Covering transformations

If $p : A \to B$ is a covering map, then a homeomorphism $T : A \to A$ with $pT = p$ is called a covering transformation. Suppose $p : A \to B$ is a covering map, $p(a_1) = p(a_2) = b$ and $G_i = p_*(\pi_1(A, a_i)) \subset \pi_1(B, b)$. Suppose that $T : A \to A$ is a covering transformation with $T(a_1) = a_2$. Show that $G_1 = G_2$.

Suppose $p : A \to B$ is a covering map with $p(a_1) = p(a_2) = b$. Show that there is a covering transformation $T : A \to A$ with $T(a_1) = a_2$ iff $p_*(\pi_1(A, a_1)) = p_*(\pi_1(A, a_2))$.

Show that the set of all covering transformations of $p : A \to B$ forms a group, which we denote $G_p$, under the operation of composition of homeomorphisms.

Show that if $T_1, T_2$ are covering transformations with $T_1(a) = T_2(a)$ for some $a \in A$, then $T_1 = T_2$.

Consider the homeomorphism $T_n : \mathbb {R} \to \mathbb {R}$, $T_n(x) = x + n$, $n \in \mathbb {Z}$. Let $p : \mathbb {R} \to S^1$ be the standard covering map $p(t) = e^{2\pi i t}$.

Show that if $T : \mathbb {R} \to \mathbb {R}$ is a covering transformation, then $T = T_n$ for some $n$ (Exercise 4.3.5). Conclude that $G_p \simeq \mathbb {Z}$ for $(\mathbb {R}, p, S^1)$.

Find the group of covering transformations of the covering $p : S^2 \to P$, $p(x) = [x]$ (where $P = S^2/x \sim -x$ and $[x]$ denotes the equivalence class of $x$).

Find the group of covering transformations of the covering $p_m : S^1 \to S^1$, $p_m(z) = z^m$ (see Exercise 4.1.3).

Consider the covering space of $S^1 \vee S^1$ from Exercise 4.2.8. Show that there is no covering transformation sending $v_1$ to $v_2$.

Prove the following theorem: Suppose $p : A \to B$ is a covering space with $p(a) = b$ and $\pi_1(A, a) \simeq \left\{ e\right\}$. Then the group of covering transformations $G_p$ is isomorphic to the fundamental group $\pi_1(B, b)$.

Follow the outline below. Define a map $r : G_p \to \pi_1(B, b)$ as follows. Given $T \in G_p$, let $s$ be a path in $A$ connecting $a$ and $T(a)$, and let $r(T) = [ps]$. Prove that $r$ is an isomorphism as follows.

Use Exercise 4.3.11 to give another argument for Exercise 4.3.9.

The normalizer $N(H)$ of a subgroup $H \subset G$ is the subgroup $\left\{ g \in G : gHg^{-1} = H\right\}$. Prove the following theorem, which generalizes Theorem 4.3.1 (Exercise 4.3.10): there is an isomorphism between the group of covering transformations $G_p$ of $p : A \to B$ and the quotient $N(H)/H$ of $H = p_*(\pi_1(A, a)) \subset \pi_1(B, b)$.

Show that there is a bijection between

and $N(H)/H$ (notation as in Exercise 4.3.13).

A group $G$ of homeomorphisms of a space $A$ is called properly discontinuous if for every $a \in A$ there is an open set $U$ containing $a$ such that $g(U)$ is disjoint from $U$ whenever $g \neq e$. Show that if $G$ is a group of homeomorphisms which is properly discontinuous, then for every $a \in A$ there is an open set $U$ containing $a$ such that $g_0(U) \cap g_1(U) = \emptyset$ for all $g_0 \neq g_1 \in G$.

Recall that a covering space $p : A \to B$ is called regular if $p_*(\pi_1(A, a))$ is a normal subgroup of $\pi_1(B, b)$; by Exercise 4.3.15, this is equivalent to the group of covering transformations acting transitively on $p^{-1}(b)$. Recall also that a group $G$ of homeomorphisms of a space $A$ is properly discontinuous (Exercise 4.3.16) if for every $a \in A$ there is an open set $U$ containing $a$ such that $g(U) \cap U = \emptyset$ whenever $g \neq e$.

Suppose that $A$ is path connected and locally path connected and $G$ is a group of homeomorphisms of $A$. Let $A/G$ denote the quotient space formed from $A$, where $a \sim g(a)$ for $g \in G$.

Prove the following theorem (characterization of regular covering spaces): suppose $p : A \to B$ is a regular covering space (Exercise 4.3.17) and $G$ is its group of covering transformations. Then $p = hq$, where $q : A \to A/G$ is the quotient covering space and $h$ is a homeomorphism induced by $p$ via the quotient construction.