Hopfian and strongly hopfian manifolds

Young Im; Yongkuk Kim

Fundamenta Mathematicae (1999)

  • Volume: 159, Issue: 2, page 127-134
  • ISSN: 0016-2736

Abstract

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Let p: M → B be a proper surjective map defined on an (n+2)-manifold such that each point-preimage is a copy of a hopfian n-manifold. Then we show that p is an approximate fibration over some dense open subset O of the mod 2 continuity set C’ and C’ ∖ O is locally finite. As an application, we show that a hopfian n-manifold N is a codimension-2 fibrator if χ(N) ≠ 0 or H 1 ( N ) ≅ ℤ 2

How to cite

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Im, Young, and Kim, Yongkuk. "Hopfian and strongly hopfian manifolds." Fundamenta Mathematicae 159.2 (1999): 127-134. <http://eudml.org/doc/212324>.

@article{Im1999,
abstract = {Let p: M → B be a proper surjective map defined on an (n+2)-manifold such that each point-preimage is a copy of a hopfian n-manifold. Then we show that p is an approximate fibration over some dense open subset O of the mod 2 continuity set C’ and C’ ∖ O is locally finite. As an application, we show that a hopfian n-manifold N is a codimension-2 fibrator if χ(N) ≠ 0 or $H_1(N) ≅ ℤ_2$},
author = {Im, Young, Kim, Yongkuk},
journal = {Fundamenta Mathematicae},
keywords = {strongly Hopfian manifold; approximate fibration; mod 2 continuity set; codimension-2 fibrator; Hopfian group; hyper-Hopfian group; residually finite group; proper map},
language = {eng},
number = {2},
pages = {127-134},
title = {Hopfian and strongly hopfian manifolds},
url = {http://eudml.org/doc/212324},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Im, Young
AU - Kim, Yongkuk
TI - Hopfian and strongly hopfian manifolds
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 2
SP - 127
EP - 134
AB - Let p: M → B be a proper surjective map defined on an (n+2)-manifold such that each point-preimage is a copy of a hopfian n-manifold. Then we show that p is an approximate fibration over some dense open subset O of the mod 2 continuity set C’ and C’ ∖ O is locally finite. As an application, we show that a hopfian n-manifold N is a codimension-2 fibrator if χ(N) ≠ 0 or $H_1(N) ≅ ℤ_2$
LA - eng
KW - strongly Hopfian manifold; approximate fibration; mod 2 continuity set; codimension-2 fibrator; Hopfian group; hyper-Hopfian group; residually finite group; proper map
UR - http://eudml.org/doc/212324
ER -

References

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  1. [1] G. Baumslag and D. Solitar, Some two-generator and one relator non-hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199-201. Zbl0108.02702
  2. [2] N. Chinen, Manifolds with nonzero Euler characteristic and codimension-2 fibrators, Topology Appl. 86 (1998), 151-167. Zbl0930.57020
  3. [3] N. Chinen, Finite groups and approximate fibrations, ibid., to appear. 
  4. [4] D. S. Coram and P. F. Duvall, Approximate fibrations, Rocky Mountain J. Math. 7 (1977), 275-288. Zbl0367.55019
  5. [5] D. S. Coram and P. F. Duvall, Approximate fibrations and a movability condition for maps, Pacific J. Math. 72 (1977), 41-56. Zbl0368.55016
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  7. [7] R. J. Daverman, Submanifold decompositions that induce approximate fibrations, Topology Appl. 33 (1989), 173-184. Zbl0684.57009
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  10. [10] R. J. Daverman, 3-manifolds with geometric structure and approximate fibrations, Indiana Univ. Math. J. 40 (1991), 1451-1469. Zbl0739.57007
  11. [11] J. C. Hausmann, Geometric Hopfian and non-Hopfian situations, in: Lecture Notes in Pure and Appl. Math. 105, Marcel Dekker, New York, 1987, 157-166. 
  12. [12] J. C. Hausmann, Fundamental group problems related to Poincaré duality, in: CMS Conf. Proc. 2, Amer. Math. Soc., Providence, R.I., 1982, 327-336. Zbl0555.57005
  13. [13] J. Hempel, 3-manifolds, Ann. of Math. Stud. 86, Princeton Univ. Press, Princeton, N.J., 1976. 
  14. [14] R. Hirshon, Some results on direct sum of hopfian groups, Ph.D. Dissertation, Adelphi Univ., 1967. 
  15. [15] Y. H. Im, Products of surfaces that induce approximate fibrations, Houston J. Math. 21 (1995), 339-348. Zbl0841.57031
  16. [16] Y. Kim, Strongly hopfian manifolds as codimension-2 fibrators, Topology Appl., to appear. Zbl0930.57021
  17. [17] Y. Kim, Manifolds with hyperhopfian fundamental group as codimension-2 fibrators, ibid., to appear. Zbl0947.57025
  18. [18] A. N. Parshin and I. R. Shafarevich, Algebra VII, Encyclopaedia Math. Sci. 58, Springer, 1993. 
  19. [19] J. Roitberg, Residually finite, hopfian and co-hopfian spaces, in: Contemp. Math. 37, Amer. Math. Soc., 1985, 131-144. Zbl0562.55008

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