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

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Discussion Overview

The discussion centers on the mathematical concept of clean intersections of manifolds, specifically addressing how to demonstrate that a transverse intersection is clean while exploring the implications of the converse not being true. The scope includes theoretical aspects of differential geometry and manifold theory.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants seek definitions and clarifications regarding the terms "transverse" and "clean" in the context of manifold intersections.
  • One participant defines a clean intersection as occurring when the tangent spaces of the intersecting manifolds at a point span the tangent space of the larger manifold, and the intersection itself is an embedded manifold.
  • A participant references a link to a resource that they find puzzling, suggesting it may contain inaccuracies regarding clean intersections.
  • Another participant suggests that the proof of the forward statement regarding transverse intersections being clean is straightforward, while implying that the converse is trivial, referencing the implicit function theorem as a basis for their reasoning.
  • A specific reference to a textbook (Guillemin and Pollack) is made for further reading on the topic.

Areas of Agreement / Disagreement

Participants express differing levels of understanding and interpretation of the concepts involved, with no consensus reached on the definitions or implications of clean intersections versus transverse intersections.

Contextual Notes

There are unresolved definitions and assumptions regarding the terms used, as well as potential inaccuracies in referenced materials that may affect the clarity of the discussion.

huyichen
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How to show that a transverse intersection is clean, but not conversely?
 
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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?
 
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
 
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..
 
Last edited:

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