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Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex.

Does $M$ admit a finite good cover? (i.e. a finite cover by contractible opens whose multiple intersections are also contractible)

I expect the answer to be yes, but I don’t see an argument as of now.

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  • $\begingroup$ Do you really need contractible opens, or do you accept open sets that are disjoint unions of contractibles? $\endgroup$
    – David Roberts ♦
    Commented Mar 28, 2019 at 23:11
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    $\begingroup$ @DavidRoberts Let’s say $M$ is connected. But yes I would content myself of a cover made my opens that are finite disjoint unions of contractibles, all whose multiple intersections are also finite disjoint unions of contractibles. $\endgroup$
    – John P.
    Commented Mar 28, 2019 at 23:31
  • $\begingroup$ Tom's answer here seems pretty close to giving a counter-example to what you want: mathoverflow.net/questions/48505/… $\endgroup$
    – Sean Lawton
    Commented Mar 29, 2019 at 0:43

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