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State sum invariants of 3-manifolds and quantum $$6j$$-symbols. (English) Zbl 0779.57009
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 $$| M|_ \varphi$$ from Heegard diagrams is discussed.
It is shown that for each $$r$$-th root of unity $$q$$, $$r>2$$, the quantum $$6j$$-symbols (associated with the quantized enveloping algebra $$U_ q(\text{sl}_ 2(\mathbb{C})))$$ provide the necessary algebraic input. Thus for each such $$q$$ the authors have constructed a TQFT. For these TQFTs explicit calculations of $$| M |$$ for $$S^ 3$$, $$\mathbb{R} P^ 3$$, $$L (3,1)$$ and $$S^ 2 \times S^ 1$$ and topological interpretations (Betti numbers) of $$| M |$$ for $$r=3$$ and arbitrary 3-manifolds are given.
Reviewer: U.Kaiser (Siegen)

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 81T25 Quantum field theory on lattices 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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