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.
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