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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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 \({\mathbb{RP}}^2 \times S^1\) 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 \({\mathbb{RP}}^2\) . Justify your answer.

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Note \(\pi_1 {\mathbb{RP}}^2 = {\mathbf{Z}}/2{\mathbf{Z}}\), 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

  • \(S^2 \vee S^2\)
  • \({\mathbb{RP}}^2 \vee S^2\)

And so all possibilities for regular covering spaces are given by

  • \(\bigvee^{2k} S^2\) “beads” wrapped into a necklace for any \(k \geq 1\)
  • \({\mathbb{RP}}^2 \vee (\bigvee^k S^2) \vee {\mathbb{RP}}^2\)
  • \(\vee^\infty S^2\), 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
  • \({\mathbb{RP}}^2 \vee S^2 \vee {\mathbb{RP}}^2\).

7 (Fall ’17) #topology/qual/completed

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

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  • \({\mathbb{RP}}_3 \vee S^2 \vee {\mathbb{RP}}^3\), 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 \(\tilde {\mathbb{RP}}^3 = S^3\).
  • \({\mathbb{RP}}^2 \vee S^3 \vee {\mathbb{RP}}^2\), 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 {\mathbb{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,

Image

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, \mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu) .\end{align*}

12 (Fall ’14) #topology/qual/work

Prove that every continuous map \(f : {\mathbb{RP}}^2 \to S^1\) 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