Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ 4$-manifolds, $ 3$-fold covering spaces and ribbons

Author: José María Montesinos
Journal: Trans. Amer. Math. Soc. 245 (1978), 453-467
MSC: Primary 57M10; Secondary 57N15
MathSciNet review: 511423
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that a PL, orientable 4-manifold with a handle presentation composed by 0-, 1-, and 2-handles is an irregular 3-fold covering space of the 4-ball, branched over a 2-manifold of ribbon type. A representation of closed, orientable 4-manifolds, in terms of these 2-manifolds, is given. The structure of 2-fold cyclic, and 3-fold irregular covering spaces branched over ribbon discs is studied and new exotic involutions on $ {S^4}$ are obtained. Closed, orientable 4-manifolds with the 2-handles attached along a strongly invertible link are shown to be 2-fold cyclic branched covering spaces of $ {S^4}$. The conjecture that each closed, orientable 4-manifold is a 4-fold irregular covering space of $ {S^4}$ branched over a 2-manifold is reduced to studying $ \gamma \char93 {S^1} \times {S^2}$ as a nonstandard 4-fold irregular branched covering of $ {S^3}$.

References [Enhancements On Off] (What's this?)

  • [1] C. Gordon, On the higher-dimensional Smith conjecture, Proc. London Math. Soc. (3) 28 (1974), 98-110. MR 0356073 (50:8544)
  • [2] H. M. Hilden, Three-fold branched coverings of $ {S^3}$, Amer. J. Math. 98 (1976), 989-997. MR 0425968 (54:13917)
  • [3] H. M. Hilden and J. M. Montesinos, A method of constructing 3-manifolds and its application to the computation of the $ \mu $-invariant, Proc. Sympos. in Pure Math., vol. 32, Amer. Math. Soc., Providence, R.I., 1977, pp. 477-485.
  • [4] P. Kim and J. Toilefson, Splitting the PL involutions on nonprime 3-manifolds (to appear).
  • [5] W. B. R. Lickorish, A representation of orientable, combinatorial 3-manifolds, Ann. of Math. (2) 76 (1962), 531-540. MR 0151948 (27:1929)
  • [6] B. Mazur, A note on some contractible 4-manifolds, Ann. of Math. (2) 73 (1961), 221-228. MR 0125574 (23:A2873)
  • [7] J. M. Montesinos, Heegaard diagrams for closed 4-manifolds, Proc. Georgia Geometric Topology Conf., 1977.
  • [8] I. Berstein and A. L. Edmonds, On the construction of branched coverings of low-dimensional manifolds (preprint). MR 517687 (80b:57003)
  • [9] -, The degree and branch set of a branched covering (preprint).
  • [10] T. Yajima, On a characterization of knot groups of some spheres in $ {R^4}$, Osaka J. Math. 6 (1969), 435-446. MR 0259893 (41:4522)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M10, 57N15

Retrieve articles in all journals with MSC: 57M10, 57N15

Additional Information

Keywords: 4-manifolds, 3-manifolds, 3-fold irregular covering spaces, 2-fold cyclic covering spaces, handle presentation, ribbons, exotic involutions, Mazur manifolds, knots, strongly-invertible knots
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society