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Thread: Algebraic Topology-The Brouwer Degree

  1. #1
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    Algebraic Topology-The Brouwer Degree

    Hi there,

    I'll be glad to receive some guidance to the attached question.
    This is what I tried:

    As for part (a): the chain $\displaystyle \sum_{i=0}^{6} [a_i, a_{i+1} ] +[a_7,a_0] $ which generates $\displaystyle sd K $ is being mapped to the generator $\displaystyle 2 ( [a_0,a_2 ] +[a_2,a_4] + [a_4,a_6]+[a_6,a_0] ) $ which generates K. As far as I know, it implies that $\displaystyle deg f =2$, as we need.

    My problem is with part (b) :
    If we take one of the $\displaystyle a_i$ ' s and denote by $\displaystyle \pi$ the radial projection, we get that:
    $\displaystyle h( \pi ( a_i) ) = f(a_i) $ . But why do we need this radial projection in the first place? isn't $\displaystyle |K|$ the unit sphere anyway? How can I define this radial projection explicitly for all the sphere and show that it's an homotopy? ( I thought of making it a linear homotopy, but it will only imply that it's an homotopy for every quadrant separately... Is this radial projection just $\displaystyle \frac{f(x)}{||f(x)||} $?" ) .

    Please help me understand where am I wrong and solve this (b) part .


    Thanks in advance !
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