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Dylan Wilson
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Proper maps and transversality

I'll begin with the question, which is intrinsically interesting:

Let M be a manifold with some submanifold Y. Suppose that $W \rightarrow M$ is a smooth, proper map. Does there exist another map $W \rightarrow M$ homotopic to the original that is ALSO proper and transverse to the submanifold Y?

Let me note that I am definitely not assuming the manifolds are compact.

Why I care: This question came up while thinking about the geometric description of complex cobordism given by Quillen in "Elementary proofs of some results of cobordism theory using Steenrod operations." I have a geometric description of the coboundary map in the Mayer-Vietoris sequence but as of now it relies on the answer to the above question being "yes."

Dylan Wilson
  • 13.5k
  • 9
  • 64
  • 108