(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Difficult homotopy question

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