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Branched standard spines of 3-manifolds. (English) Zbl 0873.57002
Lecture Notes in Mathematics. 1653. Berlin: Springer. viii, 132 p. (1997).
This book contains extensive results on the construction of combinatorial realizations of some categories of 3-manifolds with extra structure. The interest in effective combinatorial presentations was increased by the development of the theory of quantum invariants. On one hand the existence and structure of these invariants has been predicted, starting from Witten’s interpretation of the Jones polynomial. On the other hand an effective and rigorous construction of the invariants has only been given via combinatorial presentations such as surgery with the Kirby calculus (for the Reshetikhin-Turaev-Witten invariants), or spines and triangulations with the appropriate moves (for the Turaev-Viro invariants). The authors show that their combinatorial presentation of spin manifolds is suitable for an effective implementation and computation of the spin-refined version of the Turaev-Viro invariants. In particular, they give some hints on the possible relations with the theory of foliations and contact structures. They also remark that their combed calculus dually represents the set of homotopy classes of oriented plane distributions on 3-manifolds.

MSC:
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R25 Vector fields, frame fields in differential topology
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