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Global Surgery Formula for the Casson-Walker Invariant. (AM-140)
Christine Lescop

Paperback | 1996 | $65.00 / £44.95 | ISBN: 9780691021324
150 pp. | 6 x 9
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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. 1Introduction and statements of the results5
Ch. 2The Alexander series of a link in a rational homology sphere and some of its properties21
Ch. 3Invariance of the surgery formula under a twist homeomorphism35
Ch. 4The formula for surgeries starting from rational homology spheres60
Ch. 5The invariant [lambda] for 3-manifolds with nonzero rank81
Ch. 6Applications and variants of the surgery formula95
Appendix: More about the Alexander series117
Bibliography147
Index149

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