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Aug 24 at 21:19 comment added Grisha Taroyan It seems plausible because of your argument that the difference between cohomology with Schwartz coefficients and compactly supported cohomology obstructs the existence of a collar neighbourhood of the ends.
Aug 24 at 21:18 comment added Grisha Taroyan Thank you so much! This argument is quite illuminating. Unfortunately, I am interested in open subsets of $\mathbb{R}^n$ where the complement has an arbitrary topology. Thus, the case you treat is, in a sense, complimentary to what I'm usually considering. Thank you again!
Aug 5 at 13:27 history answered Igor Khavkine CC BY-SA 4.0