Three-dimensional topology includes two vast domains: the study of geometric structures on 3-manifolds and the study of topological invariants of 3-manifolds, knots, etc. This book belongs to the second domain. We shall study an invariant called the maximal abelian torsion and denoted T. It is defined for a compact smooth (or piecewise-linear) manifold of any dimension and, more generally, for an arbitrary finite CW-complex X. The torsion T(X) is an element of a certain extension of the group ring Z[Hl(X)]. The torsion T can be naturally considered in the framework of simple homotopy theory. In particular, it is invariant under simple homotopy equivalences and can distinguish homotopy equivalent but non homeomorphic CW-spaces and manifolds, for instance, lens spaces. The torsion T can be used also to distinguish orientations and so-called Euler structures. Our interest in the torsion T is due to a particular role which it plays in three-dimensional topology. First of all, it is intimately related to a number of fundamental topological invariants of 3-manifolds. The torsion T(M) of a closed oriented 3-manifold M dominates (determines) the first elementary ideal of 7fl (M) and the Alexander polynomial of 7fl (M). The torsion T(M) is closely related to the cohomology rings of M with coefficients in Z and ZjrZ (r ; ::: 2). It is also related to the linking form on Tors Hi (M), to the Massey products in the cohomology of M, and to the Thurston norm on H2(M).
Inhaltsverzeichnis
I Generalities on Torsions. - I. 1 Torsions of chain complexes and CW-spaces. - I. 2 Combinatorial Euler structures and their torsions. - I. 3 The maximal abelian torsion. - I. 4 Smooth Euler structures and their torsions. - II The Torsion versus the Alexander-Fox Invariants. - II. 1 The first elementary ideal. - II. 2 The case b1 ? 2. - II. 3 The case b1 = 1. - II. 4 Extension to 3-manifolds with boundary. - II. 5 The Alexander polynomials. - III The Torsion versus the Cohomology Rings. - III. 1 Determinant and Pfaffian for alternate trilinear forms. - III. 2 The integral cohomology ring. - III. 3 Square volume forms and refined determinants. - III. 4 The cohomology ring mod r. - IV The Torsion Norm. - IV. 1 The torsion polytope and the torsion norm. - IV. 2 Comparison with the Thurston norm. - IV. 3 Proof of Theorem 2. 2. - V Homology Orientations in Dimension Three. - V. 1 Relative torsions of chain complexes. - V. 2 Induced homology orientations. - V. 3 Homology orientations and link exteriors. - V. 4 Homology orientations and surgery. - VI Euler Structures on 3-manifolds. - VI. 1 Gluing of smooth Euler structures and the class c. - VI. 2 Euler structures on solid tori and link exteriors. - VI. 3 Gluing of combinatorial Euler structures and torsions. - VII A Gluing Formula with Applications. - VII. 1 A gluing formula. - VII. 2 The Alexander-Conway function and surgery. - VII. 3 Proof of Formula (I. 4. e). - VII. 4 The torsion versus the Casson-Walker-Lescop invariant. - VII. 5 Examples and computations. - VIII Surgery Formulas for Torsions. - VIII. 1 Two lemmas. - VIII. 2 A surgery formula for ? -torsions. - VIII. 3 A surgery formula for the Alexander polynomial. - VIII. 4 A surgery formula for ? (M) in the case b1(M) ? 1. - VIII. 5 Realization of the torsion. - IX The Torsion Function. - IX. 1 The torsion function, basic Euler structures, and gluing. - IX. 2 Moments of the torsion function. - IX. 3 Axioms for the torsion function. - IX. 4 A surgery formula for the torsion function. - IX. 5 Formal expansions in Q(H) with applications. - X Torsion of Rational Homology Spheres. - X. 1 The torsion and the first elementary ideal. - X. 2 The torsion versus the linking form. - X. 3 The torsion versus the cohomology ring mod r. - X. 4 A gluing formula. - X. 5 A surgery formula. - X. 6 The torsion function and its moments. - XI Spinc Structures. - XI. 1 Spinc structures on 3-manifolds. - XI. 2 The torsion function versus the Seiberg-Witten invariants. - XI. 3 Spin structures on 3-manifolds. - XII Miscellaneous. - XII. 1 Torsions of connected sums. - XII. 2 The torsion versus the Massey products. - XII. 3 Genus estimates for ? r-surfaces. - Open Problems.
