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

by Kevin Walker

Overview

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.

Product Details

ISBN-13:	9780691025322
Publisher:	Princeton University Press
Publication date:	03/23/1992
Series:	Annals of Mathematics Studies , #126
Pages:	150
Product dimensions:	6.00(w) x 9.25(h) x (d)

Table of Contents

• Frontmatter, pg. i
• Contents, pg. v
• 0. Introduction, pg. 1
• 1. Topology of Representation Spaces, pg. 6
• 2. Definition of λ, pg. 27
• 3. Various Properties of λ, pg. 41
• 4. The Dehn Surgery Formula, pg. 81
• 5. Combinatorial Definition of λ, pg. 95
• 6. Consequences of the Dehn Surgery Formula, pg. 108
• A. Dedekind Sums, pg. 113
• B. Alexander Polynomials, pg. 122
• Bibliography, pg. 129

Editorial 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."

Bulletin of the American Mathematical Society

"[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."—Bulletin of the American Mathematical Society

From the Publisher