While proving that the quotient space B ¯ n / S n − 1 is homeomorphic to S n , I needed to construct a continuous function p : B ¯ n → S n . I figured that by fixing two points R = ( 0 , 0 , … , r ) and L = ( 0 , 0 , … , − r ) in the closed n-ball B ¯ n of radius r, I could then use the function p ( x ) = { ( x 1 , … , x n , r 2 − | | x | | 2 ) : ( x n − r ) 2 ≤ ( x n + r ) 2 ( x 1 , … , x n , − r 2 − | | x | | 2 ) : ( x n + r ) 2 &lt; ( x n − r ) 2 (where | | x | | is the standard norm in R n ) to determine the sign of the ( n + 1 )th component of the image of x ∈ B ¯ n . However, knowing that the closed unit interval can be mapped to the circle by the parametrization x ↦ ( cos ⁡ ( 2 π x ) , sin ⁡ ( 2 π x ) ), I can't help but to wonder whether some similar construction could be made from B ¯ n to S n ?That is, do you happen to know some nice/comfortable continuous mappings from a general closed n-ball to n-sphere, similar to the parametrization of a unit circle by a unit interval?

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While proving that the quotient space             B      ¯        n        /        S          n      −      1       is homeomorphic to       S    n  , I needed to construct a continuous function   p  :            B      ¯        n    →      S    n  . I figured that by fixing two points   R  =  (  0  ,  0  ,  …  ,  r  ) and   L  =  (  0  ,  0  ,  …  ,  −  r  ) in the closed   n-ball             B      ¯        n   of radius   r, I could then use the function   p  (  x  )  =      {                            (                      x            1                    ,          …          ,                      x            n                    ,                                    r              2                        −                          |                                      |                        x                          |                                                      |                            2                                )          :          (                      x            n                    −          r                      )            2                    ≤          (                      x            n                    +          r                      )            2                                                           (                      x            1                    ,          …          ,                      x            n                    ,          −                                    r              2                        −                          |                                      |                        x                          |                                                      |                            2                                )          :          (                      x            n                    +          r                      )            2                    &amp;lt;          (                      x            n                    −          r                      )            2                                  (where       |        |    x      |        |   is the standard norm in             R        n  ) to determine the sign of the   (  n  +  1  )th component of the image of   x  ∈            B      ¯        n  . However, knowing that the closed unit interval can be mapped to the circle by the parametrization   x  ↦  (  cos  ⁡  (  2  π  x  )  ,  sin  ⁡  (  2  π  x  )  ), I can&#039;t help but to wonder whether some similar construction could be made from             B      ¯        n   to       S    n  ?That is, do you happen to know some nice/comfortable continuous mappings from a general closed   n-ball to   n-sphere, similar to the parametrization of a unit circle by a unit interval?

recajossikpfmq · Accepted Answer

There is no map that &quot;wraps&quot; the Cartesian space around a sphere analogous to the covering of the unit circle by a line (because the sphere is simply-connected in dimension greater than 1), but there is the map defined by  x  ↦            (        sin  ⁡  (      |    x      |    )      x                  |            x              |              ,  cos  ⁡  (      |    x      |    )            )        ,which sends the open ball of radius π about the origin diffeomorphically to the unit sphere with the &quot;south pole&quot;   (  0  ,  …  ,  0  ,  −  1  ) removed, and which collapses the boundary of the ball to the south pole. (This map is closely related to the Riemannian exponential map of a round sphere.)

While proving that the quotient space <mover> B <mo accent="false">¯<

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