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Conditions for Solution to Pullback Equation on Forms?

  1. Jan 23, 2014 #1

    WWGD

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    Hi, All:

    I have a quotient map given by the mapping torus (S,h) , where S is a compact surface with nonempty boundary, and h: S→S is a homeomorphism. Let I=[0,1].
    The mapping torus ## S_h## of the pair (S,h) is defined as the quotient q: $$ q:S \times I/~$$ , where (x,0)~(h(x),1), i.e., we glue ## S \times I## along h.

    Now, let ω be a 1-form on S . I'm trying to see under what conditions we can solve the pullback equation:
    $$ q* β =ω $$ ,

    i.e., I want to find a form β on ## S_f## which pulls-back to ω . Now, the problem is that the induced map ## q*: Hm( S \times I/~)→ S \times I##on cohomology is not necessarily onto -- q is not a homeomorphism, for one, tho the gluing map is almost as nice as can be, since it is a homeomorphism.

    Can anyone think of conditions on ω under which there is a form β with ω= q* β ?

    Thanks.
     
    Last edited: Jan 23, 2014
  2. jcsd
  3. Jan 28, 2014 #2

    WWGD

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    Never mind, thanks, I got it.
     
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