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Srong topology, but really a question on covering spaces

  1. Oct 8, 2009 #1


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    1. The problem statement, all variables and given/known data

    Prove that the set of maps {f in Cs1(M,N)|f is a covering space} is open in the strong topology.

    2. Relevant equations

    The strong topology has as base neighborhoods sets of functions that are disjointly uniformly near f, along with their derivative, on compact subsets of a covering set of M

    3. The attempt at a solution

    So, I'm mostly having trouble with the covering space part. I understand how the strong topology works, and how proofs for things like the set of immersions, or the set of closed immersions, or the set of submersions is open work. But I understand those objects, and covering spaces are something I had to look up online.

    From what I understand, around each point x in N I have an open subset of N, U, such that f-1(U) is a bunch of disjoint sets each homeomorphic to U. I tried taking all of those as my cover of N, but then neither my cover of N nor M is locally finite and I wasn't able to figure out how to actually deduce anything. Then I tried saying take a locally finite chart of N {Vi}, then pick a point x in N. Around that point there is a U as above, and f-1(U) is disjoint sets homeomorphic to U. Intersecting U with all the Vi's that x lie in give something homeomorphic to a subset of Rn for some n (by the definition of a chart), and looking at f-1(U intersect with those Vi) is disjoint subsets in M that must be homeomorphic to that same open subset.

    Then if I pick a chart of M of {Ui such that f(Ui is a subset of Vi and Ki compact subsets that cover M, I can pick my ei small enough so that |f-g|<ei on Ki means that g(Ki) is a subset of Vi also. So my goal now is to find an open set around x that I can pull back to M via the inverse of g and get a bunch of homeomorphic copies of it. But I'm not really sure how to proceed at this point, since I don't know techniques to show that g is in fact a covering space, or what the criterion for it is besides the definition.

    TL;DR I need to know what is required to make something a covering space/what techinques are good to show something is a covering space
  2. jcsd
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