An Extension of Casson's Invariant. (AM-126), Volume 126 / / Kevin Walker.

• Superior document: Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
• VerfasserIn: Walker, Kevin,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2016]; ©1992
• Year of Publication: 2016
• Language: English
• Series: Annals of Mathematics Studies ; 126
• Physical Description: 1 online resource (150 p.)

This book describes an invariant, l, of oriented rational homology 3-spheres which is a generalization of work of Andrew Casson in the integer homology sphere case. Let R(X) denote the space of conjugacy classes of representations of p(X) into SU(2). Let (W,W,F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is declared to be an appropriately defined intersection number of R(W) and R(W) inside R(F). The definition of this intersection number is a delicate task, as the spaces involved have singularities. A formula describing how l transforms under Dehn surgery is proved. The formula involves Alexander polynomials and Dedekind sums, and can be used to give a rather elementary proof of the existence of l. It is also shown that when M is a Z-homology sphere, l(M) determines the Rochlin invariant of M.

Table of Contents

Frontmatter -- Contents -- 0. Introduction -- 1. Topology of Representation Spaces -- 2. Definition of λ -- 3. Various Properties of λ -- 4. The Dehn Surgery Formula -- 5. Combinatorial Definition of λ -- 6. Consequences of the Dehn Surgery Formula -- A. Dedekind Sums -- B. Alexander Polynomials -- Bibliography