New aspects of complexity theory for 3-manifolds. (English. Russian original) Zbl 1473.57058
Russ. Math. Surv. 73, No. 4, 615-660 (2018); translation from Usp. Mat. Nauk 73, No. 4, 53-102 (2018).
The notion of Matveev complexity of a 3-manifold was originally introduced by S. Matveev in [Sov. Math. Dokl. 38 (1), 75–78 (1989; Zbl 0674.57012), translation from Dokl. Akad. Nauk SSSR 301, No. 2, 280–283 (1988)], by making use of the idea of representing the manifold by a spine of it. For each 3-manifold \(M\), its complexity \(c(M)\) is the minimum number of true vertices in an almost simple spine of \(M\).
The present work is a detailed and updated survey on the theory of complexity, both for closed 3-manifolds and for 3-manifolds with boundary. Infinite families of closed orientable manifolds and hyperbolic manifolds with totally geodesic boundary are presented, and the exact values of the Matveev complexity are given for them. New methods for computing complexity are described, based on calculation of the Turaev-Viro invariants and hyperbolic volumes of 3-manifolds.
The work contains a table with the number of different combinatorial tetrahedral tessellations and the number of different tetrahedral manifolds with at most 25 tetrahedra in the orientable case and at most 21 tetrahedra in the non-orientable case. In addition, knots and links are listed whose complements are tetrahedral manifolds with at most 12 tetrahedra.
A rich bibliography (89 titles) completes the paper.
The present work is a detailed and updated survey on the theory of complexity, both for closed 3-manifolds and for 3-manifolds with boundary. Infinite families of closed orientable manifolds and hyperbolic manifolds with totally geodesic boundary are presented, and the exact values of the Matveev complexity are given for them. New methods for computing complexity are described, based on calculation of the Turaev-Viro invariants and hyperbolic volumes of 3-manifolds.
The work contains a table with the number of different combinatorial tetrahedral tessellations and the number of different tetrahedral manifolds with at most 25 tetrahedra in the orientable case and at most 21 tetrahedra in the non-orientable case. In addition, knots and links are listed whose complements are tetrahedral manifolds with at most 12 tetrahedra.
A rich bibliography (89 titles) completes the paper.
Reviewer: Maria Rita Casali (Modena)
MSC:
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 57K30 | General topology of 3-manifolds |
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
Citations:
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