Let $A_{1}$ be the circle $x^{2}+y^{2}=1, B_{1}$ be the circle $(x-2)^{2}+y^{2}=1, A_{2}$ be the circle $(x-4)^{2}+y^{2}=1$ and $B_{2}$ be the circle $(x+2)^{2}+y^{2}=1 .$
Define $\tilde{X}=A_{1} \cup B_{1} \cup A_{2} \cup B_{2}$ and $X=A_{1} \cup B_{1} .$ Define a map $p: \tilde{X} \rightarrow X$ as follows:
i) $p(\cos (t), \sin (t))=(\cos (2 t), \sin (2 t))$ for $(\cos (t), \sin (t)) \in A_{1}$ (Circle $A_{1}$ wraps twice around $\left.A_{1}\right)$
ii) $p(-\cos (t)+2, \sin (t))=(-\cos (2 t)+2, \sin (2 t))$ for $(-\cos (t)+2, \sin (t)) \in B_{1}$ (Circle $B_{1}$ wraps twice around $B_{1}$ ),
iii) $\left.p\right|_{A_{2}},$ and $\left.p\right|_{B_{2}}$ is mapped homeomorphically onto $A_{1}$ and $B_{1}$ respectively. Now $p$ is a covering map. (You do not need to prove this.)
a) Give a presentation for $\pi_{1}(\tilde{X},(1,0))$ and $\pi_{1}(X,(1,0))$. Carefully write down the generators.
b) Compute $p_{*}\left(\pi_{1}(\tilde{X},(1,0)\right)$ as a subgroup of $\pi_{1}(X,(1,0))$
c) Is $p_{*}\left(\pi_{1}(\tilde{X},(1,0)\right )$ a normal subgroup of $\pi_{1}(\tilde{X},(1,0)) ?$ Give reasons.
d) What is the automorphism group $A(\tilde{X}, p) ?$ Give reasons.
My try:
a) $\pi_1(X)=\Bbb Z * \Bbb Z=<a,b>(say)$ and $\pi_1(\tilde X)=\Bbb Z * \Bbb Z * \Bbb Z * \Bbb Z$
For part b) I don't know how to proceed.
c)
If $p : (\tilde{X},\tilde{v}) \rightarrow (X,v)$ is a covering map. Then $p$ is regular if and only if any two points in $p^{-1}(v)$ differ by a covering transformation.
or
If $p : (\tilde{X},\tilde{v}) \rightarrow (X,v)$ is a covering map. Then $p$ is regular if for any two points $\tilde{v}_{1}, \tilde{v}_{2}$ in $p^{-1}(v)$, a loop $\ell$ based at $v$ in $X$ lifts to a loop $\tilde{\ell}$ in $\tilde{X}$ based at $\tilde{v}_{1}$ if and only if it lifts to a loop in $\tilde{X}$ based at $\tilde{v}_{2}$.
Then, we know that $(\tilde{X},\tilde{v})$ is a regular covering space of $(X,v)$ if and only if $p_{*}(\pi_{1}(\tilde{X},\tilde{v}))$ is a normal subgroup of $\pi_{1}(X,v)$. Moreover we also have one to one correspondence between $C(\tilde{X}, p, X) \to N(H_0)/H_0$ where $H_0=p_{*}(\pi_1(\tilde{X},(1,0)))$ and $C(\tilde{X}, p, X)$ is the group of covering transformations and if $H_0$ is normal then we will have one to one correspondence between $C(\tilde{X}, p, X) \to \pi_1(X)/H_0$.
One footnote: I am reading Munkres basically and these are the questions depending on that. Hatcher will be a bit tough for me right now. Please give me a detailed explanation for part or full question. Then I will have an idea.