An unexpected surgery construction of a lens space

James Bailey and Dale Rolfsen

Vol. 71 (1977), No. 2, 295–298

DOI: 10.2140/pjm.1977.71.295

Abstract

A useful method of constructing-3-dimensional manifolds is to remove the interior of a tubular neighbourhood V ⊂ S³ of a knot K in the 3-sphere and sew it back differently, via a homeomorphism h : ∂V → ∂V . This surgery construction, due to M. Dehn, yields the manifold

where x ∈ ∂V ⊂ V is identified with h(x) ∈ ∂V ⊂ S³ − int V . For example surgery along a trivial knot yields, for various choices of h, exactly the class of lens spaces L(p,q), including the extreme cases L(1,0)≅S³ and L(0,1)≅S² × S¹.

Mathematical Subject Classification

Primary: 57A10, 57A10

Secondary: 55A25

Milestones

Received: 9 April 1975

Revised: 4 May 1976

Published: 1 August 1977