# covering spaces and equivalency of these three propositions [closed]

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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• What have you tried ? Do you know how to prove any of the implications ? (I would advise A) => B) => C) => A) ) – Max Jan 8 at 22:07
• I try to work on it with conjugacy classes but i wasn't successful to reach the answers – pershina olad Jan 8 at 22:11

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 $$p_*(\gamma\delta\gamma^{-1})$$ ?

• 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 – pershina olad Jan 10 at 12:29
• $x_0 = p(y_0)$; what part are you talking about ? – Max Jan 10 at 16:17
• 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 – pershina olad Jan 11 at 6:45
• Yes but most importantly, what is its value ? – Max Jan 11 at 10:30
• 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 – pershina olad Jan 12 at 5:51