Torsions of 3-dimensional Manifolds

1. Front Matter

   Pages i-x

   PDF

2. Generalities on Torsions

   • Vladimir Turaev

   Pages 1-11

3. The Torsion versus the Alexander-Fox Invariants

   • Vladimir Turaev

   Pages 13-30

4. The Torsion versus the Cohomology Rings

   • Vladimir Turaev

   Pages 31-51

5. The Torsion Norm

   • Vladimir Turaev

   Pages 53-64

6. Homology Orientations in Dimension Three

   • Vladimir Turaev

   Pages 65-71

7. Euler Structures on 3-manifolds

   • Vladimir Turaev

   Pages 73-80

8. A Gluing Formula with Applications

   • Vladimir Turaev

   Pages 81-97

9. Surgery Formulas for Torsions

   • Vladimir Turaev

   Pages 99-118

10. The Torsion Function

    • Vladimir Turaev

    Pages 119-137

11. Torsion of Rational Homology Spheres

    • Vladimir Turaev

    Pages 139-160

12. Spin^c Structures

    • Vladimir Turaev

    Pages 161-173

13. Miscellaneous

    • Vladimir Turaev

    Pages 175-185

14. Back Matter

    Pages 187-198

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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).

• Analysis
• Reidemeister torsion
• manifold
• topological invariant
• topology

"This is an excellent exposition about abelian Reidemeister torsions for three-manifolds."

—Zentralblatt Math

"The present monograph covers in great detail the work of the author spanning almost three decades. …[Establishing an explicit formula given a 3-manifold] is a truly remarkable feat… This monograph contains a wealth of information many topologists will find very handy. …Many of the new points of view pioneered by Turaev are gradually becoming mainstream and are spreading beyond the pure topology world. This monograph is a timely and very useful addition to the scientific literature."

--Mathematical Reviews

• Institut de Recherche Mathématique Avancée, Université Louis Pasteur — CNRS, Strasbourg, France

  Vladimir Turaev

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