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N-sphere and its equator

  1. Sep 30, 2009 #1

    TFT

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    For [tex]n\geq 2[/tex], is there a smooth map [tex]f: S^n\rightarrow E[/tex] ([tex]E[/tex] is the equator of [tex]S^n[/tex]) which has the property that the restriction of [tex]f[/tex] to [tex]E[/tex] is a diffeomorphism from [tex]E[/tex] to [tex]E[/tex]?
     
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  3. Sep 30, 2009 #2

    quasar987

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    Do you know how to prove using homology theory that there is no retraction from the n-disk to its boundary? It is the same here:

    Suppose such a function f exists. Call g the restriction of f to the northern hemisphere of S^n wich we will regard as the n-disk D^n. Call h the restriction of g to E. By hypothese, h is a diffeo and so has degree ±1. The following diagram commute:

    D^n<---E
    |............|
    |g..........| h
    |............|
    E<-------|

    that is, g o i = h where i is the inclusion of E in D^n. Passing to the realm of (n-1)-degree homology, the above diagram becomes the following comutative diagram

    0<------Z
    |............|
    |............| ±1
    |............|
    Z<-------|

    which is absurd.
     
  4. Sep 30, 2009 #3

    TFT

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    Thanks a lot!
     
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