Global Surgery Formula for the Casson-Walker Invariant. (AM-140), Volume 140

Christine Lescop

Published by Princeton University Press 1996

Global Surgery Formula for the Casson-Walker Invariant. (AM-140), Volume 140

Christine Lescop

Volume 140 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400865154

Overview
Contents

About this book

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.

Author / Editor information

Christine Lescop is Researcher in Mathematics at the Centre National de la Recherche Scientifique at the Institut Fourier in Grenoble, France.

Topics

Frontmatter Publicly Available Download PDF	i
Table of contents Publicly Available Download PDF	3
Chapter 1. Introduction and statements of the results Requires Authentication Unlicensed Licensed Download PDF	5
Chapter 2. The Alexander series of a link in a rational homology sphere and some of its properties Requires Authentication Unlicensed Licensed Download PDF	21
Chapter 3. Invariance of the surgery formula under a twist homeomorphism Requires Authentication Unlicensed Licensed Download PDF	35
Chapter 4. The formula for surgeries starting from rational homology spheres Requires Authentication Unlicensed Licensed Download PDF	60
Chapter 5. The invariant A. for 3-manifolds with nonzero rank Requires Authentication Unlicensed Licensed Download PDF	81
Chapter 6. Applications and variants of the surgery formula Requires Authentication Unlicensed Licensed Download PDF	95
Appendix. More about the Alexander series Requires Authentication Unlicensed Licensed Download PDF	117
Bibliography Requires Authentication Unlicensed Licensed Download PDF	147
Index Requires Authentication Unlicensed Licensed Download PDF	149