Vladimir G. Turaev Libri

This monograph is now available in its second revised edition due to its strong appeal and wide use. It systematically explores 3-dimensional topological quantum field theories (TQFTs), building on the author's collaboration with N. Reshetikhin and O. Viro. The subject is rooted in the discovery of the Jones polynomial of knots and Witten-Chern-Simons field theory. The algebraic study of 3-dimensional TQFTs is influenced by braided categories and quantum groups. The book is divided into three parts: Part I constructs 3-dimensional TQFTs and 2-dimensional modular functors from modular categories, yielding a broad range of knot and 3-manifold invariants, along with linear representations of surface mapping class groups. Part II employs 6j-symbols to define state sum invariants of 3-manifolds, establishing their connection to the TQFTs from Part I through the theory of shadows. Part III presents constructions of modular categories derived from quantum groups and skein modules of tangles in 3-space. This essential contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with a background in basic algebra and topology, serving as a vital resource for those eager to explore this intriguing intersection of mathematics and physics.

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