##
**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.

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) |