Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A Hurewicz-type theorem for approximate fibrations


Authors: D. S. Coram and P. F. Duvall
Journal: Proc. Amer. Math. Soc. 78 (1980), 443-448
MSC: Primary 55R65; Secondary 55P05
DOI: https://doi.org/10.1090/S0002-9939-1980-0553392-4
MathSciNet review: 553392
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper concerns conditions on point inverses which insure that a mapping between locally compact, separable, metric ANR's is an approximate fibration. Roughly a mapping is said to be $ {\pi _i}$-movable [respectively, $ {H_i}$-movable] provided that nearby fibers include isomorphically into mutual neighborhoods on $ {\pi _i}$ [resp. $ {H_i}$]. An earlier result along this line is that $ {\pi _i}$-movability for all i implies that a mapping is an approximate fibration. The main result here is that for a $ U{V^1}$ mapping, $ {\pi _i}$-movability for $ i \leqslant k - 1$ plus $ {H_k}$- and $ {H_{k + 1}}$-movability imply $ {\pi _k}$-movability of the mapping. Hence a $ U{V^1}$ mapping which is $ {H_i}$-movable for all i is an approximate fibration. Also, if a $ U{V^1}$ mapping is $ {\pi _i}$-movable for $ i \leqslant k$ and k is at least as large as the fundamental dimension of any point inverse, then it is an approximate fibration. Finally, a $ U{V^1}$ mapping $ f:{M^m} \to {N^n}$ between manifolds is an approximate fibration provided that f is $ {\pi _i}$-movable for all $ i \leqslant \max \{ m - n,\tfrac{1}{2}(m - 1)\} $.


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

  • [CD1] D. Coram and P. Duvall, Approximate fibrations, Rocky Mountain J. Math. 7 (1977), 275-288. MR 0442921 (56:1296)
  • [CD2] -, Approximate fibrations and a movabilitity condition for maps, Pacific J. Math. 72 (1977), 41-56. MR 0467745 (57:7597)
  • [E] R. D. Edwards, A locally compact ANR is a Hilbert cube manifold factor (to appear).
  • [L1] R. C. Lacher, Cell-like mappings. I, Pacific J. Math. 30 (1969), 717-731. MR 0251714 (40:4941)
  • [L2] R. C. Lacher, Cellularity criteria for maps, Michigan Math. J. 17 (1970), 385-396. MR 0279818 (43:5539)
  • [LM] R. C. Lacher and D. R. McMillan, Partially acyclic mappings between manifolds, Amer. J. Math. 94 (1972), 246-266. MR 0301743 (46:898)
  • [S] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55R65, 55P05

Retrieve articles in all journals with MSC: 55R65, 55P05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0553392-4
Keywords: Approximate fibration, UV property
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society