Chapter 1

Suppose a group \(G\) acts simplicially on a \(\Delta\) -complex \(X,\) where 'simplicially' means that each element of \(G\) takes each simplex of \(X\) onto another simplex by a linear homeomorphism. If the action is free, show it is a covering space action.

Let \(X\) be a graph in which each vertex is an endpoint of only finitely many edges. Show that the weak topology on \(X\) is a metric topology.

Show that the free product \(G * H\) of nontrivial groups \(G\) and \(H\) has trivial center, and that the only elements of \(G * H\) of finite order are the conjugates of finite-order elements of \(G\) and \(H\)

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

Show that if \(p_{1}: \tilde{X}_{1} \rightarrow X_{1}\) and \(p_{2}: \tilde{X}_{2} \rightarrow X_{2}\) are covering spaces, so is their product \(p_{1} \times p_{2}: \tilde{X}_{1} \times \tilde{X}_{2} \rightarrow X_{1} \times X_{2}\)

Show that the change-of-basepoint homomorphism \(\beta_{h}\) depends only on the homotopy class of \(h\).

Show that every graph product of trivial groups is free.

For a finite graph \(X\) define the Euler characteristic \(\chi(X)\) to be the number of vertices minus the number of edges. Show that \(\chi(X)=1\) if \(X\) is a tree, and that the rank (number of elements in a basis) of \(\pi_{1}(X)\) is \(1-\chi(X)\) if \(X\) is connected.

Show that the complement of a finite set of points in \(\mathbb{R}^{n}\) is simply-connected if \(n \geq 3\)

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

For a path-connected space \(X,\) show that \(\pi_{1}(X)\) is abelian iff all basepoint-change homomorphisms \(\beta_{h}\) depend only on the endpoints of the path \(h\).

If \(X\) is a finite graph and \(Y\) is a subgraph homeomorphic to \(S^{1}\) and containing the base point \(x_{0},\) show that \(\pi_{1}\left(X, x_{0}\right)\) has a basis in which one element is represented by the loop \(Y\).

Let \(X \subset \mathbb{R}^{3}\) be the union of \(n\) lines through the origin. Compute \(\pi_{1}\left(\mathbb{R}^{3}-X\right)\)

Construct a simply-connected covering space of the space \(X \subset \mathbb{R}^{3}\) that is the union of a sphere and a diameter. Do the same when \(X\) is the union of a sphere and a circle intersecting it in two points.

A subspace \(X \subset \mathbb{R}^{n}\) is said to be star-shaped if there is a point \(x_{0} \in X\) such that, for each \(x \in X,\) the line segment from \(x_{0}\) to \(x\) lies in \(X .\) Show that if a subspace \(X \subset \mathbb{R}^{n}\) is locally star-shaped, in the sense that every point of \(X\) has a star-shaped neighborhood in \(X,\) then every path in \(X\) is homotopic in \(X\) to a piecewise linear path, that is, a path consisting of a finite number of straight line segments traversed at constant speed. Show this applies in particular when \(X\) is open or when \(X\) is a union of finitely many closed convex sets.

Construct a connected graph \(X\) and maps \(f, g: X \rightarrow X\) such that \(f g=\mathbb{1}\) but \(f\) and \(g\) do not induce isomorphisms on \(\pi_{1} .\) [Note that \(f_{*} g_{*}=\mathbb{1}\) implies that \(f_{*}\) is surjective and \(g_{*}\) is injective.

Let \(X \subset \mathbb{R}^{2}\) be a finite graph that is the union of the edges of a convex polygon and a finite number of line segments having endpoints on these edges. (a) Show that \(\pi_{1}(X)\) is free with a basis consisting of loops formed by the boundaries of the bounded complementary regions of \(X,\) joined to a basepoint by paths in \(X .\) (b) Show this is true for all choices of paths to the basepoint.

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

Show that every homomorphism \(\pi_{1}\left(S^{1}\right) \rightarrow \pi_{1}\left(S^{1}\right)\) can be realized as the induced homomorphism \(\varphi_{*}\) of a map \(\varphi: S^{1} \rightarrow S^{1}\).

Let \(F\) be the free group on two generators and let \(F^{\prime}\) be its commutator subgroup. Find a set of free generators for \(F^{\prime}\) by considering the covering space of the graph \(S^{1} \vee S^{1}\) corresponding to \(F^{\prime}\).

Given a space \(X\) and a path-connected subspace \(A\) containing the basepoint \(x_{0}\) show that the map \(\pi_{1}\left(A, x_{0}\right) \rightarrow \pi_{1}\left(X, x_{0}\right)\) induced by the inclusion \(A \hookrightarrow X\) is surjective iff every path in \(X\) with endpoints in \(A\) is homotopic to a path in \(A\).

If \(F\) is a finitely generated free group and \(N\) is a nontrivial normal subgroup of infinite index, show, using covering spaces, that \(N\) is not finitely generated.

Show that every graph product of groups can be realized by a graph whose vertices are partitioned into two subsets, with every oriented edge going from a vertex in the first subset to a vertex in the second subset.

Let \(X\) be the quotient space of \(S^{2}\) obtained by identifying the north and south poles to a single point. Put a cell complex structure on \(X\) and use this to compute \(\pi_{1}(X)\)

Show that for a space \(X,\) the following three conditions are equivalent: (a) Every map \(S^{1} \rightarrow X\) is homotopic to a constant map, with image a point. (b) Every map \(S^{1} \rightarrow X\) extends to a map \(D^{2} \rightarrow X\) (c) \(\boldsymbol{\pi}_{1}\left(X, x_{0}\right)=0\) for all \(x_{0} \in X\)

Show that a finitely generated group has only a finite number of subgroups of a given finite index. [First do the case of free groups, using covering spaces of graphs. The general case then follows since every group is a quotient group of a free group.]

Compute the fundamental group of the space obtained from two tori \(S^{1} \times S^{1}\) by identifying a circle \(S^{1} \times\left\\{x_{0}\right\\}\) in one torus with the corresponding circle \(S^{1} \times\left\\{x_{0}\right\\}\) in the other torus.

We can regard \(\pi_{1}\left(X, x_{0}\right)\) as the set of basepoint- preserving homotopy classes of maps \(\left(S^{1}, s_{0}\right) \rightarrow\left(X, x_{0}\right) .\) Let \(\left[S^{1}, X\right]\) be the set of homotopy classes of maps \(S^{1} \rightarrow X\) with no conditions on basepoints. Thus there is a natural map \(\Phi: \pi_{1}\left(X, x_{0}\right) \rightarrow\left[S^{1}, X\right]\) obtained by ignoring basepoints. Show that \(\Phi\) is onto if \(X\) is path-connected, and that \(\Phi([f])=\Phi([g])\) iff \([f]\) and \([g]\) are conjugate in \(\pi_{1}\left(X, x_{0}\right) .\) Hence \(\Phi\) induces a oneto-one correspondence between \(\left[S^{1}, X\right]\) and the set of conjugacy classes in \(\pi_{1}(X)\) when \(X\) is path-connected.

Show that if a path-connected, locally path-connected space \(X\) has \(\pi_{1}(X)\) finite, then every map \(X \rightarrow S^{1}\) is nullhomotopic. IUse the covering space \(\mathbb{R} \rightarrow S^{1} .\) ]

Define \(f: S^{1} \times I \rightarrow S^{1} \times I\) by \(f(\theta, s)=(\theta+2 \pi s, s),\) so \(f\) restricts to the identity on the two boundary circles of \(S^{1} \times I .\) Show that \(f\) is homotopic to the identity by a homotopy \(f_{t}\) that is stationary on one of the boundary circles, but not by any homotopy \(f_{t}\) that is stationary on both boundary circles. [Consider what \(f\) does to the path \(\left.s \mapsto\left(\theta_{0}, s\right) \text { for fixed } \theta_{0} \in S^{1} .\right]\)

Let \(X\) be the wedge sum of \(n\) circles, with its natural graph structure, and let \(\tilde{X} \rightarrow X\) be a covering space with \(Y \subset \tilde{X}\) a finite connected subgraph. Show there is a finite graph \(Z \supset Y\) having the same vertices as \(Y,\) such that the projection \(Y \rightarrow X\) extends to a covering space \(Z \rightarrow X\).

Let \(A_{1}, A_{2}, A_{3}\) be compact sets in \(\mathbb{R}^{3} .\) Use the Borsuk-Ulam theorem to show that there is one plane \(P \subset \mathbb{R}^{3}\) that simultaneously divides each \(A_{i}\) into two pieces of equal measure.

Let \(a\) and \(b\) be the generators of \(\pi_{1}\left(s^{1} \vee s^{1}\right)\) 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 \((a b)^{4},\) and prove that this covering space is indeed the correct one.

Show, using fundamental groups and induced homomorphisms, that there is no retraction of the Möbius band onto its boundary circle.

Consider the quotient space of a cube \(I^{3}\) obtained by identifying each square face with the opposite square face via the right-handed screw motion consisting of a translation by one unit in the direction perpendicular to the face combined with a one-quarter twist of the face about its center point. Show this quotient space \(X\) is a cell complex with two 0-cells, four 1-cells, three 2-cells, and one 3-cell. Using this structure, show that \(\pi_{1}(X)\) is the quaternion group \(\\{\pm 1, \pm i, \pm j, \pm k\\},\) of order eight.

Given a space \(X\) with basepoint \(x_{0} \in X,\) we may construct a CW complex \(L(X)\) having a single 0 -cell, a 1 -cell for each loop in \(X\) based at \(x_{0},\) and a 2 -cell for each map of a standard triangle \(T\) into \(X\) taking the three vertices to the basepoint. Such a 2-cell is attached to the three 1 -cells that are the loops obtained by restricting the map to the three edges of \(T .\) Show that \(\pi_{1}(L(X))\) is isomorphic to \(\pi_{1}\left(X, x_{0}\right)\) via an isomorphism induced by a natural map \(L(X) \rightarrow X\).

If \(X_{0}\) is the path-component of a space \(X\) containing the basepoint \(x_{0},\) show that the inclusion \(X_{0} \hookrightarrow X\) induces an isomorphism \(\pi_{1}\left(X_{0}, x_{0}\right) \rightarrow \pi_{1}\left(X, x_{0}\right)\).

Given maps \(X \rightarrow Y \rightarrow Z\) such that both \(Y \rightarrow Z\) and the composition \(X \rightarrow Z\) are covering spaces, show that \(X \rightarrow Y\) is a covering space if \(Z\) is locally path-connected, and show that this covering space is normal if \(X \rightarrow Z\) is a normal covering space.

Show that \(\pi_{1}\left(\mathbb{R}^{2}-\mathbb{Q}^{2}\right)\) is uncountable.

For a path-connected, locally path-connected, and semilocally simply-connected space \(X,\) call a path-connected covering space \(\tilde{X} \rightarrow X\) 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 a 'universal' abelian covering space is unique up to isomorphism. Describe this covering space explicitly for \(X=S^{1} \vee S^{1}\) and \(X=S^{1} \vee S^{1} \vee S^{1}\)

Show that the subspace of \(\mathbb{R}^{3}\) that is the union of the spheres of radius \(1 / n\) and center \((1 / n, 0,0)\) for \(n=1,2, \cdots\) is simply- connected.

Let \(X\) be the space obtained from a torus \(S^{1} \times S^{1}\) by attaching a Möbius band via a homeomorphism from the boundary circle of the Möbius band to the circle \(S^{1} \times\left\\{x_{0}\right\\}\) in the torus. Compute \(\pi_{1}(X),\) describe the universal cover of \(X,\) and describe the action of \(\pi_{1}(X)\) on the universal cover. Do the same for the space \(Y\) obtained by attaching a Möbius band to \(\mathbb{R P}^{2}\) via a homeomorphism from its boundary circle to a circle in \(\mathrm{RP}^{2}\) lifting to the equator in the covering space \(S^{2}\) of \(\mathrm{RP}^{2}\).

Given covering space actions of groups \(G_{1}\) on \(X_{1}\) and \(G_{2}\) on \(X_{2},\) show that the action of \(G_{1} \times G_{2}\) on \(X_{1} \times X_{2}\) defined by \(\left(g_{1}, g_{2}\right)\left(x_{1}, x_{2}\right)=\left(g_{1}\left(x_{1}\right), g_{2}\left(x_{2}\right)\right)\) is a covering space action, and that \(\left(X_{1} \times X_{2}\right) /\left(G_{1} \times G_{2}\right)\) is homeomorphic to \(X_{1} / G_{1} \times X_{2} / G_{2}\)

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

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

For a universal cover \(p: \tilde{X} \rightarrow X\) we have two actions of \(\pi_{1}\left(X, x_{0}\right)\) on the fiber \(p^{-1}\left(x_{0}\right),\) namely the action given by lifting loops at \(x_{0}\) and the action given by restricting deck transformations to the fiber. Are these two actions the same when \(X=S^{1} \vee S^{1}\) or \(X=S^{1} \times S^{1} ?\) Do the actions always agree when \(\pi_{1}\left(X, x_{0}\right)\) is abelian?

Let \(Y\) be path-connected, locally path-connected, and simply-connected, and let \(G_{1}\) and \(G_{2}\) be subgroups of Homeo(Y) defining covering space actions on \(Y .\) Show that the orbit spaces \(Y / G_{1}\) and \(Y / G_{2}\) are homeomorphic iff \(G_{1}\) and \(G_{2}\) are conjugate subgroups of Homeo(Y).

Draw the Cayley graph of the group \(\mathbb{Z} * \mathbb{Z}_{2}=\left\langle a, b | b^{2}\right\rangle\)

Show that the normal covering spaces of \(S^{1} \vee S^{1}\) are precisely the graphs that are Cayley graphs of groups with two generators. More generally, the normal covering spaces of the wedge sum of \(n\) circles are the Cayley graphs of groups with \(n\) generators.

Consider covering spaces \(\boldsymbol{p}: \tilde{\boldsymbol{X}} \rightarrow X\) with \(\tilde{\boldsymbol{X}}\) and \(X\) connected CW complexes, the cells of \(\tilde{X}\) projecting homeomorphically onto cells of \(X\). Restricting \(p\) to the 1-skeleton then gives a covering space \(\tilde{X}^{1} \rightarrow X^{1}\) over the 1 -skeleton of \(X .\) Show: (a) Two such covering spaces \(\tilde{X}_{1} \rightarrow X\) and \(\tilde{X}_{2} \rightarrow X\) are isomorphic iff the restrictions \(\tilde{x}_{1}^{1} \rightarrow x^{1}\) and \(\tilde{X}_{2}^{1} \rightarrow X^{1}\) are isomorphic. (b) \(\widetilde{x} \rightarrow X\) is a normal covering space iff \(\tilde{X}^{1} \rightarrow X^{1}\) is normal. (c) The groups of deck transformations of the coverings \(\tilde{X} \rightarrow X\) and \(\tilde{X}^{1} \rightarrow X^{1}\) are isomorphic, via the restriction map.