Published by Princeton University Press 1992

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

Kevin Walker

Volume 126 in the series Annals of Mathematics Studies

About this book

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.

Reviews

"[This is] a monograph describing Walker's extension of Casson's invariant to Q HS . . . This is a fascinating subject and Walker's book is informative and well written . . . it makes a rather pleasant introduction to a very active area in geometric topology."

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	v
0. Introduction Requires Authentication Unlicensed Licensed Download PDF	1
1. Topology of Representation Spaces Requires Authentication Unlicensed Licensed Download PDF	6
2. Definition of λ Requires Authentication Unlicensed Licensed Download PDF	27
3. Various Properties of λ Requires Authentication Unlicensed Licensed Download PDF	41
4. The Dehn Surgery Formula Requires Authentication Unlicensed Licensed Download PDF	81
5. Combinatorial Definition of λ Requires Authentication Unlicensed Licensed Download PDF	95
6. Consequences of the Dehn Surgery Formula Requires Authentication Unlicensed Licensed Download PDF	108
A. Dedekind Sums Requires Authentication Unlicensed Licensed Download PDF	113
B. Alexander Polynomials Requires Authentication Unlicensed Licensed Download PDF	122
Bibliography Requires Authentication Unlicensed Licensed Download PDF	129