© 2002

Torsions of 3-dimensional Manifolds


Part of the Progress in Mathematics book series (PM, volume 208)

Table of contents

  1. Front Matter
    Pages i-x
  2. Vladimir Turaev
    Pages 1-11
  3. Vladimir Turaev
    Pages 31-51
  4. Vladimir Turaev
    Pages 53-64
  5. Vladimir Turaev
    Pages 65-71
  6. Vladimir Turaev
    Pages 73-80
  7. Vladimir Turaev
    Pages 81-97
  8. Vladimir Turaev
    Pages 99-118
  9. Vladimir Turaev
    Pages 119-137
  10. Vladimir Turaev
    Pages 139-160
  11. Vladimir Turaev
    Pages 161-173
  12. Vladimir Turaev
    Pages 175-185
  13. Back Matter
    Pages 187-198

About this book


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

Authors and affiliations

  1. 1.Institut de Recherche Mathématique AvancéeUniversité Louis Pasteur — CNRSStrasbourgFrance

Bibliographic information


"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