Results 1 to 2 of 2

Thread: Contractible space

  1. #1
    Nov 2008

    Contractible space

    I' working on this problem, and I got stuck in the middle. Could someone give me a hand? Let X be a contractible space. I'd like to show that X is simply connected. I already showed that X is path connected. I need to show for every x_0 \in  X, \pi_1(X,x_0) is the trivial group. So, it suffices to show every loop based at x_0 is path homotopic to the path e: I\to X defined by e(x)=x_0 for every x\in X

    Let \alpha be a loop based at fixed x_0 \X. X is contractible implies there exists a homotopy H: X \times I \to X such that H(x,0)=x and H(x,1)=x_0. I tried to construct a path homotopy from \alpha to e using H, but I don't think mine works. I define G(x,t)= H(\alpha(x),t). Then G(x,0)=H(\alpha(x),0)=\alpha(x) and G(x,1)=H(\alpha(x),1)=x_0. But this is only a homotopy, I want H(0,t)=H(1,t)=x_0. I can't get it with this map.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Mar 2010
    Beijing, China
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Question on null space/column space/row space of a matrix
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Dec 1st 2011, 02:47 PM
  2. Replies: 5
    Last Post: Aug 16th 2011, 03:52 PM
  3. Replies: 2
    Last Post: Jul 8th 2011, 03:16 PM
  4. Banach space with infinite vector space basis?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Mar 24th 2011, 07:23 PM
  5. Replies: 15
    Last Post: Jul 23rd 2010, 12:46 PM

Search Tags

/mathhelpforum @mathhelpforum