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Homework Help: Difficult homotopy question

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data

    (i) Let [tex] A= (I\times I)/J[/tex] be the identification space of the unit square in which all points in the subspace [tex]J=(I\times \left\{1\right\})\cup(\left\{0,1\right\}\times I)[/tex] are identified. Use the circles [tex] C_t=\left\{(x,y)\in D^2|(x-t)^2+y^2=(1-t)^2 , t\in I\right\} [/tex] to construct a homeomorphism [tex] f:A \rightarrow D^2 [/tex] such that [tex] f[s,0]=(\cos 2\pi s,\sin 2\pi s), f[J]=(1,0),f[I\times \left\{t\right\}]=C_t [/tex]

    (ii) Regard [tex]S^1[/tex] as the identification space of I in which the points [tex] \left\{ 0,1 \right\} [/tex] are identified via the homeomorphism [tex] I/{\lef\t{ 0,1 \right\} } \rightarrow S^1 ; \rightarrow (\cos2 \pi s, \sin 2\pi s) [/tex]. Use (i) to prove that a based loop [tex] \omega: (S^1,(1,0)) \rightarrow (X,x) [/tex] is homotopic rel{(1,0)} to the constant based loop [tex] e_x: (S^1,(1,0) \rightarrow (X,x) [/tex] if and only if [tex] \omega [/tex] extends to a based map [tex] \Omega : (D^2,(1,0)) \rightarrow (X,x) [/tex]

    2. Relevant equations



    3. The attempt at a solution

    (i) The space A is the square with 3 sides identified to a point. The suggested homeomorphism maps the remaining side to the circle [tex] S^1 [/tex] and all other horizontal lines to circles centred at (t,0). Choose [tex] f[s,t] = (1-t)(\cos2 \pi s +t,\sin2\pi s) [/tex]. This satisfies the first and third of the conditions required in (i), but I'm not sure about the second. I'm not sure if the question requires a proof that this is a homeomorphism.

    (ii) I'm not sure about this part. Homeomorphic spaces are automatically homotopy equivalent. We can consider a loop [tex] \omega[/tex] as a function [tex] \alpha (t) = \omega (\cos 2\pi t, \sin 2 \pi t), \alpha (0) = \alpha(1) =x) [/tex] since the loop is based at x. So I think if I show that it has to extend to a map on the space A given above then it automatically has to extend to a map on the disk.
     
  2. jcsd
  3. Feb 24, 2010 #2
    Can no-one help with this one? I'm wondering what they mean by saying that the map "extends" to the disk. The homotopy given by considering the loop as a line with equal end-points is a map on the (identification space of) square, which I might try to relate to the disk in (i) through homeomorphism.
     
  4. Feb 24, 2010 #3
    Looks like that is exactly what you should try. "Extends" means that there is a continuous map of pointed spaces [tex]\Omega: (D^2, (1, 0)) \to (X, x))[/tex] such that [tex]\Omega[/tex] restricts to [tex]\omega[/tex] on [tex]S^1[/tex]: [tex]\Omega|_{S^1} = \omega[/tex].
     
  5. Feb 24, 2010 #4
    Your map does not satisfy the second condition in part (i): J contains the points (s,1) for all s, and your map takes all of these to (0,0). I think you are on the right track though. But you will need to show that this is a homeomorphism to answer the question fully.
     
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