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Maps with Discrete Branch Sets Between Manifolds of Codimension One

Published online by Cambridge University Press:  20 November 2018

J. G. Timourian
J. G. Timourian*
Affiliation:
Syracuse University, Syracuse, New York; University of Tennessee, Knoxville, Tennessee
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Let M n and Np be separable manifolds of dimensions n and p, respectively, with n ≧ p, and without boundary unless otherwise indicated. A mapƒ: M → N is proper if, for each compact set K ⊂ N, f –l(K) is compact. It is topologically equivalent to g: X → Y if there exist homeomorphisms α of M onto X and β of N onto Y such that βƒα–1 = g. At x ∈ M, ƒ is locally topologically equivalent to g if, for every neighbourhood W ⊂ M of x, there exist neighbourhoods U ⊂ W of x and V of ƒ(x) such that ƒ | U: U → V is topologically equivalent to g.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 660 - 668
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Church, P. T., Differentiable open maps on manifolds, Trans. Amer. Math. Soc. 109 (1963), 87–100.Google Scholar
2. Church, P. T., Factorization of differentiable maps with branch set dimension at most n — 3, Trans. Amer. Math. Soc. 115 (1965), 370–387 Google Scholar
3. Church, P. T. and Hemmingsen, E., Light open maps on n-manifolds, Duke Math. J. 27 (1960), 527–536.Google Scholar
4. Church, P. T. and Hemmingsen, E., Light open maps on n-manifolds. Ill, Duke Math. J. 30 (1963), 379–390 Google Scholar
5. Church, P. T. and Timourian, J. G., Fiber bundles with singularities, J. Math. Mech. 18 (1968), 71–90.Google Scholar
6. Dieudonné, J., Foundations of modern analysis (Academic Press, New York, 1960).Google Scholar
7. Hurewicz, W. and Wallman, H., Dimension theory, 2nd ed. (Princeton Univ. Press, Princeton, N.J., 1948).Google Scholar
8. Nathan, W. D., Open mappings on manifolds, Doctoral dissertation, Syracuse University, Syracuse, New York, 1968.Google Scholar
9. Palais, R. S., Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959), 125–141.Google Scholar
10. Steenrod, N., The topology of fiber bundles (Princeton Univ. Press, Princeton, N.J., 1951).Google Scholar
11. Stoi'low, S., Sur les transformations continues et la topologie des fonctions analytiques, Ann. Sci. École Norm. Sup. (III) 45 (1928), 347–382.Google Scholar
12. Timourian, J. G., Fiber bundles with discrete singular set, J. Math. Mech. 18 (1968), 61–70.Google Scholar
13. Timourian, J. G., Singular fiberings of manifolds, Doctoral dissertation, Syracuse University, Syracuse, New York, 1967.Google Scholar
14. Whyburn, G. T., Analytic topology, 2nd éd., Amer. Math. Soc. Colloq. Publ., Vol. 28 (Amer. Math. Soc, Providence, R.I., 1963).Google Scholar