Introduction to Combinatorial Torsions

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Springer Science & Business Media, 01.01.2001 - 124 Seiten
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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.
 

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Inhalt

Algebraic Theory of Torsions
1
2 Computation of the torsion
7
3 Generalizations and functoriality of the torsion
12
4 Homological computation of the torsion
16
Topological Theory of Torsions
23
6 The ReidemeisterFranz torsion
30
7 The Whitehead torsion
35
8 Simple homotopy equivalences
40
13 The maximal abelian torsion
64
14 Torsions of manifolds
69
15 Links
81
16 The Fox Differential Calculus
83
17 Computing rM3 from the Alexander polynomial of links
92
Refined Torsions
97
19 The Conway link function
102
20 Euler structures
108

9 Reidemeister torsions and homotopy equivalences
43
10 The torsion of lens spaces
44
11 Milnors torsion and Alexanders function
51
12 Group rings of finitely generated abelian groups
58
21 Torsion versus SeibergWitten invariants
112
References
117
Index
121
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