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Is there an analogue of Ehresmann fibration theorem for (finite) CW complexes ?

Note is not true that an open surjective (necessary proper) cellular map of finite CW or simplicial complexes is necessarily a Serre fibration: ramified coverings of surfaces are counterexamples, as pointed out in comments.

Relevant questions are: Ehresmann fibration theorem for manifolds with boundary, and Ehresmann's fibration theorem in the C1 class, and Simplicial approximation of a fibration, and Which maps of simplicial sets geometrically realize to fibrations?.

user420620
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    A ramified cover between surfaces is open, and it is not a Serre fibration right? – Nicolast Dec 19 '21 at 20:04
  • I'm probably missing something but why say $z^2:\mathbb{C}\to\mathbb{C}$ is not a Serre fibration ? It does have the path lifting property, right, though ? By the way, I forgot to say "surjective" and now I added it. – user420620 Dec 20 '21 at 11:52
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    Its fibres do not have constant (weak) homotopy type. – Tyrone Dec 20 '21 at 12:12
  • sorry, of course... i'll edit the question accordingly. – user420620 Dec 20 '21 at 19:22

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I think it's not quite clear what you're looking for, but one possible thing to say is that being a Serre fibration is a local condition. Just as Ehresmann's theorem gives an infinitesimal condition which implies being a Serre fibration for manifolds, this principle gives you a local condition. Indeed, I suppose Ehresmann's theorem follows from this fact along with the implicit function theorem.

Also, regarding the formulation of the question, note that every Serre fibration bretween CW complexes is a Hurewicz fibration.

Tim Campion
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