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Proof in Homotopy Theory

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that if Y is contractible, then any two maps from X to Y are homotopic.


    I feel like I have a very, very, very sloppy proof :(

    2. Relevant equations



    3. The attempt at a solution
    Assume Y is contractible to a point y0 held fixed, then there is a map F:Y x [0,1] -> Y such that
    (i) F(y,0)=y0, for all y in Y
    (ii) F(y,1)=y, for all y in Y
    (iii) F(y0,t)=y0, for all t in [0,1]

    Let F(y,t) be the homotopy of Y to a point y0
    Claim: any map f:X -> Y is homotopic to y0 by the homotopy ft(x)=F(f(x),t)
    Proof of Claim: Since f(x)=y in Y, ft=F(y,t), where y is in Y and t is in [0,1],
    And since Y is contractible, f:X -> Y is homotopic to y0

    Therefore, any two maps from X to Y are homotopic.
     
  2. jcsd
  3. May 4, 2010 #2

    lanedance

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    Homework Helper

    I don't know much about homotopies but you reasoning seems ok, it seems pretty clear that any map f is homotopic to the constant to y0.

    The only assumption I can see in there, maybe worth checking you can use, is that a homotopy of functions is an equivalence class, ie.
    f ~ y0
    g ~ y0
    implies
    f ~ g
     
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