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Introduction to Combinatorial Torsions Vladimir Turaev

Introduction to Combinatorial Torsions By Vladimir Turaev

Introduction to Combinatorial Torsions by Vladimir Turaev


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Summary

Offers an introduction to combinatorial torsions of cellular spaces and manifolds with emphasis on torsions of 3-dimensional manifolds. This book describes the results of G Meng, C H Taubes and the author on the connections between the refined torsions and the Seiberg-Witten invariant of 3-manifolds.

Introduction to Combinatorial Torsions Summary

Introduction to Combinatorial Torsions by Vladimir Turaev

This book is an extended version of the notes of my lecture course given at ETH in spring 1999. The course was intended as an introduction to combinatorial torsions and their relations to the famous Seiberg-Witten invariants. Torsions were introduced originally in the 3-dimensional setting by K. Rei- demeister (1935) who used them to give a homeomorphism classification of 3-dimensional lens spaces. The Reidemeister torsions are defined using simple linear algebra and standard notions of combinatorial topology: triangulations (or, more generally, CW-decompositions), coverings, cellular chain complexes, etc. The Reidemeister torsions were generalized to arbitrary dimensions by W. Franz (1935) and later studied by many authors. In 1962, J. Milnor observed 3 that the classical Alexander polynomial of a link in the 3-sphere 8 can be interpreted as a torsion of the link exterior. Milnor's arguments work for an arbitrary compact 3-manifold M whose boundary is non-void and consists of tori: The Alexander polynomial of M and the Milnor torsion of M essentially coincide.

Introduction to Combinatorial Torsions Reviews

[The book] contains much of the needed background material in topology and algebra...Concering the considerable material it covers, [the book] is very well-written and readable.

--Zentralblatt Math

Table of Contents

I Algebraic Theory of Torsions.- 1 Torsion of chain complexes.- 2 Computation of the torsion.- 3 Generalizations and functoriality of the torsion.- 4 Homological computation of the torsion.- II Topological Theory of Torsions.- 5 Basics of algebraic topology.- 6 The Reidemeister-Franz torsion.- 7 The Whitehead torsion.- 8 Simple homotopy equivalences.- 9 Reidemeister torsions and homotopy equivalences.- 10 The torsion of lens spaces.- 11 Milnor's torsion and Alexander's function.- 12 Group rings of finitely generated abelian groups.- 13 The maximal abelian torsion.- 14 Torsions of manifolds.- 15 Links.- 16 The Fox Differential Calculus.- 17 Computing ?(M3) from the Alexander polynomial of links.- III Refined Torsions.- 18 The sign-refined torsion.- 19 The Conway link function.- 20 Euler structures.- 21 Torsion versus Seiberg-Witten invariants.- References.

Additional information

NLS9783764364038
9783764364038
3764364033
Introduction to Combinatorial Torsions by Vladimir Turaev
New
Paperback
Birkhauser Verlag AG
2001-01-01
124
N/A
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