Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is not empty) is a submanifold of $X$ with dimension $\dim U+\dim V-n$. Now if $X$ is a Whitney stratification and $U,V$ are Whitney substratifications, i.e. $U$, $V$ are subsets of $X$ with Whitney stratification structure, and each stratum of $U$ and $V$ contained in a single stratum of $X$. Assume that $U$ is transverse to $V$ in $X$, i.e. we require the usual transversal conditions on each stratum, then is it true that $U\cap V$ is a Whitney substratification of $X$ with dimension $\dim U + \dim V - n$? Where can I find some reference on this topic?

share|improve this question
add comment

2 Answers

I wanted to just put a comment but mathoverflow did not allow me.. hope it will soon.

The answer is yes and in fact there are better results.

Check this article:

P. Orro and D. Trotman, 'Regularity of the transverse intersection of two regular stratifications', London Mathematical society, Lecture Note Series 380, Cambridge University Press (2010), p. 298-304.

share|improve this answer
add comment

The Answer is yes. For details see Chap. I, $\S 1$ of

Topological Stability of Smooth Mappings, Lecture Notes in Mathematics, vol. 552, Springer Verlag, 1976.

The answer can be found in Proposition (1.3) Chap.I of the above reference.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.