REVIEW OF: "Benedetti Riccardo - Petronio Carlo, Spin structures on 3-manifolds via arbitrary triangulations, Algebr. Geom. Topol. 14, No. 2, 1005-1054 (2014)". [DE062726187]

In the present paper a combinatorial encoding of spin structures based on arbitrary triangulations of oriented compact 3-manifolds is introduced. The goal is achieved by means of the notion of weak branching, which turns out to be related to the notion of Z/2Z-taut structure on triangulations, introduced by Luo (see [Proc. Am. Math. Soc. 140, No. 3, 1053-1068 (2012; Zbl 1250.57028)] and [Proc. Am. Math. Soc. 141, No. 1, 335-350 (2013; Zbl 1272.57004)]). In particular, by taking into account the set of all pairs (M, s) (M being a compact oriented 3-manifold and s being a spin structure on M), the authors claim: - “given any (loose) triangulation T of M, with ideal vertices at the components of @M and possibly internal vertices, and any s, we encode s by decorating T with certain extra combinatorial structures; - we exibit combinatorial moves on decorated triangulations relating to each other any two that encode the same (M, s).” A dual version of the above encoding is also presented, in terms of special spines dual to triangulations (see [Acta Appl. Math. 19, No.2, 101-130 (1990; Zbl 0724.57012)]). A first application of the described techniques is contained in [Baseilhac-Benedetti, Analytic families of quantum hyperbolic invariants and their asymptotical behaviour, I, arXiv:1212.4261]. Further possible applications are also pointed out, concerning “an effective construction of the Roberts spin-refined TuraevViro invariants and of the related Blanchet spin-refined ReshetikhinTuraev invariants of the double of a manifold”.