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Construct an explicit isomorphism

  1. Dec 24, 2013 #1
    $\Bbb{R}P^1$ bundle isomorphic to the Mobius bundle

    I'm trying to construct an explicit isomorphism from ##E = \{([x], v) : [x] ∈ \Bbb{R}P^1, v ∈ [x]\}## to ##T = [0, 1] × R/ ∼## where ##(0, t) ∼ (1, −t)##. I verified that ##\Bbb{R}P^1## is homeomorphic to ##\Bbb{S}^1## which is homeomorphic to ##[0,1]/∼## where ##0∼1##. So this is the map I have in my mind: ##([x],v)\to (x,(1-x)v+xe^v)##. Does that work? It doesn't look very natural.
     
  2. jcsd
  3. Jan 4, 2014 #2

    WWGD

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    How about pulling back the bundle using the homeomorphism?
     
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