# Covering Spaces

## 1 (Spring 11/Spring ’14) #topology/qual/completed

• Give the definition of a covering space $$\widehat{X}$$ (and covering map $$p : \widehat{X} \to X$$) for a topological space $$X$$.

• State the homotopy lifting property of covering spaces. Use it to show that a covering map $$p : \widehat{X} \to X$$ induces an injection \begin{align*} p^\ast : \pi_1 (\widehat{X}, \widehat{x}) \to \pi_1 (X, p(\widehat{x})) \end{align*} on fundamental groups.

• Let $$p : \widehat{X} \to X$$ be a covering map with $$Y$$ and $$X$$ path-connected. Suppose that the induced map $$p^\ast$$ on $$\pi_1$$ is an isomorphism.

Prove that $$p$$ is a homeomorphism.

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• \todo[inline]{Todo}
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Homotopy lifting property:

\begin{center}
\begin{tikzcd}
&  & \tilde X \arrow[dd, "\pi"] \\
&  &                            \\
Y\times I \arrow[rr, "H"] \arrow[rruu, "\exists \tilde H", dashed] &  & X
\end{tikzcd}
\end{center}


$$\pi$$ clearly induces a map $$p_*$$ on $$\pi_1$$ by functoriality, so we’ll show that $$\ker p_*$$ is trivial. Let $$\gamma: S^1 \to \tilde X \in \pi_1(\tilde X)$$ and suppose $$\alpha \coloneqq p_*(\gamma) = [e] \in \pi_1(X)$$. We’ll show $$\gamma \simeq[e]$$ in $$\pi_1(\tilde X)$$.

Since $$\alpha = [e]$$, $$\alpha \simeq{\operatorname{const.}}$$ and thus there is a homotopy $$H: I\times S^1 \to X$$ such that $$H_0 = {\operatorname{const.}}(x_0)$$ and $$H_1 = \gamma$$. By the HLP, this lifts to $$\tilde H: I\times S^1 \to \tilde X$$. Noting that $$\pi^{-1}({\operatorname{const.}}(x_0))$$ is still a constant loop, this says that $$\gamma$$ is homotopic to a constant loop and thus nullhomotopic.

Since both spaces are path-connected, the degree o the covering map $$\pi$$ is precisely the index of the included fundamental group. This forces $$\pi$$ to be a degree 1 covering and hence a homeomorphism.

## 2 (Fall ’06/Fall ’09/Fall ’15) #topology/qual/work

• Give the definitions of covering space and deck transformation (or covering transformation).

• Describe the universal cover of the Klein bottle and its group of deck transformations.

• Explicitly give a collection of deck transformations on \begin{align*}\left\{{(x, y) \mathrel{\Big|}-1 \leq x \leq 1, -\infty < y < \infty}\right\}\end{align*} such that the quotient is a Möbius band.

• Find the universal cover of $${\mathbf{RP}}^2 \times S^1$$ and explicitly describe its group of deck transformations.

## Spring 2021 #5

Identify five mutually non-homeomorphic connected spaces $$X$$ for which there is a covering map $$p:X\to K$$ where $$K$$ is the Klein bottle. Give an example of the covering in each case.

## 3 (Spring ’06/Spring ’07/Spring ’12) #topology/qual/work

• What is the definition of a regular (or Galois) covering space?

• State, without proof, a criterion in terms of the fundamental group for a covering map $$p : \tilde X \to X$$ to be regular.

• Let $$\Theta$$ be the topological space formed as the union of a circle and its diameter (so this space looks exactly like the letter $$\Theta$$). Give an example of a covering space of $$\Theta$$ that is not regular.

## 4 (Spring ’08) #topology/qual/work

Let $$S$$ be the closed orientable surface of genus 2 and let $$C$$ be the commutator subgroup of $$\pi_1 (S, \ast)$$. Let $$\tilde S$$ be the cover corresponding to $$C$$. Is the covering map $$\tilde S \to S$$ regular?

The term “normal” is sometimes used as a synonym for regular in this context.

What is the group of deck transformations?

Give an example of a nontrivial element of $$\pi_1 (S, \ast)$$ which lifts to a trivial deck transformation.

## 5 (Fall ’04) #topology/qual/work

Describe the 3-fold connected covering spaces of $$S^1 \lor S^1$$.

## 6 (Spring ’17) #topology/qual/completed

Find all three-fold covers of the wedge of two copies of $${\mathbf{RP}}^2$$ . Justify your answer.

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Note $$\pi_1 {\mathbf{RP}}^2 = {\mathbf{Z}}/2{\mathbf{Z}}$$, so $$\pi_1 X = ({\mathbf{Z}}/2{\mathbf{Z}})^2$$.

The pullback of any neighborhood of the basepoint needs to be locally homeomorphic to one of

• $$S^2 \vee S^2$$
• $${\mathbf{RP}}^2 \vee S^2$$

And so all possibilities for regular covering spaces are given by

• $$\bigvee^{2k} S^2$$ “beads” wrapped into a necklace for any $$k \geq 1$$
• $${\mathbf{RP}}^2 \vee (\bigvee^k S^2) \vee {\mathbf{RP}}^2$$
• $$\vee^\infty S^2$$, the universal cover

To get a threefold cover, we want the basepoint to lift to three preimages, so we can take

• $$S^2 \vee S^2 \vee S^2$$ wrapped
• $${\mathbf{RP}}^2 \vee S^2 \vee {\mathbf{RP}}^2$$.

## 7 (Fall ’17) #topology/qual/completed

Describe, as explicitly as you can, two different (non-homeomorphic) connected two-sheeted covering spaces of $${\mathbf{RP}}^2 \lor {\mathbf{RP}}^3$$, and prove that they are not homeomorphic.

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• $${\mathbf{RP}}_3 \vee S^2 \vee {\mathbf{RP}}^3$$, which has $$\pi_2 = 0 \ast {\mathbf{Z}}\ast 0 = {\mathbf{Z}}$$ since $$\pi_{i\geq 1} X = \pi_{i\geq 1}\tilde X$$ and $$\tilde {\mathbf{RP}}^3 = S^3$$.
• $${\mathbf{RP}}^2 \vee S^3 \vee {\mathbf{RP}}^2$$, which has $$\pi_2 = {\mathbf{Z}}\ast 0 \ast {\mathbf{Z}}= {\mathbf{Z}}\ast {\mathbf{Z}}\neq {\mathbf{Z}}$$

## 8 (Spring ’19) #topology/qual/completed

Is there a covering map from \begin{align*} X_3 = \left\{{x^2 + y^2 = 1}\right\} \cup \left\{{(x - 2)^2 + y^2 = 1}\right\} \cup \left\{{(x + 2)^2 + y^2 = 1}\right\} \subset {\mathbf{R}}^2 \end{align*} to $$S^1 \vee S^1$$? If there is, give an example; if not, give a proof.

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Yes,

## 9 (Spring ’05) #topology/qual/work

• Suppose $$Y$$ is an $$n$$-fold connected covering space of the torus $$S^1 \times S^1$$. Up to homeomorphism, what is $$Y$$? Justify your answer.

• Let $$X$$ be the topological space obtained by deleting a disk from a torus. Suppose $$Y$$ is a 3-fold covering space of $$X$$.

What surfaces could $$Y$$ be? Justify your answer, but you need not exhibit the covering maps explicitly.

## 10 (Spring ’07) #topology/qual/work

Let $$S$$ be a connected surface, and let $$U$$ be a connected open subset of $$S$$. Let $$p : \tilde S \to S$$ be the universal cover of $$S$$. Show that $$p^{-1}(U )$$ is connected if and only if the homeomorphism $$i_\ast : \pi_1 (U ) \to \pi_1 (S)$$ induced by the inclusion $$i : U \to S$$ is onto.

## 11 (Fall ’10) #topology/qual/work

Suppose that X has universal cover $$p : \tilde X \to X$$ and let $$A \subset X$$ be a subspace with $$p(\tilde a) = a \in A$$. Show that there is a group isomorphism \begin{align*} \ker(\pi_1 (A, a) \to \pi_1 (X, a)) \cong \pi_1 (p^{-1}A, \overline{a}) .\end{align*}

## 12 (Fall ’14) #topology/qual/work

Prove that every continuous map $$f : {\mathbf{RP}}^2 \to S^1$$ is homotopic to a constant.

Hint: think about covering spaces.

## 13 (Spring ’16) #topology/qual/work

Prove that the free group on two generators contains a subgroup isomorphic to the free group on five generators by constructing an appropriate covering space of $$S^1 \lor S^1$$.

## 14 (Fall ’12) #topology/qual/work

Use covering space theory to show that $${\mathbf{Z}}_2 \ast {\mathbf{Z}}$$ (that is, the free product of $${\mathbf{Z}}_2$$ and $${\mathbf{Z}}$$) has two subgroups of index 2 which are not isomorphic to each other.

## 15 (Spring ’17) #topology/qual/work

• Show that any finite index subgroup of a finitely generated free group is free. State clearly any facts you use about the fundamental groups of graphs.

• Prove that if $$N$$ is a nontrivial normal subgroup of infinite index in a finitely generated free group $$F$$ , then $$N$$ is not finitely generated.

## 16 (Spring ’19) #topology/qual/work

Let $$p : X \to Y$$ be a covering space, where $$X$$ is compact, path-connected, and locally path-connected.

Prove that for each $$x \in X$$ the set $$p^{-1}(\left\{{p(x)}\right\})$$ is finite, and has cardinality equal to the index of $$p_* (\pi_1 (X, x))$$ in $$\pi_1 (Y, p(x))$$.

#topology/qual/completed #topology/qual/work #5