State sum invariants of 3-manifolds and quantum \(6j\)-symbols
type: Item
instance of: scholarly article
title: State sum invariants of 3-manifolds and quantum \(6j\)-symbols
zbMATH Open document ID: 0779.57009
DOI: 10.1016/0040-9383(92)90015-A
author: Vladimir Turaev
author: Oleg Viro
published in: Topology
publication date: 1993-05-16T00:00:00Z
review text: The authors show how some algebraic input (over a commutative ring \(K\) with 1) gives rise to (unoriented) topological quantum field theories (TQFT). A TQFT here is a functor from the category of surfaces (with homomorphisms being 3-dimensional bordisms) into the category of finite- dimensional \(K\)-modules. Such a TQFT in particular assigns to each closed unoriented (maybe non-orientable) 3-manifold \(M\) an element \(| M | \in K\). The basic idea for the construction is that of admissable colorings of triangulated surfaces and 3-manifolds. Fix some finite set \(I\) and a distinguished subset of unordered triples of elements of \(I\), called admissable. Then an admissable coloring of a triangulated surface (resp. 3-manifold) is a function from the corresponding set of edges into \(I\), such that the colors of the boundary of each 2-simplex determine an admissable triple. The algebraic input now allows to assign to each compact triangulated 3-manifold \(M\) (possibly with boundary) and admissable coloring \(\varphi\) an element \(| M |_ \varphi \in K\). Using this and assuming independence of triangulations, the TQFT is constructed in a purely formal way. The \(K\)-module associated to a surface is essentially the free module on admissable colorings of \(F\). The invariant of a closed 3-manifold \(M\) is the sum \(\sum| M |_ \varphi\) over all colorings \(\varphi\) of \(M\). In order to prove that \(| M | _ \varphi\) does not depend on the triangulation the authors define equivalent notions of coloring and \(|\;|_ \varphi\) for a category of simple 2-complexes (dual to triangulations), and then translate the Alexander moves (moves relating different triangulations of a dimensionally homogeneous polyhedron) into this dual framework. This is necessary to reduce to a finite set of moves. It should be mentioned that the authors prove a relative version of the Alexander theorem. The relation to simple spines of manifolds, and a resulting method of computation of \(| M|…
reviewed by: Uwe Kaiser
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 81T25
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 167790
zbMATH Keywords: state sums
zbMATH Keywords: \(sl_ 2(\mathbb{C})\)
zbMATH Keywords: topological quantum field theories
zbMATH Keywords: admissable colorings of triangulated surfaces and 3-manifolds
zbMATH Keywords: Alexander moves
zbMATH Keywords: simple spines of manifolds
zbMATH Keywords: Heegard diagrams
zbMATH Keywords: quantized enveloping algebra
MaRDI profile type: Publication
Wikidata QID: Q60308781
full work available at URL: 0040-9383(92)90015-a
OpenAlex ID: W2019558609
description: scientific article; zbMATH DE number 167790