Introduction to Combinatorial Torsions

2001
ISBN: 978-3-7643-6403-8
Verlag: Springer

Buch, Englisch, 124 Seiten, Paperback, Format (B × H): 170 mm x 244 mm, Gewicht: 242 g

Reihe: Lectures in Mathematics. ETH Zürich

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.

Zielgruppe

Research

Autoren/Hrsg.

Turaev, Vladimir

Fachgebiete

• Naturwissenschaften Biowissenschaften Angewandte Biologie Biomathematik
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie
• Mathematik | Informatik Mathematik Topologie Analytische Topologie

Inhaltsverzeichnis

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.