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Irreducible $ 3$-manifolds whose orientable covers are not prime


Author: W. H. Row
Journal: Proc. Amer. Math. Soc. 34 (1972), 541-545
MSC: Primary 57A10
DOI: https://doi.org/10.1090/S0002-9939-1972-0296947-2
MathSciNet review: 0296947
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Abstract: J. L. Tollefson has asked if every closed covering space of a prime 3-manifold is prime. In the present paper, the author provides a negative answer by constructing infinitely many topologically distinct, irreducible, closed 3-manifolds with the property that none of their orientable covering spaces are prime. These 3-manifolds are distinguished by the maximum number of disjoint, nonparallel, 2-sided projective planes that they contain. The author does not know if every closed covering space of a prime, orientable 3-manifold is prime.


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

  • [1] Wolfgang Haken, Some results on surfaces in 3-manifolds, Studies in Modern Topology, Math. Assoc. Amer., distributed by Prentice-Hall, Englewood Cliffs, N.J., 1968, pp. 39-98. MR 36 #7118. MR 0224071 (36:7118)
  • [2] -, Erratum for some results on surfaces in 3-manifolds (mimeographed notes).
  • [3] D. R. McMillan, Jr. and Harry Row, Tangled embeddings of one-dimensional continua, Proc. Amer. Math. Soc. 22 (1969), 378-385. MR 39 #7571. MR 0246267 (39:7571)
  • [4] J. W. Milnor, A unique decomposition theorem for 3-manifolds, Amer. J. Math. 84 (1962), 1-7. MR 25 #5518. MR 0142125 (25:5518)
  • [5] J. L. Tollefson, On 3-manifolds that cover themselves, Michigan Math. J. 16 (1969), 103-109. MR 39 #4853. MR 0243532 (39:4853)
  • [6] F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56-88. MR 36 #7146. MR 0224099 (36:7146)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0296947-2
Keywords: 3-manifold, irreducible, prime, covering space, incompressible, 2-sided projective plane, cube-with-a-knotted-hole
Article copyright: © Copyright 1972 American Mathematical Society

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