Vladimir Turaev Bücher

This monograph provides a systematic treatment of topological quantum field theories (TQFT's) 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 new fundamental algebraic and topological concepts that emerged in this theory. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFT's and 2-dimensional modular functors from so-called modular categories. This gives new knot and 3-manifold invariants as well as linear representations of the mapping class groups of surfaces. In Part II the machinery of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFT's constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and Kauffman's skein modules. This book is accessible to graduate students in mathematics and physics with a knowledge of basic algebra and topology. It will be an indispensable source for everyone who wishes to enter the forefront of this rapidly growing and fascinating area at the borderline of mathematics and physics. Most of the results and techniques presented here appear in book form for the first time.

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 Müger 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 construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic tothe Reshetikhin-Turaev surgery graph TQFT derived from the center of that category. The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.

Three-dimensional topology encompasses the study of geometric structures and topological invariants of 3-manifolds and knots. This work focuses on the invariant known as maximal abelian torsion, denoted T, applicable to compact smooth or piecewise-linear manifolds and finite CW-complexes X. The torsion T(X) is an element of an extension of the group ring Z[Hl(X)] and can be analyzed within the context of simple homotopy theory. It remains invariant under simple homotopy equivalences, distinguishing homotopy equivalent but non-homeomorphic CW-spaces and manifolds, such as lens spaces. Additionally, T can differentiate orientations and Euler structures. The significance of torsion T lies in its crucial role in three-dimensional topology. It is closely linked to several fundamental topological invariants of 3-manifolds. Specifically, the torsion T(M) of a closed oriented 3-manifold M determines the first elementary ideal of the fundamental group π1(M) and the Alexander polynomial of π1(M). Furthermore, T(M) is associated with the cohomology rings of M with coefficients in Z and Z/rZ (for r ≥ 2), the linking form on Tors Hi(M), Massey products in the cohomology of M, and the Thurston norm on H2(M).

„[The book] contains much of the needed background material in topology and algebra…Concering the considerable material it covers, [the book] is very well-written and readable.“ --Zentralblatt Math