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."



Proper maps and transversality
Regarding the coboundary map in complex cobordism, I think that this was done by Dold in "Geometric Cobordism and the Fixed Point Transfer" (at least for oriented cobordism) in 2.10.


