Surgery formulae for Casson’s invariant and extensions to homology lens spaces. (English) Zbl 0691.57004

A rational invariant is defined which generalizes Casson’s invariant for homology-3-spheres to homology lens spaces (i.e. 3-manifolds obtained by arbitrary Dehn-surgery on knots in homology-3-spheres, or equivalently, with finite cyclic first homology), involving a “Dedekind-sum” arithmetic function. This depends on a formula which calculates the Casson invariant of homology-3-spheres resulting from Dehn-surgery on a 2-component link and, up to framed link homotopy, also on an n-component link (Casson’s original formula determines his invariant for a homology- 3-sphere obtained by surgery on a knot), and also on a calculation of the Casson invariant of Seifert fibered homology-3-spheres (independently obtained by Fintushel-Stern).
Several interesting applications of this profound study of Casson-type invariants are given, for example the following. If K is a knot in \(S^ 3\) such that the evaluation at 1, \(\Delta_ K''(1)\), of the second derivative of the Alexander polynomial of K is not 0 then the manifolds K(p/q) obtained by p/q-surgery on K are all distinct, and, for each \(p\geq 1\), one obtains at most one lens space.
Reviewer: B.Zimmermann


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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