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Fundamental Group

  1. Apr 21, 2010 #1
    I am doing some revision and trying to do fundamental groups and I was wondering if the fundamental group of the following space is {1} i.e. all loops based p are homotopic.

    fundamental group of (X,p) = D^2\{(x,0) : 0<=x<=1} where p=(-1,0)
     
  2. jcsd
  3. Apr 21, 2010 #2
    Yes. Just consider a loop [itex]f : [0,1] \to X[/itex]. This is homotopic to the constant map c(t)=p by the usual straight-line homotopy [itex](t,i) \mapsto (1-i)f(t) + ip[/itex]. You can verify yourself that this works.

    The intuitive idea when dealing with fairly nice subsets of [tex]\mathbb{R}^2[/tex] is that the fundamental group is trivial if and only if the space has no holes (since then you can wrap a loop around that hole).
     
  4. Apr 22, 2010 #3
    Ok thanks I get that as I thought.

    I am trying to show that the fundamental group of a product of Topological Spaces is isomorphic to the product of fundamental groups:

    pi1(X x Y , (p,q)) -> pi1(X,p) x pi1(Y,q)

    I can understand this but want to construct the isomorphism and am struggling?
     
  5. Apr 22, 2010 #4
    Just construct it in the obvious way. Given two loops [itex]f_1 : I \to X[/itex] and [itex]f_2 : I \to Y[/itex] you can construct a path [itex]I \to X \times Y[/itex] where [itex]t \mapsto (f_1(t),f_2(t))[/itex]. Conversely given a path [itex]f : I \to X \times Y[/itex] you can construct paths [itex]I \to X[/itex] and [itex]I \to Y[/itex] by projecting onto X and Y. You can show that both of these give homomorphisms of the fundamental groups and that they are inverses.
     
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