## Introduction to Combinatorial TorsionsThis 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. |

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### Contents

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 |

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### Common terms and phrases

3-manifold abelian group acyclic chain Alexander polynomial Ao(X based chain complex boundary homomorphism cellular chain cellular subdivision chain complex choice claim closed connected oriented compute Conway function Corollary CW-complex CW-decomposition cyclic group defined deformation retract denoted direct sum element elementary expansions esh-equivalent Eul(M Eul(X Euler structures Figure finite connected CW-complex free abelian fundamental family given H/Tors H Hence homology orientation homotopy equivalence implies induced integer invariant invertible isomorphism k-cells knot Lemma lens spaces Let F lift locally path-connected manifold map f matrix maximal abelian torsion Milnor torsion module obtain open cells ordered oriented pl-triangulation projection proof of Theorem Reidemeister torsions ring homomorphism Section short exact sequence simple homotopy simple homotopy equivalence Spin"-structure subcomplex T-chain Te(X Theorem 2.2 Ti(X Tors H triangulation unique factorization domain universal covering vect(M Whitehead torsion Z/pZ