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A topological study of some groups arising from cellular quotients. (English) Zbl 1095.57002

The present paper takes into account the family of closed connected orientable 3-manifolds \(M_n\), \(n \geq 1\), first constructed in [A. C. Kim and A. I. Kostrikin, Dokl. Math. 51, No. 1, 36–38 (1995; Zbl 0907.57001) and Sb. Math. 188, No. 2, 173–194 (1997; Zbl 0906.20021)] by pairwise identification of the boundary faces of combinatorial polyhedra \(P_n\), \(n \geq 1\), homeomorphic to 3-cells. By studying suitable presentations of the fundamental group \(G_n\) of \(M_n\) (\(n \geq 1\)), together with their derived quotients and split extensions, and the corresponding spines, the authors completely classify the manifolds \(M_n\). In particular, for all \(n \geq 1\), \(M_n\) is proved to be the \(3n\)-fold cyclic covering of the 3-sphere branched over the Whitehead link, where the branching indices of its components are \(3\) and \(3n\).

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
57M50 General geometric structures on low-dimensional manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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References:

[1] Cavicchioli A., Algebra Colloquium 5 pp 203–
[2] DOI: 10.1007/s002290050227 · Zbl 0947.57013 · doi:10.1007/s002290050227
[3] DOI: 10.1142/S021821650300241X · Zbl 1031.57002 · doi:10.1142/S021821650300241X
[4] DOI: 10.1142/S0129167X95000109 · Zbl 0840.57007 · doi:10.1142/S0129167X95000109
[5] DOI: 10.1080/00927879508825526 · Zbl 1002.57503 · doi:10.1080/00927879508825526
[6] DOI: 10.1515/9783110857726.169 · doi:10.1515/9783110857726.169
[7] Kim A. C., Doklady Akad. Nauk 340 pp 158–
[8] DOI: 10.4213/sm193 · doi:10.4213/sm193
[9] Kim Y., Comm. Korean Math. Soc. 11 pp 933–
[10] DOI: 10.1016/S0001-8708(71)80003-8 · Zbl 0223.30008 · doi:10.1016/S0001-8708(71)80003-8
[11] Mednykh A. D., Geometry and Analysis (Sci. Ser. A Math. Sci., N.S.) 8 pp 1–
[12] DOI: 10.1017/S0305004100043279 · doi:10.1017/S0305004100043279
[13] Rolfsen D., Mathematics Lecture Series, No. 7, in: Knots and Links (1976)
[14] Seifert H., A Textbook of Topology: Topology of 3-Dimensional Fibered Spaces (1980) · Zbl 0469.55001
[15] DOI: 10.1007/BF01388735 · Zbl 0604.57001 · doi:10.1007/BF01388735
[16] Takahashi M., Yokohama Math. J. 25 pp 91–
[17] Thurston W. P., Lect. Notes, in: The Geometry and Topology of 3-Manifolds (1980)
[18] DOI: 10.1007/BF01474572 · Zbl 0007.02806 · doi:10.1007/BF01474572
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