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[SOLVED] homotopic curves
Apparently if [itex]\gamma_4 = \gamma_2 +\gamma_3 \gamma_1\gamma_3[/itex], then [itex]\gamma_4[/itex] is homotopic to [itex]\gamma_5[/itex] in any region containing [itex]\gamma_1[/itex],[itex]\gamma_2[/itex], and the region between them minus z.
I am not convinced that this is true.
I can picture how you could transform [itex]\gamma_4[/itex] into [itex]\gamma_5[/itex] one another by first moving the two [itex]\gamma_3[/itex]s apart, then kind of going around the circle and finally contracting the curve, but I am not convinced that this is a continuous function from the unit interval cross the unit interval. Specifically, I am not convinced that you can just move the two [itex]\gamma_3[/itex]s apart in a continuous way. How can you rigorously show that there is a homotopy?
Homework Statement
Apparently if [itex]\gamma_4 = \gamma_2 +\gamma_3 \gamma_1\gamma_3[/itex], then [itex]\gamma_4[/itex] is homotopic to [itex]\gamma_5[/itex] in any region containing [itex]\gamma_1[/itex],[itex]\gamma_2[/itex], and the region between them minus z.
I am not convinced that this is true.
I can picture how you could transform [itex]\gamma_4[/itex] into [itex]\gamma_5[/itex] one another by first moving the two [itex]\gamma_3[/itex]s apart, then kind of going around the circle and finally contracting the curve, but I am not convinced that this is a continuous function from the unit interval cross the unit interval. Specifically, I am not convinced that you can just move the two [itex]\gamma_3[/itex]s apart in a continuous way. How can you rigorously show that there is a homotopy?
Homework Equations
The Attempt at a Solution
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