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Dylan Wilson
Dylan Wilson
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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. John Francis sent me a nice proof in the affirmative if I'm willing to assume that everything in sight is an open submanifold of a compact one, so for all practical purposes I have what I need. However, I'd still like to know if this result is true more generally, and I can find nothing in the literature.

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."

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. John Francis sent me a nice proof in the affirmative if I'm willing to assume that everything in sight is an open submanifold of a compact one, so for all practical purposes I have what I need. However, I'd still like to know if this result is true more generally, and I can find nothing in the literature.

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."

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."

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Dylan Wilson
Dylan Wilson
  • 13.5k
  • 9
  • 64
  • 108

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. John Francis sent me a nice proof in the affirmative if I'm willing to assume that everything in sight is an open submanifold of a compact one, so for all practical purposes I have what I need. However, I'd still like to know if this result is true more generally, and I can find nothing in the literature.

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."