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Nov 10, 2016 at 10:56 comment added David Roberts @SimonRose that is a special case of a project I'm vainly wishing to attract a masters student with (the fact my academic position is as tenuous as it can get isn't helping...) cf my question mathoverflow.net/questions/156142/…, which should cover orbifolds by taking almost free actions of compact Lie groups.
Nov 10, 2016 at 8:51 comment added Simon Rose Can't you just choose a fine enough refinement such that all intersections are contractible?
Nov 10, 2016 at 8:14 comment added Włodzimierz Holsztyński What about the intersections $\ U_i\cap U_j\ $ etc.
Nov 10, 2016 at 8:04 history answered Simon Rose CC BY-SA 3.0