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## "Invariants of piecewise-linear 3-manifolds", John Barrett, Bruce Westbury, 1996

*Reviewed March 13, 2024*

*Citation:* Barrett, John, and Bruce Westbury. "Invariants of piecewise-linear 3-manifolds." Transactions of the American Mathematical Society 348.10 (1996): 3997-4022.

*Web:* https://arxiv.org/abs/hep-th/9311155

*Tags:* TQFT, Monoidal-categories, Foundational

This paper is where the invariant which we now call the Turaev-Viro invariant
was first defined. The original Turaev-Viro invariant was introduced in

> Turaev, Vladimir G., and Oleg Ya Viro. "State sum invariants of 3-manifolds and quantum 6j-symbols." Topology 31.4 (1992): 865-902.

The original construction only applied to sl_2(C) quantum groups. This new
construction applies to arbitrary spherical categories. In fact, it
was precisely for this paper that spherical categories were introduced.
When Barrett and Westbury were trying to generalize Turaev-Viro's construction
to its most general form, they were naturally led to defining the notation
of a spherical category. That paper was published three years prior:

> Barrett, John W., and Bruce W. Westbury. "Spherical categories." Advances in Mathematics 143.2 (1999): 357-375.

As is usual with papers from this time, results
are re-cast in the language of Hopf algebras
whenever possible. Reshetikhin-Turaev's construction
was originally performed for "modular Hopf algebras".
In this paper, there is an in-depth discussion
of "spherical Hopf algebras". The category
of representations of a spherical Hopf algebra
is a spherical category.

The way this paper proves that their quantity is an
invariant is by working with Pachner moves, which
at this time new:

> Pachner, Udo. "PL homeomorphic manifolds are equivalent by elementary shellings." European journal of Combinatorics 12.2 (1991): 129-145.

The authors credit the idea of working with triangulations
and then proving invariance under a finite set of moves to
earlier work on the recouping theory of Lie groups.
Indeed, reading these works one finds that
they serve as a primordial root for much of quantum
topology. A good reference is Moussouris' PhD thesis:

> Moussouris, J. Quantum models of space-time based on recoupling theory. Diss. University of Oxford, 1984.

It is good to note that the Turaev-Viro model has since been shown to go down to the level
of a 3-2-1 extended TQFT, which in a sense completes the definition of the Turaev-Viro model:

> Kirillov Jr, Alexander, and Benjamin Balsam. "Turaev-Viro invariants as an extended TQFT." arXiv preprint arXiv:1004.1533 (2010).