Introduction to Combinatorial Torsions

Vladimir G. Turaev

ISBN 10: 3764364033 / ISBN 13: 9783764364038

Published by Springer, Basel, Birkhäuser Basel, Birkhäuser Jan 2001, 2001

About this Item

This item is printed on demand - it takes 3-4 days longer - Neuware -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. 124 pp. Englisch.

Seller Inventory # 9783764364038

Bibliographic Details

• Title: Introduction to Combinatorial Torsions
• Publisher: Springer, Basel, Birkhäuser Basel, Birkhäuser Jan 2001
• Publication Date: 2001
• Language: English
• ISBN 10: 3764364033
• ISBN 13: 9783764364038
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 244W x 195H x 37D millimeters 244 millimeters width by 195 millimeters height by 37 millimeters depth
• Weight: 260 grams

About this title

Synopsis

This book is an introduction to combinatorial torsions of cellular spaces and manifolds with special emphasis on torsions of 3-dimensional manifolds. The first two chapters cover algebraic foundations of the theory of torsions and various topological constructions of torsions due to K. Reidemeister, J.H.C. Whitehead, J. Milnor and the author. We also discuss connections between the torsions and the Alexander polynomials of links and 3-manifolds. The third (and last) chapter of the book deals with so-called refined torsions and the related additional structures on manifolds, specifically homological orientations and Euler structures. As an application, we give a construction of the multivariable Conway polynomial of links in homology 3-spheres. At the end of the book, we briefly describe the recent 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. The exposition is aimed at students, professional mathematicians and physicists interested in combinatorial aspects of topology and/or in low dimensional topology. The necessary background for the reader includes the elementary basics of topology and homological algebra.

Review

"[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