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Three-dimensional topology includes two vast domains: the study of geometric structures on 3-manifolds and the study of topological invariants of 3-manifolds, knots, etc. This book belongs to the second domain. We shall study an invariant called the maximal abelian torsion and denoted T. It is defined for a compact smooth (or piecewise-linear) manifold of any dimension and, more generally, for an arbitrary finite CW-complex X. The torsion T(X) is an element of a certain extension of the group ring Z Hl(X)]. The torsion T can be naturally considered in the framework of simple homotopy theory....

From the reviews: "This is an excellent exposition about abelian Reidemeister torsions for three-manifolds." -Zentralblatt Math

"This monograph contains a wealth of information many topologists will find very handy. ...Many of the new points of view pioneered by Turaev are gradually becoming mainstream and are spreading beyond the pure topology world. This monograph is a timely and very useful addition to the scientific literature." -Mathematical Reviews

The third edition of this monograph provides a systematic treatment of topological quantum field theories in three dimensions, inspired by the discovery of the Jones polynomial of knots, the Witten-Chern-Simons field theory, and the theory of quantum groups. The author, one of the leading experts in the subject, gives a rigorous and self-contained exposition of fundamental algebraic and topological concepts that emerged in this theory.

This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories. Starting with basic definitions, it proceeds to the forefront of current research.

Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories. It then presents deep theorems of Muger on the center of a pivotal fusion category. These theorems are proved in Part 2 using the theory of Hopf monads. In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and...

In this well-written presentation, motivated by numerous examples and problems, the authors introduce the basic theory of braid groups, highlighting several definitions that show their equivalence; this is followed by a treatment of the relationship between braids, knots and links. Important results then treat the linearity and orderability of the subject. Relevant additional material is included in five large appendices.

Braid Groups will serve graduate students and a number of mathematicians coming from diverse disciplines.

This 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 the algebraic foundations of the theory of torsions and various topological constructions of torsions. The third and last chapter deals with so-called refined torsions and the related additional structures. The book includes the necessary amount of standard background material to be accessible to the beginner.