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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

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