Cover -- Title -- Copyright -- Table of contents -- Chapter 1: Introduction and statements of the results -- 1.1 Introduction -- 1.2 Conventions -- 1.3 Surgery presentations and associated functions -- 1.4 Introduction of the surgery formula F -- 1.5 Statement of the theorem -- 1.6 Sketch of the proof of the theorem and organization of the book -- 1.7 Equivalent definitions for F -- Chapter 2: The Alexander series of a link in a rational homology sphere and some of its properties -- 2.1 The background -- 2.2 A definition of the Alexander series -- 2.3 A list of properties for the Alexander series -- 2.4 Functions of the linking numbers of a link -- 2.5 The first terms of the Alexander series -- Chapter 3: Invariance of the surgery formula under a twist homeomorphism -- 3.1 Introduction -- 3.2 Variation of the different pieces of FM under an ω-twist: the statements -- 3.3 Proofs of 3.2.13 and 3.2.16 -- 3.4 More linking functions: semi-open graphs and functions α -- 3.5 Variation of the ζ-coefficients under an ω-twist -- 3.6 Proof of 3.2.11 (variation of the piece containing the ζ-coefficients under the ω-twist) -- Chapter 4: The formula for surgeries starting from rational homology spheres -- 4.1 Introduction -- 4.2 Sketch of the proof of Proposition T2 -- 4.3 Proof of Lemma 4.2.2 -- 4.4 Proof of Lemma 4.2.3 -- 4.5 Proof of Lemma 4.2.5 -- 4.6 The Walker surgery formula -- 4.7 Comparing T2 with the Walker surgery formula -- Chapter 5: The invariant λ for 3-manifolds with nonzero rank -- 5.1 Introduction -- 5.2 The coefficients a1 of homology unlinks in rational homology spheres (after Hoste) -- 5.3 Computing λ for manifolds with rank at least 2 -- Chapter 6: Applications and variants of the surgery formula -- 6.1 Computing λ for all oriented Seifert fibered spaces using the formula.

6.2 The formula involving the figure-eight linking -- 6.3 Congruences and relations with the Rohlin invariant -- 6.4 The surgery formula in terms of one-variable Alexander polynomials -- Appendix: More about the Alexander series -- A.1 Introduction -- A.2 Complete definition of the Reidemeister torsion of (N,o(N)) up to positive units -- A.3 Proof of the symmetry property of the Reidemeister torsion -- A.4 Various properties of the Reidemeister torsion -- A.5 A systematic way of computing the Alexander polynomials of links in S^3 -- A.6 Relations with one-variable Alexander polynomials -- Bibliography -- Index.