A note on the construction of simply-connected -manifolds as branched covering spaces of

Author:
Joan S. Birman

Journal:
Proc. Amer. Math. Soc. **55** (1976), 440-442

MathSciNet review:
0394629

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Abstract | References | Additional Information

Abstract: Let be a knot in , and let be a transitive representation into the symmetric group on letters. The pair defines a unique closed, connected orientable -manifold , which is represented as an -sheeted covering space of , branched over . A procedure is given for representing by a Heegard splitting, and a formula is given for computing the genus of that Heegard splitting of . This formula is then applied to the -sheeted irregular covering spaces studied by Hilden (Bull. Amer. Math. Soc. **80** (1974), 1243-1244) and Montesinos (Bull. Amer. Math. Soc. **80** (1974), 845-846), and, also, Tesis (Univ. de Madrid, 1971) to show that these particular covering spaces cannot yield counterexamples to the Poincaré Conjecture if the branch set has bridge number .

**[1]**Joan S. Birman and Hugh M. Hilden,*The homeomorphism problem for 𝑆³*, Bull. Amer. Math. Soc.**79**(1973), 1006–1010. MR**0319180**, 10.1090/S0002-9904-1973-13303-8**[2]**Joan S. Birman and Hugh M. Hilden,*Heegaard splittings of branched coverings of 𝑆³*, Trans. Amer. Math. Soc.**213**(1975), 315–352. MR**0380765**, 10.1090/S0002-9947-1975-0380765-8**[3]**R. H. Fox,*Construction of simply connected 3-manifolds*, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 213–216. MR**0140116****[4]**Hugh M. Hilden,*Every closed orientable 3-manifold is a 3-fold branched covering space of 𝑆³*, Bull. Amer. Math. Soc.**80**(1974), 1243–1244. MR**0350719**, 10.1090/S0002-9904-1974-13699-2**[5]**-,*Three-fold branched coverings of*, Amer. J. Math. (to appear).**[6]**J. M. Montesinos,*Sobre la conjetura de Poincaré y los recubridores ramificados sobre un nudo*, Tesis, Universidad de Madrid, 1971.**[7]**José M. Montesinos,*A representation of closed orientable 3-manifolds as 3-fold branched coverings of 𝑆³*, Bull. Amer. Math. Soc.**80**(1974), 845–846. MR**0358784**, 10.1090/S0002-9904-1974-13535-4**[8]**José M. Montesinos,*A representation of closed orientable 3-manifolds as 3-fold branched coverings of 𝑆³*, Bull. Amer. Math. Soc.**80**(1974), 845–846. MR**0358784**, 10.1090/S0002-9904-1974-13535-4**[9]**George Springer,*Introduction to Riemann surfaces*, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1957. MR**0092855****[10]**Friedhelm Waldhausen,*Über Involutionen der 3-Sphäre*, Topology**8**(1969), 81–91 (German). MR**0236916**

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0394629-3

Keywords:
-manifold,
homotopy sphere,
branched covering space,
knot,
Heegard splitting

Article copyright:
© Copyright 1976
American Mathematical Society