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On transversality

Published online by Cambridge University Press:  20 January 2009

J. W. Bruce
J. W. Bruce
Affiliation:
School of Mathematics, University of Newcastle Upon Tyne, Newcastle Upon Tyne NE1 7RU
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The notion of transversality has proved of immense value in differential topology. The Thom transversality lemma and its many variants show that transversality is a dense,and often open, property. In one parameter families the occurrence of non-transversality is inevitable; for example one cannot pull two linked curves in ℝ3 apart without a non transverse intersection. The aim of this note is to prove the following. In any generic family of mappings each map in the family fails to satisfy some fixed transversality conditions at worst at isolated points, and even at these points in rather special sorts of way. So, returning to the above example, given two space curves C1 and C2 without a (necessarily non-transverse) intersection we expect, in any genericisotopy of C2, that it will meet C1 if at all, at isolated points In particular generically we do not expect C1 and C2, any time, to have an arc in common

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 1 , February 1986 , pp. 115 - 123
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

REFERENCES

1.Golubitsky, M. and Guillemfn, V., Stable mappings and their singularities, Graduate Texts in Mathematics 14 (Springer-Verlag, 1973).CrossRefGoogle Scholar
2.Guillemin, V. and Pollack, A., Differential Topology (Prentice Hall, 1974).Google Scholar
3.Kurland, H. and Robbin, J., Infinite Codimension and Transversality, Dynamical Systems, Warwick, Lecture Notes in Mathematics 468 (Springer 1974).Google Scholar
4.Loolienga, E. J. N., Isolated Singular Points on Complete Intersections, London Math. Soc. Lecture Note Series 77 (Cambridge Univ. Press, 1984).Google Scholar
5.Martinet, J., Singularities of Smooth Functions and Maps, London Math. Soc. Lecture Note Series 58 (Cambridge Univ. Press, 1982).Google Scholar
6.Wall, C. T. C., Finite determinancy of smooth mappings, Bull. London Math. Soc. 13 (1981), 481–539.CrossRefGoogle Scholar
7.Wall, C. T. C., Geometric Properties of Generic Differentiable Manifolds in Geometry and Topology, Vol. Ill, Lecture Notes in Maths. 597 (Springer, 1976).Google Scholar