Overview

This text is concerned with one of the most interesting and important topological invariants of 3-dimensional manifolds - the maximal abelian torsion - based on an original idea of Kurt Reidemeister (1935). It contains a systematic exposition of the theory of maximal abelian torsions of 3-manifolds, mainly focusing on topological properties of the torsion. The text also features a detailed description of relations between the torsion and the Alexander-Fox invariants of the fundamental group.

Full Product Details

Author:   Vladimir Turaev
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2002 ed.
Volume:   208
Dimensions:   Width: 15.50cm , Height: 1.50cm , Length: 23.50cm
Weight:   0.500kg
ISBN:  

9783764369118

ISBN 10:   3764369116
Pages:   196
Publication Date:   21 November 2002
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

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.

Reviews

This is an excellent exposition about abelian Reidemeister torsions for three-manifolds. <p>a Zentralblatt Math <p> The present monograph covers in great detail the work of the author spanning almost three decades. a ][Establishing an explicit formula given a 3-manifold] is a truly remarkable feata ] This monograph contains a wealth of information many topologists will find very handy. a ]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. <p>--Mathematical 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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