Covering Spaces

1. Show that \(\mathbb {R}P^2\) has a 2-fold cover which is \(S^2\).
2. Show that the Klein bottle has a 2-fold cover which is a torus.

Let \(a\) and \(b\) be the two generators of \(\pi _1(S^1 \vee S^1)\) corresponding to the two \(S^1\) summands. Draw a picture of the covering space of \(S^1 \vee S^1\) corresponding to the normal subgroup generated by \(a^2\), \(b^2\), and \((ab)^4\), and prove this covering space is indeed the correct one.

Draw a picture of a covering space of \(S^1\vee S^1\) whose fundamental group is the normal subgroup of \(\pi _1(S^1\vee S^1,*)\) generated by \(a^2\), \(b^2\), and \((ab)^2\), and prove your covering space is the correct one.

Let \(f:M\to N\) be a covering map between manifolds that are both nonempty, connected, and compact without boundary. Suppose \(f\) is homotopic to a constant map. Prove that \(N\) is a point.

Let \(p : \tilde {X} \to X\) be a covering map between two connected CW-complexes. If \(p\) is null-homotopic, prove that \(\tilde {X}\) is contractible. If you use a theorem make sure you state it carefully and include all the hypotheses.

Let \(X\) be a connected cell complex and \(f:\mathbb {R}P^{2n}\to X\) a covering map from the real projective space of dimension \(2n\). Show that \(f\) is a homeo- morphism.

State the homotopy lifting property of covering maps and use it to prove that the covering map \(S^n\to \mathbb {R}\mathrm {P}^n\) from the \(n\)-sphere to the real projective \(n\)-space is not nullhomotopic for any \(n\ge 1\).

Let \(X\) be a connected CW complex and \(A\subset X\) a connected subcomplex. Suppose that inclusion \(A\subset X\) is \(\pi _1\)-injective (for some, hence all, basepoints in \(A\)). Let \(p:\widetilde {X}\to X\) be the universal cover of \(X\). Prove that each component of \(p^{-1}(A)\) is simply connected.

Let \(X\) be the wedge sum of two copies of \(\mathbb {R}\mathrm {P}^2\). Show that there is only one connected degree three cover of \(X\) and determine if it is regular or irregular.

Let \(X\) be the wedge product of a circle and a projective plane. Find all connected three sheeted covers of \(X\).

Let \(X=S^1\vee S^1\), with basepoint \(*\) at the point where the two circles are glued, and with \(a,b:S^1\to X\) the generators of \(\pi _1(X,*)\) corresponding to the identity maps onto each circle in the wedge sum. Construct a covering space \(p:(E,e)\to (X,*)\) such that \(p_*\pi _1(E,e)\) is the subgroup of \(\pi _1(X,*)\) generated by \(a^2\), \(ab^{-1}\), \(ab\). Justify that your answer works in two ways:

1. using covering space theory; and
2. using the van Kampen theorem.

Let \(n\ge 1\) be a positive integer.

1. Give an elementary description of the universal cover \(X\to \mathbb {R}\mathrm {P}^n\) of \(\mathbb {R}\mathrm {P}^n\).
2. Describe a CW structure on \(\mathbb {R}\mathrm {P}^n\).
3. Compute the cohomology ring \(H^*(\mathbb {R}\mathrm {P}^n;\mathbb {Z}/2)\) and the map (from (a)) \(H^*(\mathbb {R}\mathrm {P}^n;\mathbb {Z}/2)\to H^*(X;\mathbb {Z}/2)\).

Let \(p:\widetilde {X}\to X\) be a cover of connected spaces. Prove \(p_*:\pi _1\widetilde {X}\to \pi _1X\) is injective.

Use covering spaces to write the generators of a normal rank 3 free subgroup of a rank 2 free group. Use covering spaces to write the generators of a non-normal rank 3 free subgroup of a rank 2 free group.

Show that the union of circles in \(\mathbb {R}^2\) with centers at \((0,\frac {1}{2n})\) and radius \(\frac {1}{n}\) does not have a universal cover.

Give an example of an irregular (i.e. not normal) covering space of the Klein bottle (with a proof).

Let \(X\) be a compact topological space and \(\widetilde {X}\) its universal cover. Show that \(\pi _1(X)\) is finite if and only if \(\widetilde {X}\) is compact.

Let \(X=S^1\vee S^1\) and \(x_0\in X\) be the attaching point.

1. Carefully state van Kampen’s Theorem and use it to compute \(\pi _1(X,x_0)\).
2. Find two covering spaces of \(X\) with three sheets; one of which is regular (determining a normal subgroup of \(\pi _1(X,x_0)\)) and the other of which is irregular.

What is the universal cover of \(\mathbb {R}\mathrm {P}^2\vee \mathbb {R}\mathrm {P}^2\)?

What is the universal cover of \(S^1\vee S^1\)?

Let \(X\) be a path connected space that is locally simply connected. Give the construction of \(p:\widetilde {X}\to X\), the universal cover of \(X\). You don’t have to define what a universal cover is, or show that your construction is a universal cover. Just state the description of the universal cover for \(X\) used in the standard proof that the universal cover for \(X\) exists.

State the Path Lifting Theorem for covering spaces and sketch a proof of it.

Let \(X\) be a path connected, locally path connected, and semi-locally simply connected space. (Recall that \(X\) is semi-locally simply connected if every point in \(X\) has a neighborhood \(U\) such that every closed loop in \(U\) can be contracted to a point in \(X\).) Outline the construction of the universal cover \(p:\widetilde {X}\to X\). Specifically,

1. Define \(\widetilde {X}\) as a set and the projection \(p:\widetilde {X}\to X\),
2. Define the topology on \(\widetilde {X}\) (you don’t have to check that this is a topology),
3. Give an argument that \(\widetilde {X}\) is simply connected.

Indicate where each of the three assumptions on \(X\) is used.

Construct an explicit irregular (i.e. not normal) covering space of the Klein bottle.

Let \(p:\widetilde {X}\to X\) be an \(n\)-sheeted covering map between two connected, locally path-connected spaces. Let \(C_*(X)\) and \(C_*(\widetilde {X})\) denote the singular chain complexes of \(X\) and \(\widetilde {X}\) respectively.

1. Give a definition of a chain morphism and prove that \(\tau :C_*(X)\to C_*(\widetilde {X})\) given by \[ \tau (\sigma )=\sum _{i=1}^n \widehat {\sigma }_i \] is a chain morphism, where \(\widehat {\sigma }_1,\ldots ,\widehat {\sigma }_n\) are the \(n\) lifts of \(\sigma \) to \(\widetilde {X}\), and \(\sigma \) is a singular simplex in \(X\).
2. Prove that \[ p_*:H_i(\widetilde {X};\mathbb {Q})\to H_i(X;\mathbb {Q}) \] is always surjective. Hint: Consider the composition \(p_*\tau :C_*(X)\to C_*(X)\).
3. Find an example where \[ p_*:H_i(\widetilde {X};\mathbb {Z})\to H_i(X;\mathbb {Z}) \] is not surjective.

Let \(X\) be a path connected, locally path connected, semi-locally simply connected Hausdorff space. One of the main theorems of covering space theory asserts that there is a simply connected space \(\widetilde {X}\) and a covering map \(p:\widetilde {X}\to X\). Outline the construction of \(\widetilde {X}\) and \(p\) as follows:

1. Describe \(\widetilde {X}\) as a set.
2. Describe the topology on \(\widetilde {X}\).
3. Describe \(p\). You do not need to prove that \(p\) is a covering map.
4. Outline the proof that \(\widetilde {X}\) is simply connected.

Let \(p:\widetilde {X}\to X\) be a covering map between cell complexes. Assume that \(\widetilde {X}\) is connected and that \(p\) is null-homotopic. Prove that \(\widetilde {X}\) is contractible.

Let \(G\) be a Lie group, and let \(\widetilde {G}\) be the universal cover of \(G\). Endow \(\widetilde {G}\) with the structure of a Lie group.

Let \(X=\mathbb {R}\mathrm {P}^2\vee \mathbb {R}\mathrm {P}^2\). Give an example of an irregular covering space \(\widetilde {X}\to X\). You should draw the picture of the 1-skeleton of \(\widetilde {X}\) (make sure 2-cells lift).

1. Define a covering space \(p:\widetilde {X}\to X\).
2. Let \(p:\widetilde {X}\to X\) be a covering space, \(\widetilde {x}_0\in \widetilde {X}\), \(x_0\in X\), and \(p(\widetilde {x}_0)=x_0\). Show that \(p_*:\pi _1(\widetilde {X},\widetilde {x}_0)\to \pi _1(X,x_0)\) is injective (carefully state the lifting property you are using).

Let \(X\) be a path connected, locally path connected, semi-locally simply connected Hausdorff space. One of the main theorems of covering space theory asserts that there is a simply connected space \(\widetilde {X}\) and a covering map \(p:\widetilde {X}\to X\). Outline the construction of \(\widetilde {X}\) and \(p\) as follows:

1. Describe \(\widetilde {X}\) as a set.
2. Describe the topology on \(\widetilde {X}\).
3. Describe \(p\). You do not need to prove that \(p\) is a covering map.
4. Outline the proof that \(\widetilde {X}\) is simply connected.

Is there a covering map \(p:X\to Y\) where

1. \(X=S_3\), \(Y=S_2\) where \(S_g\) is a closed orientable surface of genus \(g\)?
2. \(X\) is \(S_3\) with a small open disk removed and \(Y\) is \(S_2\) with a small open disk removed.

When \(p\) exists, what is the number of sheets?

Show that \(S^3\) (the 3-sphere) is the covering space of a manifold \(M\) which is not diffeomorphic to \(\mathbb {R}\mathrm {P}^3\). (Hint: find an appropriate finite group action on \(S^3\).)

Let \(X\) be the union of circles in \(\mathbb {R}^2\) with centers at \((0,\frac {1}{2n})\) and radius \(1/n\). Show \(X\) doesn’t have a universal cover (show that every connected covering space of \(X\), including \(X\) itself, has non-trivial \(\pi _1\)).

Let \(X\) be a compact topological space and \(\widetilde {X}\) its universal cover. Show that \(\pi _1(X)\) is finite if and only if \(\widetilde {X}\) is compact.

Give an example of an irregular (i.e. not normal) covering space of the Klein bottle (with a proof).

Let \(f:M\to N\) be a covering map between closed connected nonempty manifolds. Suppose \(f\) is homotopic to a constant map. Prove that \(N\) is a point.

Give an example of an irregular (i.e. not normal) covering space (with a proof).

For a connected CW complex \(X\), call a connected covering space \(\widetilde {X}\to X\) abelian if it is normal and has abelian deck transformation group. Show that \(X\) has an abelian covering space that is a covering space of every other abelian covering space of \(X\), and that such ‘universal’ abelian covering space is unique up to isomorphism. Describe this covering space explicitly for \(X=S^1\vee S^1\). Carefully state theorems you are using.

Show that a \(p\)-sheeted covering map \(M\to N\) between closed connected oriented smooth manifolds has degree \(\pm p\).

Two covering spaces \(p_0:X_0\to X\) and \(p_1:X_1\to X\) of a space \(X\) are isomorphic if there is a homeomorphism \(h:X_0\to X_1\) such that \(p_0=p_1\circ h\). In this question all spaces are path-connected, locally path-connected and semilocally path-connected.

1. Prove the following piece of the classification theorem for covering spaces: Let \(x\in X\), \(x_0\in X_0\) and \(x_1\in X_1\) be basepoints such that \(p_0(x_0)=p_1(x_1)=x\). The covers are isomorphic if and only if \((p_0)_*(\pi _1(X_0,x_0))\) and \((p_1)_*(\pi _1(X_1,x_1))\) are conjugate subgroups of \(\pi _1(X,x)\).
2. Show that there are non-isomorphic covers of a genus two surface of every degree.

Suppose \(p:\widetilde {X}\to X\) is a covering map. Prove that \(p\) is null homotopic if and only if \(\widetilde {X}\) is contractible.

Find a non-regular cover of \(S^1\vee S^1\), the wedge of 2 circles. You should both explicitly describe the cover and the corresponding subgroup of \(\pi _1(S^1\vee S^1)\).

Find all connected covering spaces of \(\mathbb {R}P^2\vee S^1\) of degree less than three. Which of these coverings are regular?

Give an example of a topological space that does not have a universal cover.

Let \(M\) be a Möbius strip and \(D\) a disk. Let \(f:\partial D\to \partial M\) have degree \(n\). Set \(X_n=M\coprod D/{\sim }\) where \(x\sim y\) if \(f(x)=y\). Calculate \(\pi _1(X_k)\) and construct all of its connected covering spaces.

Give an explicit example of a covering space of \(X=S^1\vee S^1\) which is not normal.

Let \(p:M\to N\) be a covering map between two closed connected manifolds of positive dimension. Show that \(p\) is not null-homotopic.

Let \(X\) be the space obtained from the torus \(S^1\times S^1\) by attaching a Möbius band via a homeomorphism from the boundary circle in the Möbius band to the circle \(S^1\times \{x_0\}\) in the torus. Compute \(\pi _1(X)\) and describe the universal cover of \(X\).

A. Let \(f:S^n\to T^n\) be continuous, \(n>1\), where \(S^n\) and \(T^n\) denote the \(n\)-sphere, and the \(n\) dimensional torus respectively. Prove that the degree of \(f\) is zero. (If you prefer, you may assume that \(f\) is smooth).

B. Let \(f:X\to Y\) be a covering map between two connected manifolds. Prove that if \(f\) is homotopic to a constant map then \(X\) is contractible.

A. Let \(F_g\) be a closed orientable surface of genus \(g\). Prove that \(\pi _1(F_g)\) is infinite, and non-abelian, provided \(g>1\).

B. Let \(G\) be the one point union of two circles (a figure eight). Draw a picture of an explicit irregular covering map \(\widetilde {G}\to G\).

Classify all connected covering spaces of \(\mathbb {R}P^2\vee \mathbb {R}P^2\) up to equivalence.

Determine all covering spaces of the two-torus \(T^2\) up to equivalence. Explain.

a. Let \(P:M\to N\) be a regular covering map between connected differentiable manifolds \(M,N\) and let \(M\) be orientable. Prove that \(N\) is orientable if and only if every deck transformation (covering space automorphism) preserves the orientation of \(M\).

b. Prove that \(\mathbb {R}P^n\) is orientable if and only if \(n\) is odd.

Take the 2-dimensional triangle \(T\) and identify all its sides as indicated below. What is the universal covering of the resulting topological space \(X\)?

Diagram description: the triangle \(T\) has all three sides identified. The left and right sides have arrows pointing toward the top vertex, and the bottom side has an arrow pointing from left to right.

Suppose that \(p:\widetilde {X}\to X\) is a covering map between connected manifolds. Prove that if \(p\) is null-homotopic, then \(\widetilde {X}\) is contractible.