# HEEGAARD SPLITTINGS OF BRANCHED CYCLIC COVERINGS OF CONNECTED SUMS OF LENS SPACES

• Kozlovskaya, Tatyana (Magadan Institute of Economy of St. Peterburg Academy of Management and Economy)
• Accepted : 2016.12.26
• Published : 2017.09.30

#### Abstract

We study relations between two descriptions of closed orientable 3-manifolds: as branched coverings and as Heegaard splittings. An explicit relation is presented for a class of 3-manifolds which are branched cyclic coverings of connected sums of lens spaces, where the branching set is an axis of a hyperelliptic involution of a Heegaard surface.

#### Acknowledgement

Supported by : Russian Foundation for Basic Research

#### References

1. J. S. Birman and H. M. Hilden, Heegaard splittings of branched coverings of $S^3$, Trans. Amer. Math. Soc. 213 (1975), 315-352.
2. A. Cattabriga, M. Mulazzani, and A. Vesnin, Complexity, Heegaard diagrams and generalized Dunwoody manifolds, J. Korean Math. Soc. 47 (2010), no. 3, 585-599. https://doi.org/10.4134/JKMS.2010.47.3.585
3. P. Cristifori, T. Kozlovskaya, and A. Vesnin, Cyclic generalizations of two hyperbolic icosahedral manifolds, Topology Appl. 159 (2012), no. 8, 2071-2081. https://doi.org/10.1016/j.topol.2012.01.016
4. P. Cristofori, M. Mulazzani, and A. Vesnin, Strongly-cyclic branched coverings of knots via (g; 1)-decompositions, Acta Math. Hungar. 116 (2007), no. 1-2, 163-176. https://doi.org/10.1007/s10474-007-6029-2
5. M. J. Dunwoody, Cyclic presentations and 3-manifolds, In: Proc. Inter. Conf., Groups-Korea 94, Walter de Gruyter, Berlin-New York 1995, 47-55.
6. L. Grasselli and M. Mulazzani, Genus one 1-bridge knots and Dunwoody manifolds, Forum Math. 13 (2001), no. 3, 379-397.
7. A. Cavicchioli, F. Hegenbarth, and A. C. Kim, A geometric study of Sieradski groups, Algebra Colloq. 5 (1998), no. 2, 203-217.
8. H. Helling, A. Kim, and J. Mennicke, A geometric study of Fibonacci groups, J. Lie Theory 8 (1998), no. 1, 1-23.
9. T. Kozlovskaya and A. Vesnin, Branched cyclic coverings of lens spaces, Sib. Math. J. 52 (2011), no. 3, 426-435. https://doi.org/10.1134/S0037446611030050
10. S. V. Matveev and A. T. Fomenko, Constant energy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds, Russian Mathematical Surveys 43 (1988), no. 1, 3-24. https://doi.org/10.1070/RM1988v043n01ABEH001554
11. J. P. Mayberry and K. Murasugi, Torsion-groups of abelian coverings of links, Trans. Amer. Math. Soc. 271 (1982), no. 1, 143-173. https://doi.org/10.1090/S0002-9947-1982-0648083-1
12. J. W. Morgan and H. Bass (eds.), The Smith conjecture, Academic Press Inc., Orlando, FL, 1984, Papers presented at the symposium held at Columbia University, New York, 1979.
13. M. Mulazzani, On p-symmetric Heegaard splittings, J. Knot Theory Ramifications 9 (2000), no. 8, 1059-1067. https://doi.org/10.1142/S0218216500000633
14. C. Weber and H. Seifert, Die Beiden Dodekaederaume, Math. Z. 37 (1933), no. 1, 237-253. https://doi.org/10.1007/BF01474572
15. J. Weeks, Hyperbolic structures on 3-manifolds, Ph. D. Thesis, Princeton University, 1985.