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Finding information on S^3

  1. Mar 26, 2013 #1
    Hello everyone,

    I am looking for some resources that could help me understand interesting properties of S^3 (or S^n really). I am doing a small project for my topology course and in my studies I have found that I lack knowledge about S^3. Are there any good texts or online lecture notes somewhere that cover this topic? In particular I might find certain quotient spaces of S^3 useful. Thus far I have been out of luck both online and at my school library.
    Thank you all in advance for any suggestions!

  2. jcsd
  3. Mar 27, 2013 #2


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    The Hopf fibration of ##S^3## illustrates a lot of important geometry, while also being relevant to certain quotient spaces, like ##S^3/S^1 = S^2## or the Lens spaces ##S^3/(\mathbb{Z}/p)##. That wiki seems to be very detailed, but Nakahara's Geometry, Topology and Physics should also have a review, along with all of the background that is relevant.
  4. Mar 27, 2013 #3


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    On R^4 there is a multiplicative group structure that make R^4 into a skew field Q called the quaternions.

    Sitting naturally inside R^4 as the unit sphere, one can check that S^3 is a subgroup for this multiplicative group structure on R^4.

    Therefor, S^3 has a natural group structure. This group structure is furthermore compatible with its topology in the sense that the group operations of multiplication and taking inverse are continuous. We say S^3 is a topological group. In fact, S^3 is, aside from S^1, the only sphere admitting a topological group structure!

    In looking for group acting on S^3, you may thus look for multiplicative subgroups of S^3, or of Q itself and intersect them with S^3.
  5. Mar 28, 2013 #4
    Excellent! This is precisely the kind of thing I was hoping for! The Hopf Fibration looks very interesting and I believe I will spend some time learning about it now. Also, thanks for the book recommendation, I will check it out soon. Thanks again!

    Oh! Dang! I feel as though I should have known this since I know of the definition of S^3 in terms of quaternions. Thank you for pointing it out though! I have a feeling this too will be very useful for me. Thanks!
  6. Mar 28, 2013 #5


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    look at the book, "Three Dimensional Geometry and Topology" Voume 1 by William Thurston
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