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Circle x sphere = ?

  1. Jan 31, 2010 #1
    circle x sphere = ???

    Is the product space [tex]S^1 \times S^2[/tex] related (e.g. homeomorphic or homotopy equivalent) to a more familiar topological space? I am currently looking at maps from [tex]S^1 \times S^2[/tex] into other spaces, and I am having a really hard time visualizing what I am doing. Any thoughts appreciated.
     
  2. jcsd
  3. Feb 1, 2010 #2
    Re: circle x sphere = ???

    Since you can visualize [tex]S^1[/tex] and [tex]S^2[/tex] by themselves, you should be able to get good impression of the whole space. Or du you need to actually see it?

    How about this: Consider [tex]S^2[/tex] as the unit sphere [tex]|x|=1[/tex] in [tex]\mathbb{R}^3[/tex]. Then just make it thicker, and identify points on the outer edge with points on the inner edge (along the radius).

    Torquil
     
  4. Feb 1, 2010 #3
    Re: circle x sphere = ???

    Well, I am looking at smooth maps [tex]f: S^1 \times S^2 \rightarrow S^2[/tex]. Then [tex]f^{-1}(z)[/tex] is a union of circles, which may or may not be linked. How exactly would I go about visualizing knots in [tex]S^1 \times S^2[/tex], and especially their relative position to each other?
     
  5. Feb 1, 2010 #4
    Re: circle x sphere = ???

    Sorry, I don't have anything useful to say about that.

    Torquil
     
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