.. _Exercise 1-3:

Exercises
==================

.. _Exercise 1-3-1:

**1.** For a covering space :math:`p:\tilde{X} \rightarrow X` and a subspace :math:`A\subset X`, let :math:`\tilde{A}=p^{-1}(A)`. Show that 
the restriction :math:`p:\tilde{A} \rightarrow A` is a covering space.

.. _Exercise 1-3-2:

**2.** Show that if :math:`p_1:\tilde{X}_1 \rightarrow X_1` and :math:`p_2:\tilde{X}_2 \rightarrow X_2` are covering spaces, so is their product
:math:`p_1 \times p_2: \tilde{X}_1 \times \tilde{X}_2 \rightarrow X_1 \times X_2`.

.. _Exercise 1-3-3:

**3.** Let :math:`p:\tilde{X} \rightarrow X` be a covering space with :math:`p^{-1}(x)` finite and nonempty for all :math:`x \in X`.
Show that :math:`\tilde{X}` is compact Hausdorff iff :math:`X` is compact Hausdorff.

.. _Exercise 1-3-4:

**4.** Construct a simply-connected overing space of the space :math:`X \subset \mathbb{R}^3` that is the union
of a sphere and a diameter. Do the same when :math:`X` is the union of a sphere and a circle
intersecting it in two points.

.. _Exercise 1-3-5:

**5.** Let :Math:`X` be the subspace of :math:`\mathbb{R}^2` consisting of the four sides of the square :math:`[0,1] \times [0,1]`
together with the segments of the vertical lines :math:`x=\frac{1}{2},\frac{1}{3},\frac{1}{4},\cdots` inside the square.
Show that for every covering spce :math:`\tilde{X} \righytarrow X` there is some neighborhood of the left
edge of :math:`X` that lifts homeomorphically to :math:`\tilde{X}`. Deduce that :math:`X` has no simply-connected 
covering space.

.. _Exercise 1-3-6:

.. container::

    **6.** Let :Math:`X` be the shrinking wedge of circles in :ref:`Example 1.25 <Example 1.25>`, and let :Math:`\tilde{X}` be its covering
    space shown in the figure below.

    .. image:: fig/ex-1-3-6.png
        :align: center
        :width: 100%
    
    Construct a two-sheeted covering space :math:`Y \rightarrow \tilde{X}` such that the composition :math:`Y \rightarrow \tilde{X} \rightarrow X`
    of the two covering spaces is not a covering space. Note that a composition of two
    covering spaces does have the unique path lifting property, however.

.. _Exercise 1-3-7:

.. container::

    .. image:: fig/ex1-3-7.png
        :align: right
        :width: 20%
    
    Let :math:`Y` be the *quasi-circle* shown in the figure, a closed subspace
    of :math:`\mathbb{R}^2` consisting of a portion of the graph of :math:`y=\sin (1/x)`, the
    segment :math:`[-1,1]` in the :math:`y`-axis, and an arc connecting these two
    pieces. Collapsing the segment of :math:`Y` in the :math:`y`-axis to a point
    gives a quotient map :math:`f:Y\rightarrow S^1`. Show that :math:`f` does not lift to
    the covering space :math:`\mathbb{R} \rightarrow S^1`, even though :math:`\pi_1(Y)=0`. Thus local
    path-connectedness of :math:`Y` is a necessary hypothesis in the lifting criterion.

.. _Exercise 1-3-8:

**8.** Let :math:`\tilde{X}` and :math:`\tilde{Y}` be simply-connected covering spaces of the path-connected, locally
path-connected space :math:`X` and :math:`Y`. Show that if :math:`X \simeq Y` then :math:`\tilde{X} \simeq \tilde{Y}`. [:ref:`Exercise 11 in Chapter 0 <Exercise 0-11>` 
may be helpful.]

.. _Exercise 1-3-9:

**9.** Show that if a path-connected, locally path-connected space :math:`X` has :math:`\pi_1(X)` finite,
then evety map :math:`X \rightarrow S^1` is nullhomotopic. [Use the covering space :math:`\mathbb{R} \rightarrow S^1`.]

.. _Exercise 1-3-10:

**10.** Find all the connected :math:`2`-sheeted and :math:`3`-sheeted covering spaces of :math:`S^1 \vee S^1`, up to 
isomorphism of covering spaces without basepoints.

.. _Exercise 1-3-11:

**11.** Construct finite graphs :math:`X_1` and :math:`X_2` having a common finite-sheeted covering space
:math:`\tilde{X}_1=\tilde{X}_2`, but such that there is no space having both :math:`X_1` and :math:`X_2` as covering spaces.

.. _Exercise 1-3-12:

**12.** Let :math:`a` and :math:`b` be the generators of :math:`\pi_1(S^1 \vee S^1)` corresponding to the two :math:`S^1` 
summands. Draw a picture of the covering space of :math:`S^1 \vee S^1` corresponding to the 
normal subgroup generated by :math:`a^2,\,b^2`, and :math:`(ab)^4`, and prove that this covering space
is indeed the correct one.

.. _Exercise 1-3-13:

**13.** Determine the covering space of :math:`S^1 \vee S^1` corresponding to the subgroup of 
:Math:`\pi_1(S^1 \vee S^1)` generated by the cubes of all elements. The covering space is :math:`27`0sheeted
and can be drawn on a torus so that the complementary regions are nine triangles
with edges labeled :math:`aaa`, nine triangles with edges labeled :math:`bbb`, and nine hexagons
with edges labeled :math:`ababab`. [For the analogous problem with sixth powers instead
of cubes, the resulting covering space would have :math:`2^{28}3^{25}` sheets! And for :math:`k^{th}` powers
with :math:`k` sufficiently large, the covering space would have infinitely many sheets. The
underlying group theory question here, whether the quotient of :Math:`\mathbb{Z} * \mathbb{Z}` obtained by
factoring out all :math:`k^{th}` powers is finite, is known as Burnside's problem. It can also be 
asked for a free group on :math:`n` generators.]

.. _Exercise 1-3-14:

**14.** Find all the connected covering spaces of :math:`\mathbb{R}P^2 \vee \mathbb{R}P^2`.

.. _Exercise 1-3-15:

**15.** Let :math:`p:\tilde{X} \rightarrow X` be a simply-connected covering space of :math:`X` and let :math:`A \subset X` be a 
path-connected, locally path-connected subspace, with :math:`\tilde{A} \subset \tilde{X}` a path-component of 
:math:`p^{-1}(A)`. Show that :math:`p:\tilde{A}\rightarrow A` is the covering space corresponding to the kernel of the
map :math:`\pi_1(A) \rightarrow \pi_1(X)`.

.. _Exercise 1-3-16:

**16.** Given maps :math:`X \rightarrow Y \rightarrow Z` such that both :math:`Y\rightarrow Z` and the composition :Math:`X\rightarrow Z` are
covering spaces, show that :Math:`X \rightarrow Y` is a covering space if :math:`Z` is locally path-connected,
and show that this covering space is normal if :math:`X \rightarrow Z` is a normal covering space.

.. _Exercise 1-3-17:

**17.** Given a group :math:`G` and a normal subgroup :math:`N`, show that there exists a normal
covering space :math:`\tilde{X} \rightarrow X` with :math:`\pi_1(X) \approx,\, \pi_1(\tilde{X}) \approx N`, and deck transformation group
:math:`G(\tilde{X}) \approx G/N`.

.. _Exercise 1-3-18:

**18.** For a path-connected, locally path-connected, and semilocally simply-connected 
space :math:`X`, call a path-connected covering space :math:`\tilde{X} \rightarrow X` *abelian* if it is normal and has
abelian deck transformation group. Show that :math:`X` has an abelian covering space that is 
a covering space of every other abelian covering space of :math:`X`, and that such a 'universal'
abelian covering space is unique up to isomorphism. Describe this covering space
explicitly for :Math:`X=S^1 \vee S^1` and :Math:`X=S^1 \vee S^1 \vee S^1`.

.. _Exercise 1-3-19:

**19.** Use the preceding problem to show that a closed orientable surface :math:`M_g` of a genus
:math:`g` has a connected normal covering space with deck transformation group isomorphic
to :math:`\mathbb{Z}^n` (the product of :math:`n` copies of :math:`\mathbb{Z}`) iff :math:`n \leq 2g`. For :math:`n=3` and :Math:`g \geq 3`, describe such
a covering space explicitly as a subspace of :Math:`\mathbb{R}^3` with translations of :Math:`\mathbb{R}^3` as deck
transformations. Show that such a covering space in :Math:`\mathbb{R}^3` exists iff there is an embedding 
of :Math:`M_g` in the :Math:`3`-torus :math:`T^3 = S^1 \times S^1 \times S^1` such that the induced map :Math:`\pi_1(M_g) \rightarrow \pi_1(T^3)`
is surjective.

.. _Exercise 1-3-20:

**20.** Construct nonnormal covering spaces of the Klein bottle by a Klein bottle and by 
a torus.

.. _Exercise 1-3-21:

**21.** Let :math:`X` be the space obtained from a torus :Math:`S^1 \times S^1` by attaching a Möbius band via a 
homeomorphism from the boundary circle of the Möbius band to the circle :math:`S^1 \times \{x_0\}`
in the torus. Compute :Math:`\pi_1(X)`, describe the universal cover of :Math:`X`, and describe the 
action of :Math:`\pi_1(X)` on the universal cover. Do the same for the space :math:`Y` obtained by
attaching a Möbius band to :Math:`\mathbb{R}P^2` via a homeomorphism from its boundary circle to
the circle in :math:`\mathbb{R}P^2` formed by the :math:`1`-skeleton of the usual CW structure on :math:`\mathbb{R}P^2`.

.. _Exercise 1-3-22:

**22.** Given covering space actions of groups :math:`G_1` on :math:`X_1` and :math:`G_2` on :math:`X_2`, show that the 
action of :math:`G_1 \times G_2` on :math:`X_1 \times X_2` defined by :math:`(g_1,g_2)(x_1,x_2)=(g_1(x_1),g_2(x_2))` is a covering
space action, and that :math:`(X_1 \times X_2)/(G_1 \times G_2)` is homeomorphic to :math:`X_1 / G_1 \times X_2/G_2`.

.. _Exercise 1-3-23:

**23.** Show that if a group :math:`G` acts freely and properly discontinuously on a Hausdorff
space :math:`X`, then the action is a covering space action. (Here 'properly discontinuously'
means that each :math:`x \in X` has a neighborhood :math:`U` such that :math:`\{g\in G \mid U \cap g(U) \neq \emptyset\}` is
finite.) In particular, a free action of a finite group on a Hausdorff space is a covering 
space action.

.. _Exercise 1-3-24:

.. container::

    **24.** Given a covering space action of a group :math:`G` on a path-connected, locally path-connected
    space :math:`X`, then each subgroup :math:`H \subset G` determines a composition of covering
    spaces :math:`X \rightarrow X/H \rightarrow X/G`. Show:

    (a) Every path-connected covering space between :math:`X` and :math:`X/G` is isomorphic to :Math:`X/H`
        for some subgroup :math:`H \subset G`.

    (b) Two such covering spaces :math:`X/H_1` and :math:`X/H_2` of :math:`X/G` are isomorphic iff :math:`H_1` and :math:`H_2`
        are conjugate subgroups of :math:`G`.

    (c) The covering space :math:`X/H \rightarrow X/G` is normal iff :math:`H` is a normal subgroup of :math:`G`, in
        which case the group of deck transformations of this cover is :math:`G/H`.

.. _Exercise 1-3-25:

**25.** Let :Math:`\varphi: \mathbb{R}^2 \rightarrw \mathbb{R}^2` be the linear transformation :math:`\varphi (X,y) = (2x, y/2)`. This generates
an action of :math:`\mathbb{Z}` on :math:`X=\mathbb{R}^2-\{0\}`. Show this action is a covering space action and
compute :math:`\pi_1(X/\mathbb{Z})`. Show the orbit space :math:`X/\mathbb{Z}` is non-Hausdorff, and describe how it is
a union of four subspaces homeomorphic to :Math:`S^1 \times \mathbb{R}`, coming from the complementary
components of the :math:`x`-axis and the :math:`y`-axis.

.. _Exercise 1-3-26:

.. container::

    **26.** For a covering space :math:`p:\tilde{X} \rightarrow X` with :math:`X` connecetd, locally path-connected, and 
    semilocally simply-connected, show:

    (a) The components of :math:`\tilde{X}` are in one-to-one correspondence with the orbits of the 
        action of :math:`\pi_1(X,x_0)` on the fiber :math:`p^{-1}(x_0)`.
    (b) Under the Galois correspondence between connected covering spaces of :math:`X` and 
        subgroups of :math:`\pi_1(X,x_0)`, the subgroup corresponding to the component of :Math:`\tilde{X}`
        containing a given lift :math:`\tilde{x}_0` of :math:`x_0` is the *stabilizer* of :math:`\tilde{x}_0`, the subgorup consisting
        of elements whose action on the fiber leaves :math:`\tilde{x}_0` fixed.

.. _Exercise 1-3-27:

**27.** For a universal cover :math:`p:\tilde{X} \rightarrow X` there are two actions of :math:`\pi_1(X,x_0)` on the fiber
:math:`p^{-1}(x_0)`. The first is the action defined on page 69 in which the element of :math:`\pi_1(X,x_0)`
determined by a loop :math:`\gamma` sends :math:`\tilde{\gamma}(1)` to :math:`\tilde{\gamma}(0)` for each lift :math:`\tilde{\gamma}` of :math:`\gamma` to :math:`\tilde{X}`, and the second is
the action given by restricting deck transformations to the fiber (see :ref:`Proposition 1.39 <Proposition 1.39>`).
Show that these two actions are different when :math:`X=S^1\vee S^1` and when :math:`X=S^1 \times S^1`
and determine when the two actions are the same. [This is a revised version of the 
original form of this exercise.]

.. _Exercise 1-3-28:

**28.** Show that for a covering space action of a group :math:`G` on a simply-connected space :math:`Y`,
:math:`\pi_1(Y/G)` is isomorphic to :math:`G`. [If :math:`Y` is locally path-connected, this is a special case of 
part (c) of :ref:`Proposition 1.40 <Proposition 1.40>`]

.. _Exericse 1-3-29:

**29.** Let :math:`Y` be path-connected, locally path-connected, and simply-connected, and let
:math:`G_1` and :math:`G_2` be subgroups of :math:`\text{Homeo}(Y)` defining covering space actions on :math:`Y`. Show
that the orbit spaces :math:`Y/G_1` and :math:`Y/G_2` are homeomorphic iff :math:`G_1` and :math:`G_2` are conjugate
subgroups of :math:`\text{Homeo}(Y)`.

.. _Exercise 1-3-30:

**30.** Draw the Cayley graph of the group :math:`\mathbb{Z} * \mathbb{Z}_2 = \langle a,b \mid b^2 \rangle`.

.. _Exercise 1-3-31:

**31.** Show that the normal covering spaces of :math:`S^1 \vee S^1` are precisely the graphs that
are Cayley graphs of groups with two generators. More generally, the normal covering
spaces of the wedge sum of :math:`n` circles are the Cayley graphs of groups with :math:`n`
generators.

.. _Exercise 1-3-32:

.. container::

    **32.** Consider covering spaces :math:`p:\tilde{X} \rightarrow X` with :math:`\tilde{X}` and :math:`X` connected CW complexes,
    the cells of :math:`\tilde{X}` projecting homeomorphically onto cells of :math:`X`. Restricting :math:`p` to the 
    :math:`1`-skeleton then gives a covering space :math:`\tilde{X}^1 \rightarrow X^1` over the :math:`1`-skeleton of :math:`X`. Show:

    (a) Two such covering spaces :math:`\tilde{X}_1 \rightarrow X` and :math:`\tilde{X}_2 \rightarrow X` are isomorphic iff the restrictions
        :math:`\tilde{X}^1_1 \rightarrow X^1` and :math:`\tilde{X}^1_2 \rightarrow X^1` are isomorphic.
    (b) :math:`\tilde{X} \rightarrow X` is a normal covering space iff :math:`\tilde{X}^1 \rightarrow X^1` is normal.
    (c) The groups of deck transformations of the coverings :math:`\tilde{X} \rightarrow X` and :math:`\tilde{X}^1 \rightarrow X^1` are 
        isomorphic, via the restriction map.

.. _Exercise 1-3-33:

**33.** In :ref:`Example 1.44 <Example 1.44>` let :math:`d` be the greatest common divisor of :math:`m` and :math:`n`, and let
:math:`m'=m/d` and :math:`n'=n/d`. Show that the graph :math:`T_{m,n}/K` consists of :math:`m'` vertices
labeled :math:`a,\,n'` vertices labeled :math:`b`, together with :math:`d` edges joining each :math:`a` vertex to
each :math:`b` vertex. Deduce that the subgroup :math:`K \subset G_{m,n}` is free on :Math:`dm'n'-m'-n'+1`
generators.