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This question is really important for me since the answer will give me the a way for solving similar proofs at algebraic topology lessons,so i need your help.. I need to prove these following statements are equivalent : Suppose that $P:Y\longrightarrow X $ be a covering space. Prove that these three statements are equivalent :

A) $ {P_*}(\pi_1(Y,{y_0}))\lhd \pi_1(X,P(y_0))$

B) For any two $y_0,y_1\in P^{-1}(x_0)$ we have $ {P_*}(\pi_1(Y,{y_0}))= {P_*}(\pi_1(Y,{y_1}))$

C) For any loop $\omega$ in $X$, all of its lifts $\widetilde\omega$ in $Y$ are loops or none of them is a loop.

I know I should try to show the equivalency by conjugacy classes but still I can not prove it, really thank anyone who help me with solving this problem

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  • $\begingroup$ What have you tried ? Do you know how to prove any of the implications ? (I would advise A) => B) => C) => A) ) $\endgroup$ – Max Jan 8 at 22:07
  • $\begingroup$ I try to work on it with conjugacy classes but i wasn't successful to reach the answers $\endgroup$ – pershina olad Jan 8 at 22:11
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Let me give you indications, rather than a full solution :

A) => B) : note that if $\alpha $ is a path from $y_0$ to $y_1$, then $p_*(\pi_1(Y,y_0))$ and $p_*(\pi_1(Y,y_1))$ are conjugate with $p_*(\alpha)$ (which is a loop based at $x_0$).

B)=> C) : if $\omega$ lifts to a loop $\tilde{\omega}$, then it is in $p_*(\pi_1(Y,y_0))$ for some $y_0$; and so for all $y_0$. What does it mean ?

C) => A) : let $\alpha$ be a loop based at $x_0$, $\beta$ a loop based at $y_0$, and consider $\alpha p_*(\beta)\alpha^{-1}$. If you lift $\alpha$ to a path $\gamma$ starting at $y_0$, then you can lift $p_*(\beta)$ to a loop $\delta$ based at $\gamma (1)$. Then what is $p_*(\gamma\delta\gamma^{-1})$ ?

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  • $\begingroup$ Awesome answer thank you max,but in C==>A I'm not sure i understand ...what you asked for is a member at fundamental group of X at P(y0)? thank for your help and support Max . @Max $\endgroup$ – pershina olad Jan 10 at 12:29
  • $\begingroup$ $x_0 = p(y_0)$; what part are you talking about ? $\endgroup$ – Max Jan 10 at 16:17
  • $\begingroup$ I mean the last question you asked from me at the proof C==>A, p∗(γδγ−1) , the answer of your question is that it belongs to the fundamental group of X at P(y0)? @Max $\endgroup$ – pershina olad Jan 11 at 6:45
  • $\begingroup$ Yes but most importantly, what is its value ? $\endgroup$ – Max Jan 11 at 10:30
  • $\begingroup$ mmm I am a little weak in algebraic constructions because the field i study is dynamical system in geometry but i think the abelian property which gives the normal subgroup is your goal to reach me to the main concept,am i right? @Max $\endgroup$ – pershina olad Jan 12 at 5:51

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