This book presents a new result in 3dimensional topology. It is well known that any closed oriented 3manifold can be obtained by surgery on a framed link inS^{3}. InGlobal Surgery Formula for the CassonWalker Invariant,a function F of framed links inS^{3}is described, and it is proven that F consistently defines an invariant, lamda (l), of closed oriented 3manifolds.lis then expressed in terms of previously known invariants of 3manifolds. For integral homology spheres,lis the invariant introduced by Casson in 1985, which allowed him to solve old and famous questions in 3dimensional topology.lbecomes 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 oflunder any surgery starting from a rational homology sphere. Thus F yields a global surgery formula for the Casson invariant.

Front Matter Front Matter (pp. i2) 
Table of Contents Table of Contents (pp. 34) 
Chapter 1 Introduction and statements of the results Chapter 1 Introduction and statements of the results (pp. 520)In 1985, A. Casson defined an integer invariant for oriented integral homology 3spheres by introducing an appropriate way of counting the conjugacy classes of the SU(2)representations of their group. He proved that his invariant λ_{c}satisfies the following interesting properties :
λ_{c}vanishes on homotopy spheres,
λ_{c}is additive under connected sum,
λ_{c}changes sign under orientation reversal,
a simple surgery formula describes the variation of λ_{c}under a surgery on a knot transforming an integral homology sphere into another one, and,
λ_{c}lifts the Rohlin μinvariant from
$\mathbb{Z}/2\mathbb{Z}$ to$\mathbb{Z}$ (recall that if M is a$\mathbb{Z}$ homology 3sphere,... 
Chapter 2 The Alexander series of a link in a rational homology sphere and some of its properties Chapter 2 The Alexander series of a link in a rational homology sphere and some of its properties (pp. 2134)Alexander polynomials are classical invariants in knot theory and have been extensively studied.
The Alexander polynomial of a link in a rational homology sphere can be defined in the powerful and very appropriate context of Reidemeister torsion theory as a Reidemeister torsion of the exterior X of the link (up to a welldetermined factor for a knot) and, following [Tu], it can be given a suitable sign, and hence a suitable normalization, if X is equipped with an orientation of
${{\text{H}}_{\text{1}}}\text{(X;}\mathbb{R}\text{)}\oplus {{\text{H}}_{\text{2}}}\text{(X;}\mathbb{R}\text{)}$ .The normalization of this Alexander polynomial for oriented links in S^{3}was frrst pointed out by Conway and...

Chapter 3 Invariance of the surgery formula under a twist homeomorphism Chapter 3 Invariance of the surgery formula under a twist homeomorphism (pp. 3559)This chapter is entirely devoted to the proof of Proposition 3.1.1, that is the invariance of 𝔽_{M}under the ωtwist described in Definition 1.6.4.
Proposition 3.1.1:(Invariance of𝔽under an ωtwist)
With the notation of Definition1.6.4,
\[{{\mathbb{F}}_{\text{M}}}\text{(}\mathbb{L}\text{)=}{{\mathbb{F}}_{\text{M}}}\text{(}\tilde{\mathbb{L}}\text{)}\] As we observed in the remarks before Equality 1.6.5, the invariance of 𝔽_{S3}under these twist homeomorphisms is sufficient to ensure that 𝔽_{S3}defines an invariant of closed oriented 3mtnifolds. Proving the invariance of 𝔽_{M}for any rational homology sphere M is not more difficult, and it will allow us to consider only generic presentations when when proving the general surgery...

Chapter 4 The formula for surgeries starting from rational homology spheres Chapter 4 The formula for surgeries starting from rational homology spheres (pp. 6080)§4.2 to §4.5 prove the surgery formula T2 satisfied by λ, that is:
Proposition T2:
For any rational homology sphereM,and for any surgery presentationℍinM,the surgery formulaF(M,ℍ)is satisfied:
\[\text{(F(M,}\mathbb{H}\text{)) }\!\!\lambda\!\!\text{ (}{{\text{ }\!\!\chi\!\!\text{ }}_{\text{M}}}\text{(}\mathbb{H}\text{))=}\frac{\left {{\text{H}}_{\text{1}}}\text{(}{{\text{ }\!\!\chi\!\!\text{ }}_{\text{M}}}\text{(}\mathbb{H}\text{))} \right}{\left {{\text{H}}_{\text{1}}}\text{(M)} \right}\text{ }\!\!\lambda\!\!\text{ (M)+}{{\text{F}}_{\text{M}}}\text{(}\mathbb{H}\text{)}\] Section 4.7 relates the onecomponent surgery formula to the Walker formula, recalled in §4.6. This relationship implies that λ satisfies Property T5.0:
Property T5.0:
IfM isa rational homology sphere,and ifλ_{w}denotes the Walker invariant as described in[W]:
\[\text{ }\!\!\lambda\!\!\text{ (M)=}\frac{\left {{\text{H}}_{\text{1}}}\text{(M)} \right}{\text{2}}{{\text{ }\!\!\lambda\!\!\text{ }}_{\text{W}}}\text{(M)}\] By definition, Proposition T2 holds for all surgery presentations in S^{3}. Our approach to the proof of Proposition T2 will be to...

Chapter 5 The invariant λ for 3manifolds with nonzero rank Chapter 5 The invariant λ for 3manifolds with nonzero rank (pp. 8194)In this section, we compute λ(M) for any oriented closed 3manifold M with positive rank. (The rank of a closed 3manifold M is its first Betti number β_{1}(M).) In order to do this, we frrst give M a surgery presentation as in:
Any oriented closed 3manifoldMcan be obtained by surgery from a rational homology sphereRaccording to the instructions of a presentation𝕃such that: The linking matrix of𝕃is null and the components of the underlying linkLof𝕃are nullhomologous.
The linkLhas thenβ_{1}(M)components, and
\[\left {{\text{H}}_{\text{1}}}\text{(R;}\mathbb{Z}\text{)} \right\text{=}\left \text{Torsion(}{{\text{H}}_{\text{1}}}\text{(M;}\mathbb{Z}\text{))} \right\] PROOF: Let β =...

Chapter 6 Applications and variants of the surgery formula Chapter 6 Applications and variants of the surgery formula (pp. 95116)Subsections 6.1 to 6.4 are independent. §6.1 and §6.3 present applications of the surgery formula while §6.2 and §6.4 prove the equivalence between Definition 1.4.8 of the surgery function 𝔽 and the definitions of 𝔽 given in §1.7.
§6.1 is devoted to computing λ for all oriented Seifert fibered spaces using the formula. These spaces are described in [Seif] and [Mo] and by the surgery presentations (taken from [Mo] Fig.12 p.146) of Figures 6.1 and 6.2 below.
The CassonWalker invariant of Seifert fibered rational homology spheres has already been computed in [L1] (it fortunately gave the same result); and the...

Appendix: More about the Alexander series Appendix: More about the Alexander series (pp. 117146) 
Bibliography Bibliography (pp. 147148) 
Index Index (pp. 149151)