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Degree of a Map

  1. Aug 24, 2011 #1

    WWGD

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    Hi: More on Prelims:

    We have a map f: S^3 -->S^3 ; S^3 is the 3-sphere , given by:

    (x1,x2,x3,x4)-->(-x2,-x3,-x4,-x1).

    We're asked to find its degree, and to determine if f is homotopic to the identity.

    I computed that f^4 ( i.e., fofofof ) is the identity, and we have that degree is

    multiplicative, so that deg(f)^4=1 , so that we can narrow the choices to degf=+/- 1.

    Now, I know we can also compute the induced map on top homology, and see if f is

    preserving- or reversing- orientation, but I cannot tell which it is; I am trying to

    use a 4-simplex , and see if this map preserves or reverses the orientation, but

    I cannot see it clearly.

    Another choice I am thinking of using is that f , seen as a map from R^4 to itself,

    is a linear map, so that we can calculate Det f , to see if f reverses or preserves

    orientation, and then maybe argue that f restricted to the subspace S^3 (unfortunately,

    S^3 is not a subspace of R^4 ) has the same effect of preserving/reversing

    orientation. Any Ideas/Suggestions/Comments?

    see well how to do that
     
  2. jcsd
  3. Aug 24, 2011 #2
    You're on the right track: you calculate the determinant of the matrix in R^4, then you check whether it flips the sign on the radial vector field.
     
  4. Aug 24, 2011 #3
    I think you can also use this fact:

    Given two maps f,g: S^n --->S^n

    such that f(x)=/-g(x), then f,g are homotopic thru:

    H(x,t)=[(1-t)f(x)+tg(x)]/|| (1-t)f(x)+tg(x) ||

    So, your f, and Id are then homotopic and so degf=1
     
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