1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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
    f ~ g
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Proof in Homotopy Theory
  1. Set theory, proof (Replies: 3)

  2. Graph theory trail proof (Replies: 11)

  3. Graph Theory Proof (Replies: 3)