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S1 isomorphic to S1/Z2?

  1. Nov 11, 2009 #1
    S1 isomorphic to S1/Z2??

    1. The problem statement, all variables and given/known data
    This isn't really a homework problem, but whatever. So in my textbook, it says that "of course [tex]S^1/Z_2[/tex] is isomorphic to [tex]S^1[/tex]." I want to know why.


    2. Relevant equations



    3. The attempt at a solution
    Is [tex]S^1[/tex] even a group? Well I guess it can be a group, with multiplication as the operation ([tex]S^1[/tex] viewed as a subset of [tex]C[/tex]). But then the only map I can think of between the two is the projection [tex]p:S^1\rightarrow S^1/Z_2[/tex], where [tex]x\in S^1[/tex] is mapped to its obit under the action of [tex]Z_2[/tex]. But this map is clearly not injective since [tex]p(x)=p(-x)[/tex] for any [tex]x \in S^1[/tex].

    If this is a typo in my book, then are the sets homeomorphic (maybe they wanted to use the symbol for homeomorphism instead of isomorphism)? What then is the map between them?
     
  2. jcsd
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