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.
"[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
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