Involutions and isotopies of lens spaces. (English) Zbl 0605.57022

Hodgson, Craig; Rubinstein, J. H.

Knot theory and manifolds, Proc. Conf., Vancouver/Can. 1983, Lect. Notes Math. 1144, 60-96 (1985).

[For the entire collection see Zbl 0564.00014.]
The authors obtain a classification of all involutions of a (3- dimensional) lens space which have 1-dimensional fixed point set. More precisely, they prove that each such involution is conjugate, by a diffeomorphism isotopic to the identity, to an isometry of the spherical metric of the lens space. As a corollary, they determine the group of isotopy classes of diffeomorphisms of each lens space. Another corollary is that a knot in \(S^ 3\) whose double branched covering is a lens space is necessarily a 2-bridge knot.

Reviewer: F.Bonahon

MSC:

57S25	Groups acting on specific manifolds
57M12	Low-dimensional topology of special (e.g., branched) coverings
57M25	Knots and links in the \(3\)-sphere (MSC2010)
57N10	Topology of general \(3\)-manifolds (MSC2010)

Keywords:

involutions of lens space; 1-dimensional fixed point set; isometry; 2- bridge knot

Citations:

Zbl 0564.00014