Synopses & Reviews
This book presents a new result in 3-dimensional topology. It is well known that any closed oriented 3-manifold can be obtained by surgery on a framed link in
S ^{3}. In Global Surgery Formula for the Casson-Walker Invariant, a function F of framed links in S
^{3} is described, and it is proven that F consistently defines an invariant, lamda (l), of closed oriented 3-manifolds. l is then expressed in terms of previously known invariants of 3-manifolds. For integral homology spheres, l is the invariant introduced by Casson in 1985, which allowed him to solve old and famous questions in 3-dimensional topology. l becomes simpler as the first Betti number increases.
As an explicit function of Alexander polynomials and surgery coefficients of framed links, the function F extends in a natural way to framed links in rational homology spheres. It is proven that F describes the variation of l under any surgery starting from a rational homology sphere. Thus F yields a global surgery formula for the Casson invariant.
Synopsis
This book presents a new result in 3-dimensional topology. It is well known that any closed oriented 3-manifold can be obtained by surgery on a framed link in
S^{3}. In
Global Surgery Formula for the Casson-Walker Invariant, a function F of framed links in
S^{3} is described, and it is proven that F consistently defines an invariant, lamda (
l), of closed oriented 3-manifolds.
l is then expressed in terms of previously known invariants of 3-manifolds. For integral homology spheres,
l is the invariant introduced by Casson in 1985, which allowed him to solve old and famous questions in 3-dimensional topology.
l becomes simpler as the first Betti number increases.
As an explicit function of Alexander polynomials and surgery coefficients of framed links, the function F extends in a natural way to framed links in rational homology spheres. It is proven that F describes the variation of l under any surgery starting from a rational homology sphere. Thus F yields a global surgery formula for the Casson invariant.
Description
Includes bibliographical references (p. [147]-148) and index.
Table of Contents
Ch. 1 | Introduction and statements of the results | 5 |
Ch. 2 | The Alexander series of a link in a rational homology sphere and some of its properties | 21 |
Ch. 3 | Invariance of the surgery formula under a twist homeomorphism | 35 |
Ch. 4 | The formula for surgeries starting from rational homology spheres | 60 |
Ch. 5 | The invariant [lambda] for 3-manifolds with nonzero rank | 81 |
Ch. 6 | Applications and variants of the surgery formula | 95 |
| Appendix: More about the Alexander series | 117 |
| Bibliography | 147 |
| Index | 149 |