Suppose that $E,B$ are arc-connected and locally arc-connected spaces. Let $p:E\to B$ be a covering space such that $p(e_0)$.

(a) Given an element $[\alpha]\in \pi_1(B, b_0)$ and $x\in p^{-1}(b_0)$, let us define $x\cdot [\alpha]=\widetilde{\alpha}(1)$, where $\widetilde{\alpha}$ is the lifting of $\alpha$ that starts in $x$. Show that this assignment is well defined and corresponds to an action to the right of $\pi_1(B, b_0)$ in the set $p^{-1}(b_0)$.

(b) Let $H=p_*(\pi_1(E,e_0))$. Use the previous numeral to show that there is a bijection between the sets $\pi_1(B,b_0)/H$ and $p^{-1}(b_0)$. It follows that the number of sheets of a covering space coincides with the index of $H$ in $\pi_1(B,b_0)$.

I already solved (a), but I do not know how to solve (b), how could I do this? Thank you very much.


Using part (a) and the orbit-stabilizer theorem, you are left to show that the action of $\pi_1(B,b_0)$ on the fibre $p^{-1}(b_0)$ is transitive and that $H$ is the stabilizer of a point in the fibre.

Regarding transitivity: Given $x,y\in p^{-1}(b_0)$, by path-connectedness of $E$ there is a path $\gamma$ from $x$ to $y$, hence $p\circ\gamma$ represents an element $[\alpha]\in\pi_1(B,b_0)$. By uniqueness of path-lifting, $\alpha$ lifts to $\gamma$, i.e. we have $\tilde{\alpha}=\gamma$, which shows $x\cdot[\alpha]=y$, hence the action is transitive.

Now I claim that $H$ is the stabilizer of $e_0$. Indeed, an element $[\alpha]\in \pi_1(B,b_0)$ fixes $e_0$ if and only if $\alpha$ lifts to an element in $\pi_1(E,e_0)$. This is precisely the case for $[\alpha]\in H$ by definition of $H$. Hence $H$ is the stabilizer of $e_0$.


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