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Path homotopy, degree of maps

  1. Mar 18, 2009 #1
    Let f and g be loops in the unit circle S^1 with base point (1,0). We show if deg(f)=deg(g), then f and g are path homotopic.

    I know f,g: [0,1]-->S^1 and there must be unique paths f* and g* (lifting paths) s.t. f*(0)=0 and g*(0)=0 and s.t. pof*=f, pog*=g where p:R-->s^1 is defined by p(t)=(cos 2 pi t, sin 2 pi t).

    I also know that deg(f)= f*(1), deg(g)=g*(1) which are integers.

    I was thinking about coming up with a path homotopy F:[0,1]x[0,1]-->S^1 s.t. for each t in [0,1]; F(t,0)=f(t) and F(t,1)=g(t)
    for each s in [0,1]; F(0,s)=(1,0) and F(1,s)=(1,0).

    But I am not sure what path homotopy would work here.
  2. jcsd
  3. Mar 19, 2009 #2
    Hint: Construct a homotopy between f* and g* in R using convex combination. From this you can obtain a homotopy bewteen f and g.
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