Introduction to combinatorial torsions. Notes taken by Felix Schlenk.

*(English)*Zbl 0970.57001
Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser. viii, 123 p. (2001).

The book gives an introduction to combinatorial torsion. It starts with a careful presentation of the algebraic foundations of the theory of torsions, including ways to compute it in the algebraic setting, in particular describing homological computations of torsion.

The second part is concerned with various versions of (topological) torsion for CW-complexes, manifolds, and homotopy equivalences between such spaces. In particular, Reidemeister-Franz torsion and the classification of lens spaces, Whitehead torsion with its relation to simple homotopy equivalence, Milnor torsion and the Alexander polynomial, also the Fox calculus to compute the Alexander polynomial, and maximal abelian torsion (a generalization of Milnor torsion) are defined and discussed. The book discusses also special relations which hold for torsions of manifolds.

The Reidemeister torsion of a manifold has certain well-known indeterminacies. The last part of the book discusses additional structures on manifolds, namely Euler structures and homology orientations, which allow the definition of the so-called sign-refined torsion, where these indeterminacies are eliminated. The sign-refined Milnor torsion computes the Conway polynomial. The theory of the Conway polynomial is also developed in the book.

At the end of the volume, the connection between torsions and the Seiberg-Witten invariants of 3-manifolds is briefly described.

The book is not intended to be a systematic treatise on torsions or on simple homotopy theory. However, concerning the (considerable) material it covers, it is very well written and readable. Despite its small size, it contains also much of the needed background material in topology and homological algebra.

The second part is concerned with various versions of (topological) torsion for CW-complexes, manifolds, and homotopy equivalences between such spaces. In particular, Reidemeister-Franz torsion and the classification of lens spaces, Whitehead torsion with its relation to simple homotopy equivalence, Milnor torsion and the Alexander polynomial, also the Fox calculus to compute the Alexander polynomial, and maximal abelian torsion (a generalization of Milnor torsion) are defined and discussed. The book discusses also special relations which hold for torsions of manifolds.

The Reidemeister torsion of a manifold has certain well-known indeterminacies. The last part of the book discusses additional structures on manifolds, namely Euler structures and homology orientations, which allow the definition of the so-called sign-refined torsion, where these indeterminacies are eliminated. The sign-refined Milnor torsion computes the Conway polynomial. The theory of the Conway polynomial is also developed in the book.

At the end of the volume, the connection between torsions and the Seiberg-Witten invariants of 3-manifolds is briefly described.

The book is not intended to be a systematic treatise on torsions or on simple homotopy theory. However, concerning the (considerable) material it covers, it is very well written and readable. Despite its small size, it contains also much of the needed background material in topology and homological algebra.

Reviewer: Thomas Schick (Münster)

##### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57Q10 | Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. |

57R57 | Applications of global analysis to structures on manifolds |

19J10 | Whitehead (and related) torsion |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |