Basic examples and properties

Show that whenever $p : A \to B$ is a covering map and $C$ is a path-connected, locally path-connected Hausdorff space, then $P : A \times C \to B \times C$, $P(a, c) = (p(a), c)$ is a covering map.

Show that if $p_1 : A_1 \to B_1$, $p_2 : A_2 \to B_2$ are covering maps, then so is $P : A_1 \times A_2 \to B_1 \times B_2$, $P(a_1, a_2) = (p_1(a_1), p_2(a_2))$.

Show that $p_m : S^1 \to S^1$, $p_m(z) = z^m$ (complex multiplication) --- or, equivalently, $p(\cos x, \sin x) = (\cos mx, \sin mx)$ --- is a covering map.

Show that if $A$ is compact, and $p : A \to B$ is a covering map, then for any $b \in B$, $p^{-1}(b)$ consists of a finite number of points.

Using the framework of Exercise 4.1.4 (that $A$ is compact and $p : A \to B$ is a covering map), show that since $B$ is assumed path connected, then the number of points in $p^{-1}(b)$ does not depend on $b$.

Recall that the order of a covering map is the (constant) number of points in each inverse image $p^{-1}(x)$; when this number is a finite $k$, the covering is called a $k$-fold cover.

Consider the cover $P_{2,1} : T \to T$ given by Exercise 4.1.6. Figure 4.1(a) pictures this cover. Now change the base space to $T^{(2)}$ by using the same construction used to get $T$ from $S^2$ (doing surgery by removing two disks and replacing them with a cylinder), and in the top space $T$ do this surgery twice, as pictured in Figure 4.1(b).

Constructing a cover p : T^{(3)} \to T^{(2)}.

Continuing with the surgery construction of Exercise 4.1.7 and starting with $P_{m,1}$ but now performing $n$ surgeries on the base space, construct an $m$-fold cover $p : T^{(mn+1)} \to T^{(n+1)}$.

Recall (from the study of nonorientable surfaces) that there are 2-fold covering spaces $p : S \to P$ and $p : T \to K$, where $S, T$ are the sphere and torus and $P, K$ are the projective plane and Klein bottle. New covering spaces can be obtained from these by doing surgery (removing a disk from both the covering space and the base space, in a way compatible with the covering map, and replacing each by a handle) as in Exercises 4.1.7-4.1.8. Show that there is a 2-fold covering space $p_n : T^{(2n)} \to P^{(2n+1)}$.

Show that there is a 2-fold covering space $p_n : T^{(2n+1)} \to P^{(2n+2)}$.

Start with $K = S^1 \times S^1 / (z, w) \sim (-z, \overline{w})$ (the usual construction of the Klein bottle $K$ as a quotient of the torus). For $n$ odd, form a new space $K_n = S^1 \times S^1 / (z, w) \sim (e^{\pi i/n} z, \overline{w})$; note that $n = 1$ gives back the usual construction of $K$. There is a natural map from $K$ to $K_n$ induced by the identity map of $T = S^1 \times S^1$ to itself (well defined because $n$ is odd, so that $(z, w) \sim (-z, \overline{w})$ is part of the equivalence relation defining $K_n$). Show that $K_n$ is homeomorphic to $K$.

For each of the finite covering spaces $p : A \to B$ of surfaces constructed by surgery in Exercises 4.1.6-4.1.11, verify that there is a formula

relating the Euler characteristics of $A$ and $B$, where $k$ is the order of the cover.

Assuming the formula $\chi (A) = k\chi (B)$ for surfaces (Exercise 4.1.12), show that the only surfaces without boundary that can cover themselves with order $k > 1$ are $T$ (the torus) and $K$ (the Klein bottle).

Suppose that $p : A \to B$ is a covering map of finite order $k$ from one surface to another and we have a handle decomposition of $B$ where each handle lies within an evenly covered neighborhood.

Let $W$ be the wedge $S^1_a \vee S^1_b$ of two circles, labeled $a, b$. A covering space of order $2$ can be pictured by labeling the cover with $a_1, a_2, b_1, b_2$. Figure 4.2 shows one example, in which each circle of $W$ is double covered by a single circle.

A double cover of S^1 \vee S^1.

There are two other nonequivalent 2-fold covering spaces of $W$, other than the one pictured above. Give pictures of them.

Figure 4.3 shows a 3-fold cover of $W = S^1_a \vee S^1_b$. The edges labeled $a$ are mapped to one circle and those labeled $b$ are mapped to the other.

A three-fold cover of S^1 \vee S^1.

Give pictures of two different nonequivalent 3-fold covering spaces of $W$ from the one given in Figure 4.3.

Prove the following lemma: Suppose $p : A \to B$ is a covering map and $U \subset B$ is a path-connected open set which is evenly covered, with $p^{-1}(U) = \bigcup_i U_i$, and $b \in U$. Let $a_i \in U_i$ satisfy $p(a_i) = b$. If $X$ is connected and $f : X \to B$ is continuous with $f(X) \subset U$, $f(x) = b$, then there is a unique lift $\tilde{f}_i$ of $f$ with $\tilde{f}_i(x) = a_i$. Moreover, $\tilde{f}_i(X) \subset U_i$.

Prove the unique path lifting theorem: Suppose $p : A \to B$ is a covering map with $p(a) = b$. Given a path $f : I \to B$ with $f(0) = b$, there is a unique path $\tilde{f} : I \to A$ with $\tilde{f}(0) = a$ and $p\tilde{f} = f$.

Suppose $r : I \to B$ is a path in $B$ with $r(0) = b_1$ and $r(1) = b_2$. Define a map from $p^{-1}(b_1)$ to $p^{-1}(b_2)$ as follows: for each $a \in p^{-1}(b_1)$, let $r_a$ be the unique path lifting of $r$ with $r_a(0) = a$. Then send $a$ to $r_a(1)$. Show that this defines a bijection between $p^{-1}(b_1)$ and $p^{-1}(b_2)$.

Prove the following theorem: Suppose that $F : (I \times I, \left\{ 0\right\} \times I, 1 \times I) \to (B, b_1, b_2)$ is a homotopy between $f_0, f_1$, and $p(a) = b_1$.

Utilize Exercise 4.1.20 to define a map from $\pi_1(B, b)$ to $p^{-1}(b)$ as follows: assign to $[f]$ the point $\tilde{f}(1)$, where $\tilde{f}$ is the unique path lifting of $f$ with $\tilde{f}(0) = a$.