Introduction to Combinatorial TorsionsSpringer Science & Business Media, 01.01.2001 - 124 Seiten 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. |
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 |
121 | |
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Häufige Begriffe und Wortgruppen
3-manifold 7-chain a₁ abelian group acyclic chain complex ai+1 Alexander polynomial B₁ b₁(M based chain complex boundary homomorphism C₁ chain complex Ci+1 claim compute Conway function Corollary CW-complex CW-decomposition cyclic group defined definition deformation retract denoted direct sum element elementary expansions esh-equivalent Eul(M Eul(X Euler structures family ê Figure finite connected CW-complex follows free abelian h₁ H₁(X H₂ Hence homology orientation homotopy equivalence ideal implies induced integer invariant invertible isomorphism k-cells K₁(A knot Lemma lens spaces Let F lift locally path-connected manifold Math matrix maximal abelian covering maximal abelian torsion Milnor torsion obtain open cells ord Tors ordered oriented pl-triangulation projection proof of Theorem Reidemeister torsion ring homomorphism Section Seiberg-Witten short exact sequence simple homotopy simple homotopy equivalence subcomplex Tors H triangulation unique factorization domain universal covering vect(M Whitehead torsion Z/pZ
Verweise auf dieses Buch
Intelligence of Low Dimensional Topology 2006: Hiroshima, Japan, 22-26 July 2006 J. Scott Carter Keine Leseprobe verfügbar - 2007 |