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Manifolds with finite first homology as codimension $ 2$ fibrators


Author: Robert J. Daverman
Journal: Proc. Amer. Math. Soc. 113 (1991), 471-477
MSC: Primary 55R65; Secondary 54B15, 57M25, 57N12, 57N15, 57N65
DOI: https://doi.org/10.1090/S0002-9939-1991-1086581-0
MathSciNet review: 1086581
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Abstract: Given a map $ f:M \to B$ defined on an orientable $ (n + 2)$-manifold with all point inverses having the homotopy type of a specified closed $ n$-manifold $ N$, we seek to catalog the manifolds $ N$ for which $ f$ is always an approximate fibration. Assuming $ {H_1}(N)$ finite, we deduce that the cohomology sheaf of $ f$ is locally constant provided $ N$ admits no self-map of degree $ d > 1$ when $ {H_1}(N)$ has a cyclic subgroup of order $ d$. For manifolds $ N$ possessing additional features, we achieve the approximate fibration conclusion.


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  • [CD1] D. S. Coram and P. F. Duvall, Approximate fibrations, Rocky Mountain J. Math. 7 (1977), 275-288. MR 0442921 (56:1296)
  • [CD2] -, Approximate fibrations and a movability condition for maps, Pacific J. Math. 72 (1977), 41-56. MR 0467745 (57:7597)
  • [Da] R. J. Daverman, Submanifold decompositions that yield approximate fibrations, Topology Appl. 33 (1989), 173-184. MR 1020279 (91d:57013)
  • [DW] R. J. Daverman and J. J. Walsh, Decompositions into codimension two manifolds, Trans. Amer. Math. Soc. 288 (1985), 273-291. MR 773061 (87h:57019)
  • [Gr] M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), Math. Sciences Research Inst. Publ., Springer-Verlag, New York and Berlin, 1987, pp. 75-263. MR 919829 (89e:20070)
  • [Ha] J.-C. Hausmann, Geometric Hopfian and non-Hopfian situations, Geometry and Topology, (C. McCrory and T. Shifrin, eds.), Lecture Notes in Pure Appl. Math., vol. 105, Marcel Dekker, New York, 1987, pp. 157-166. MR 873292 (88f:57031)
  • [He1] J. Hempel, $ 3$-Manifolds, Ann. of Math. Stud., No. 86, Princeton Univ. Press, Princeton, NJ, 1976. MR 0415619 (54:3702)
  • [He2] -, Residual finiteness for $ 3$-manifolds, Combinatorial Group Theory and Topology (S. M. Gersten and J. R. Stallings, eds.), Ann. of Math. Stud., No. 111, Princeton Univ. Press, 1987, pp. 379-396.
  • [Sc] G. P. Scott, The geometries of $ 3$-manifolds, Bull. London. Math. Soc. 15 (1983), 401-487. MR 705527 (84m:57009)
  • [Sp] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1086581-0
Keywords: Approximate fibration, approximate lifting, continuity set, codimension 2 fibrator, geometric structure, Hopfain group, mapping degree
Article copyright: © Copyright 1991 American Mathematical Society

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