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Free homotopy

  1. May 12, 2009 #1
    Hello,

    Here is a short lemma:

    A path-connected space X is simply-connected iff any two loops in X are free homotopic.

    My question is whether it is allowed to use a straight-line homotopy straight away in order to construct a free homotopy? For example, let u and v be two loops and w is a curve from point a to point b. Then:


    [itex]
    \begin{equation} H_f(t,s):=
    \begin{cases} (1-3s)u(t)+3sa,\ \text{for $0\leqslant{}s\leqslant\frac{1}{3}$}; \\
    w(3s-1),\ \text{for $\frac{1}{3}\leqslant{}s\leqslant\frac{2}{3}$}; \\
    (3-3s)b+(3s-2)v(t),\ \text{for $\frac{2}{3}\leqslant{}s\leqslant1$}.
    \end{cases}
    \end{equation}
    [/itex]
    (there is an error in the latex-output: "0" instead of "sh")
    That would actually do, wouldn't it? I mean a Nullhomotopy is not necesserally a straight-line homotopy. So maybe it's a loss of generality?
     
  2. jcsd
  3. Jun 2, 2009 #2
    What if your space is X=S^2? Wouldn't your homotopy travel outside X?
     
  4. Jun 2, 2009 #3

    morphism

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    Science Advisor
    Homework Helper

    You're not guaranteed that "x+y" makes sense for x and y in X.
     
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