Torsions of 3-dimensional Manifolds

By: Vladimir Turaev

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Hardcover | 1 December 2002

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212 Pages

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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).

Industry Reviews

"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

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Non-Fiction Mathematics Topology Calculus & Mathematical Analysis Real Analysis Geometry Medicine Medicine in General Numerical Analysis

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Published: 24th October 2012

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Introduction
Generalities on Torsions
The Torsion versus the Alexander-Fox Invariants
The Torsion versus the Cohomology Rings
The Torsion Norm
Homology Orientations in Dimension Three
Euler Structures on 3-manifolds
A Gluing Formula with Applications
Surgery Formulas for Torsions
The Torsion Function
Torsion of Rational Homology Spheres
Spin[superscript c] Structures
Miscellaneous
Open Problems
Bibliography
Index
Table of Contents provided by Blackwell. All Rights Reserved.

Torsions of 3-dimensional Manifolds

At a Glance

Hardcover

Industry Reviews

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How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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