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

Paperback | 1992 | $67.50 | £56.95 | ISBN: 9780691025322
150 pp. | 6 1/8 x 9 1/4
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eBook | ISBN: 9781400882465 |
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Reviews | Table of Contents

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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."--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

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