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Maps with Locally Flat Singular Sets

Published online by Cambridge University Press:  20 November 2018

J. G. Timourian
J. G. Timourian*
Affiliation:
The University of Tennessee, Knoxville, Tennessee
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A map f : M → N is topologically equivalent tog: X → Y if there exist homeomorphisms α: M → X and β: N → Y such that βfα–1 = g. At x ∊ M, f is locally topologically equivalent to g if, for every neighbourhood W ⊂ M of x, there exist neighbourhoods U ⊂ W of x and V of f(x) such that f|U: U → V is topologically equivalent to g.

1.1. Definition. Given a map f: M → N and x ∊ M, let F be the component of f–1(f(x)) containing x. The singular set Af is defined as follows: x ∊ M – Af if and only if there are neighbourhoods U of F and V of f(x) such that f| U: U → F is topologically equivalent to the product projection map of V × F onto V.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 916 - 921
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Antonelli, P. L., Montgomery-Samelson singular fiberings of spheres, Proc. Amer. Math. Soc. 22 (1969), 247–250.Google Scholar
2. Antonelli, P. L., Structure theory for Montgomery-Samelson fiberings between manifolds. I, Can. J. Math. 21 (1969), 170–179.Google Scholar
3. Borel, A., Seminar on transformation groups, Annals of Math. Studies, No. 46 (Princeton Univ. Press, Princeton, N. J., 1960).Google Scholar
4. Browder, W., Higher torsion in H-spaces, Trans. Amer. Math. Soc. 108 (1963), 353–375.Google Scholar
5. Church, P. T. and Timourian, J. G., Fiber bundles with singularities, J. Math. Mech. 18 (1968), 71–90.Google Scholar
6. Hu, S. T., Homotopy theory (Academic Press, New York, 1959).Google Scholar
7. Hurewicz, W. and Wallman, H., Dimension theory, 2nd ed. (Princeton Univ. Press, Princeton, N. J., 1948).Google Scholar
8. Spanier, E. H., Algebraic topology (McGraw-Hill, New York, 1966).Google Scholar
9. Steenrod, N., The topology of fibre bundles, Princeton Mathematical Series, Vol. 14 (Princeton Univ. Press, Princeton, N. J., 1951).Google Scholar
10. Timourian, J. G., Fiber bundles with discrete singular set, J. Math. Mech. 18 (1968), 61–70.Google Scholar
11. Timourian, J. G., Maps with discrete branch sets between manifolds of codimension one, Can. J. Math. 21 (1969), 660–668.Google Scholar
12. Timourian, J. G., Singular maps on manifolds (to appear in Duke Math. J.).Google Scholar
13. Whyburn, G. T., Analytic topology, 2nd éd., Amer. Math. Soc. Colloq. Publ., Vol. 28 (Amer. Math. Soc, Providence, R.I., 1963).Google Scholar
14. Wilder, R. L., Monotone mappings of manifolds. II, Michigan Math. J. 5 (1958), 19–23.Google Scholar