Introduction to Combinatorial Torsions

Front Cover
Birkhäuser, Dec 6, 2012 - Mathematics - 124 pages
0 Reviews
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.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Algebraic Theory of Torsions 1 Torsion of chain complexes
1
Computation of the torsion
7
Generalizations and functoriality of the torsion
12
Homological computation of the torsion
16
Topological Theory of Torsions 5 Basics of algebraic topology
23
The ReidemeisterFranz torsion
30
The Whitehead torsion
35
Simple homotopy equivalences
40
The maximal abelian torsion
64
Torsions of manifolds
69
Links
81
The Fox Differential Calculus
83
Computing TM from the Alexander polynomial of links
92
Refined Torsions 18 The signrefined torsion 97
96
The Conway link function
102
Euler structures
108

Reidemeister torsions and homotopy equivalences
43
The torsion of lens spaces
44
Milnors torsion and Alexanders function
51
Group rings of finitely generated abelian groups
58
Torsion versus SeibergWitten invariants
112
References
117
Copyright

Other editions - View all

Common terms and phrases

Bibliographic information