How to show that a transverse intersection is clean, but not conversely?

[huyichen]

How to show that a transverse intersection is clean, but not conversely?

 

[mathwonk]

definitions please? i assume you are discussing two manifolds inside another manifold, and that transverse means the two tangent spaces span the big tangent space.

so what does clean mean?

 

[huyichen]

If K and L are embedded manifold of M, and T_p(K intersect L)=T_p K intersect T_p L and K intersect L is again a embedded manifold , then we say K intersect L is clean

 

[mathwonk]

this link may be helpful, but it puzzles me a bit, as it seems to contain false statements.
http://books.google.com/books?id=Bv...AEwAw#v=onepage&q=clean intersections&f=false

 

[mathwonk]

then the proof seems trivial. i.e. the converse statement is trivial, and the truth of the forward statement seems to be the implicit function theorem.

see guillemin and pollack, chapter 1, page 27 ff..