Author	Turaev, Vladimir. author
Title	Torsions of 3-dimensional Manifolds [electronic resource] / by Vladimir Turaev
Imprint	Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2002
Connect to	http://dx.doi.org/10.1007/978-3-0348-7999-6
Descript	X, 196 p. online resource

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 nonยญ 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)

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