4

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.

R. van Dobben de Bruyn
  • 27,218
John P.
  • 180
  • Do you really need contractible opens, or do you accept open sets that are disjoint unions of contractibles? – David Roberts Mar 28 '19 at 23:11
  • 1
    @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. – John P. Mar 28 '19 at 23:31
  • Tom's answer here seems pretty close to giving a counter-example to what you want: https://mathoverflow.net/questions/48505/finite-type-vs-finite-dimensional-cohomology – Sean Lawton Mar 29 '19 at 00:43

0 Answers0