differential topological aspects of diffeomorphisms
57R65 Surgery and handlebodies
57N10 Topology of general \(3\)-manifolds (MSC2010)