
Introduction to Combinatorial Torsions
Notes taken by Felix Schlenk
Vladimir Turaev(Author)
Birkhäuser (Publisher)
1st Edition
Published on 1. January 2001
Book
Paperback/Softback
VIII, 124 pages
978-3-7643-6403-8 (ISBN)
Description
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.
Reviews / Votes
"[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
More details
Series
Language
English
Place of publication
Basel
Switzerland
Publishing group
Springer Basel
Target group
Professional and scholarly
Research
Illustrations
13 s/w Abbildungen
VIII, 124 p. 13 illus.
Dimensions
Height: 244 mm
Width: 170 mm
Thickness: 8 mm
Weight
242 gr
ISBN-13
978-3-7643-6403-8 (9783764364038)
DOI
10.1007/978-3-0348-8321-4
Schweitzer Classification
Person
Dr. Christian Kassel is the director of CNRS (Centre National de la Recherche Scientifique in France), was the director of l'Institut de Recherche Mathematique Avancee from 2000 to 2004, and is an editor for the Journal of Pure and Applied Algebra. Kassel has numerous publications, including the book Quantum Groups in the Springer Gradate Texts in Mathematics series.
Dr. Vladimir Turaev was also a professor at the CNRS and is currently at Indiana University in the Department of Mathematics.
Content
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.