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Algebraic Topology

Allen Hatcher

Chapter 1

The Fundamental Group - all with Video Answers

Educators

Chapter Questions

06:51

Problem 1

Show that composition of paths satisfies the following cancellation property: If $f_{0} \cdot g_{0} \simeq f_{1} \cdot g_{1}$ and $g_{0} \simeq g_{1}$ then $f_{0} \simeq f_{1}$.

Linda Hand
Linda Hand
Numerade Educator
07:44

Problem 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.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:32

Problem 1

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.

Victor Salazar
Victor Salazar
Numerade Educator
08:21

Problem 1

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.

Shubh Ashish
Shubh Ashish
Numerade Educator
01:02

Problem 1

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 $H$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:05

Problem 2

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

Anthony Ramos
Anthony Ramos
Numerade Educator
01:36

Problem 2

Let $X$ be a connected CW complex and $G$ a group such that every homomorphism
(X) $\rightarrow G$ is trivial. Show that every map $X \rightarrow K(G, 1)$ is nullhomotopic.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:04

Problem 2

Show that a connected graph retracts onto any connected subgraph.

WZ
Wen Zheng
Numerade Educator
02:41

Problem 2

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 $p_{1} \times p_{2}: \tilde{X}_{1} \times \tilde{X}_{2} \rightarrow X_{1} \times X_{2}$

Amany Waheeb
Amany Waheeb
Numerade Educator
02:16

Problem 2

Let $X \subset \mathbb{R}^{m}$ be the union of convex open sets $X_{1}, \cdots, X_{n}$ such that $X_{i} \cap X_{j} \cap X_{k} \neq \varnothing$ for all $i, j, k .$ Show that $X$ is simply-connected.

Uma Kumari
Uma Kumari
Numerade Educator
10:08

Problem 3

Show that every graph product of trivial groups is free.

Chris Trentman
Chris Trentman
Numerade Educator
02:36

Problem 3

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.

WZ
Wen Zheng
Numerade Educator
04:04

Problem 3

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

WZ
Wen Zheng
Numerade Educator
05:53

Problem 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.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:51

Problem 3

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 $h$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:27

Problem 4

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.

Jerry Zhang
Jerry Zhang
Numerade Educator
01:17

Problem 4

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

Ernest Castorena
Ernest Castorena
Numerade Educator
02:49

Problem 4

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 $Y$.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:31

Problem 4

Use van Kampen's theorem to compute $A * c$ as a quotient of $A * \mathbb{Z},$ as stated in the text.

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
02:16

Problem 4

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.

Uma Kumari
Uma Kumari
Numerade Educator
02:49

Problem 5

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.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:40

Problem 5

Consider the graph of groups $\Gamma$ having one vertex, $\mathbb{Z}$, and one edge, the map $\mathbb{Z} \rightarrow \mathbb{Z}$ that is multiplication by $2,$ realized by the 2 -sheeted covering space $S^{1} \rightarrow S^{1}$. Show that $\pi_{1}(K \Gamma)$ has presentation $\left\langle a, b | b a b^{-1} a^{-2}\right\rangle$ and describe the universal cover of $K \Gamma$ explicitly as a product $T \times \mathbb{R}$ with $T$ a tree. [The group $\pi_{1}(K \Gamma)$ is the first in a family of groups called Baumslag-Solitar groups, having presentations of the form $\left\langle a, b | b a^{m} b^{-1} a^{-n}\right\rangle .$ These are HNN extensions $\mathbb{Z} *_{z} .1$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:16

Problem 5

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.

Uma Kumari
Uma Kumari
Numerade Educator
04:42

Problem 5

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 $\varphi: S^{1} \rightarrow S^{1}$.

Mengchun Cai
Mengchun Cai
Numerade Educator
04:16

Problem 5

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.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:16

Problem 6

Suppose a space $Y$ is obtained from a path-connected subspace $X$ by attaching $n$ -cells for a fixed $n \geq 3 .$ Show that the inclusion $X \hookrightarrow Y$ induces an isomorphism on $\pi_{1}$. ISee the proof of Proposition 1.26 .1 Apply this to show that the complement
of a discrete subspace of $\mathbb{R}^{n}$ is simply-connected if $n \geq 3$

Uma Kumari
Uma Kumari
Numerade Educator
01:40

Problem 6

Show that for a graph of groups all of whose edge homomorphisms are injective maps $\mathbb{Z} \rightarrow \mathbb{Z},$ we can choose $K \Gamma$ to have universal cover a product $T \times \mathbb{R}$ with $T$ a tree. Work out in detail the case that the graph of groups is the infinite sequence $\mathbb{Z} \stackrel{2}{\longrightarrow} \mathbb{Z} \stackrel{3}{\longrightarrow} \mathbb{Z} \stackrel{4}{\longrightarrow} \mathbb{Z} \rightarrow \cdots$ where the map $\mathbb{Z} \stackrel{n}{\longrightarrow} \mathbb{Z}$ is multiplication by $n .$ Show that $\pi_{1}(K \Gamma)$ is isomorphic to $Q$ in this case. How would one modify this example to get $\pi_{1}(K \Gamma)$ isomorphic to the subgroup of $Q$ consisting of rational numbers with denominator a power of $2 ?$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
22:21

Problem 6

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 $A$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:19

Problem 6

Let $X$ be the shrinking wedge of circles in Example $1.25,$ and let $\tilde{X}$ be its covering space shown in the figure below. Construct a two-sheeted covering space $Y \rightarrow \tilde{X}$ such that the composition $Y \rightarrow \tilde{X} \rightarrow X$ 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.

Sarah X
Sarah X
Numerade Educator
01:47

Problem 6

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 $F^{\prime}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:25

Problem 7

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.

Ely Crowder
Ely Crowder
Numerade Educator
08:21

Problem 7

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.

Shubh Ashish
Shubh Ashish
Numerade Educator
01:04

Problem 7

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 $\pi_{1}(X)$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:01

Problem 7

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 $x_{0} \in X$

Donald Albin
Donald Albin
Numerade Educator
03:45

Problem 7

Let $Y$ be the $q$ uasi-circle shown in the figure, a closed subspace of $\mathbb{R}^{2}$ consisting of a portion of the graph of $y=\sin (1 / x),$ the segment [-1,1] in the $y$ -axis, and an arc connecting these two pieces. Collapsing the segment of $Y$ in the $y$ -axis to a point gives a quotient map $f: Y \rightarrow S^{1}$. Show that $f$ does not lift to the covering space $\mathbb{R} \rightarrow S^{1}$, even though $\pi_{1}(Y)=0 .$ Thus local path-connectedness of $Y$ is a necessary hypothesis in the lifting criterion.

Narayan Hari
Narayan Hari
Numerade Educator
01:35

Problem 8

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.]

Nick Johnson
Nick Johnson
Numerade Educator
01:35

Problem 8

Show that a finite graph product of finitely generated groups is finitely generated, and similarly for finitely presented groups.

Nick Johnson
Nick Johnson
Numerade Educator
03:45

Problem 8

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.

Narayan Hari
Narayan Hari
Numerade Educator
08:26

Problem 8

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.

Cameron Bunney
Cameron Bunney
Numerade Educator
02:42

Problem 8

Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of the path-connected, locally path-connected spaces $X$ and $Y$. Show that if $X \simeq Y$ then $\tilde{X}=\tilde{Y}$. IExercise 10 in Chapter 0 may be helpful.]

Arwa Ali
Arwa Ali
Numerade Educator
15:25

Problem 9

In the surface $M_{g}$ of genus $g,$ let $C$ be a circle that separates $M_{g}$ into two compact subsurfaces $M_{h}^{\prime}$ and $M_{k}^{\prime}$ obtained from the closed surfaces $M_{h}$
and $M_{k}$ by deleting an open disk from each. Show that $M_{h}^{\prime}$ does not retract onto its boundary circle $C,$ and hence $M_{g}$ does not retract onto $C .$ IHint: abelianize $\pi_{1} .$ But show that $M_{g}$ does retract onto the nonseparating circle $C^{\prime}$ in the figure.
FIGURE CANT COPY

Regina Hays
Regina Hays
Numerade Educator
02:05

Problem 9

Using covering spaces, show that an index $n$ subgroup $H$ of a group $G$ has at most
$n$ conjugate subgroups $g H g^{-1}$ in $G .$ Apply this to show that there exists a normal subgroup $K \subset G$ of finite index with $K \subset H .$ [For the latter statement, consider the intersection of all the conjugate subgroups $g H g^{-1} .$ This is the maximal normal subgroup of $G$ contained in $H .$ ]

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:56

Problem 9

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 $\mathbb{R} \rightarrow S^{1} .$ ]

Chris Trentman
Chris Trentman
Numerade Educator
01:40

Problem 9

Show that a finite graph product of finite groups has a free subgroup of finite index, by constructing a finite-sheeted covering space of $K \Gamma$ from universal covers of the mapping cylinders of $K \Gamma .$ IThe converse is also true for finitely generated groups; see [Scott \& Wall 1979] for more on this.]

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:21

Problem 9

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 $\left.s \mapsto\left(\theta_{0}, s\right) \text { for fixed } \theta_{0} \in S^{1} .\right]$

Jack Chen
Jack Chen
Numerade Educator
02:49

Problem 10

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 $Z \rightarrow X$.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:19

Problem 10

Consider two arcs $\alpha$ and $\beta$ embedded in $D^{2} \times I$ as shown in the figure. The loop $y$ is obviously nullhomotopic in $D^{2} \times I,$ but show that there is no nullhomotopy of $\gamma$ in the complement of $\alpha \cup \beta$
FIGURE CANT COPY

Lauren Shelton
Lauren Shelton
Numerade Educator
06:36

Problem 10

Find all the connected 2 -sheeted and 3 -sheeted covering spaces of $S^{1} \vee S^{1},$ up to isomorphism of covering spaces without basepoints.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:01

Problem 10

Does the Borsuk-Ulam theorem hold for the torus? In other words, for every map $f: S^{1} \times S^{1} \rightarrow \mathbb{R}^{2}$ must there exist $(x, y) \in S^{1} \times S^{1}$ such that $f(x, y)=f(-x,-y) ?$

Doruk Isik
Doruk Isik
Numerade Educator
01:13

Problem 11

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.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
01:21

Problem 11

Apply the two preceding problems to show that if $F$ is a finitely generated free group and $x \in F$ is not the identity element, then there is a normal subgroup $H \subset F$ of finite index such that $x \notin H .$ Hence $x$ has nontrivial image in a finite quotient group of $F .$ In this situation one says $F$ is residually finite.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:09

Problem 11

Construct finite graphs $X_{1}$ and $X_{2}$ having a common finite-sheeted covering space $\tilde{X}_{1}=\tilde{X}_{2},$ but such that there is no space having both $X_{1}$ and $X_{2}$ as covering spaces.

Ashley Boni
Ashley Boni
Numerade Educator
03:47

Problem 11

The mapping torus $T_{f}$ of a map $f: X \rightarrow X$ is the quotient of $X \times I$ obtained by identifying each point $(x, 0)$ with $(f(x), 1) .$ In the case $X=S^{1} \vee S^{1}$ with $f$ basepoint-preserving, compute a presentation for $\pi_{1}\left(T_{f}\right)$ in terms of the induced $\operatorname{map} f_{*}: \pi_{1}(X) \rightarrow \pi_{1}(X) .$ Do the same when $X=S^{1} \times S^{1} .$ [One way to do this is to
regard $T_{f}$ as built from $X \vee S^{1}$ by attaching cells.

Regina Hays
Regina Hays
Numerade Educator
01:21

Problem 12

Let $F$ be a finitely generated free group, $H \subset F$ a finitely generated subgroup, and $x \in F-H .$ Show there is a subgroup $K$ of finite index in $F$ such that $K \supset H$ and $x \notin K .$ [Apply Exercise 10.]

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:48

Problem 12

Given a map $f: X \rightarrow Y$ and a path $h: I \rightarrow X$ from $x_{0}$ to $x_{1},$ show that $f_{*} \beta_{h}=\beta_{f h} f_{*}$ in the diagram to the right.

James Kiss
James Kiss
Numerade Educator
00:59

Problem 12

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.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:11

Problem 12

The Klein bottle is usually pictured as a subspace of $\mathbb{R}^{3}$ like the subspace $X \subset \mathbb{R}^{3}$ shown in the first figure at the right. If one wanted a model
that could actually function as a bottle, one would
delete the open disk bounded by the circle of self- intersection of $X,$ producing a subspace $Y \subset X .$ Show that $\pi_{1}(X) \approx \mathbb{Z} * \mathbb{Z}$ and that
$\pi_{1}(Y)$ has the presentation $\left\langle a, b, c | a b a^{-1} b^{-1} c b^{\varepsilon} c^{-1}\right\rangle$ for $\varepsilon=\pm 1 .$ (Changing the
sign of $\varepsilon$ gives an isomorphic group, as it happens.) Show also that $\pi_{1}(Y)$ is isomorphic to $\pi_{1}\left(\mathbb{R}^{3}-Z\right)$ for $Z$ the graph shown in the figure. The groups $\pi_{1}(X)$ and $\pi_{1}(Y)$ are not isomorphic, but this is not easy to prove; see the discussion in Example 1B.13.
FIGURE CANT COPY

James Kiss
James Kiss
Numerade Educator
08:12

Problem 13

Determine the covering space of $S^{1} \vee S^{1}$ corresponding to the subgroup of $\pi_{1}\left(S^{1} \vee S^{1}\right)$ generated by the cubes of all elements. The covering space is 27 -sheeted and can be drawn on a torus so that the complementary regions are nine triangles with edges labeled aaa, nine triangles with edges labeled $b b b,$ and nine hexagons with edges labeled ababab. IFor the analogous problem with sixth powers instead of cubes, the resulting covering space would have $2^{28} 3^{25}$ sheets! And for $k^{t h}$ powers with $k$ sufficiently large, the covering space would have infinitely many sheets. The underlying group theory question here, whether the quotient of $\mathbb{Z} * \mathbb{Z}$ obtained by factoring out all $k^{\text {th }}$ powers is finite, is known as Burnside's problem. It can also be asked for a free group on $n$ generators.

Yuou Sun
Yuou Sun
Numerade Educator
01:21

Problem 13

Let $x$ be a nontrivial element of a finitely generated free group $F .$ Show there is a finite-index subgroup $H \subset F$ in which $x$ is one element of a basis. [Exercises 4 and
10 may be helpful.]

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:25

Problem 13

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

Ely Crowder
Ely Crowder
Numerade Educator
02:09

Problem 13

The space $Y$ in the preceding exercise can be obtained from a disk with two holes by identifying its three boundary circles. There are only two essentially different ways of identifying the three boundary circles. Show that the other way yields a space $Z$ with $\pi_{1}(Z)$ not isomorphic to $\pi_{1}(Y) .$ [Abelianize the fundamental groups to show they are not isomorphic.]

Amy Jiang
Amy Jiang
Numerade Educator
08:50

Problem 14

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.

Ely Crowder
Ely Crowder
Numerade Educator
03:38

Problem 14

Show that the existence of maximal trees is equivalent to the Axiom of Choice.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:52

Problem 14

Construct infinitely many nonhomotopic retractions $S^{1} \vee S^{1} \rightarrow S^{1}$.

Chris Trentman
Chris Trentman
Numerade Educator
01:12

Problem 14

Find all the connected covering spaces of $\mathrm{RP}^{2} \vee \mathbb{R P}^{2}$

Jeffrey Utley
Jeffrey Utley
Numerade Educator
02:31

Problem 15

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

Victor Salazar
Victor Salazar
Numerade Educator
06:39

Problem 15

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 $\pi_{1}\left(X_{0}, x_{0}\right) \rightarrow \pi_{1}\left(X, x_{0}\right)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:58

Problem 15

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 $L(X) \rightarrow X$.

Anthony Ramos
Anthony Ramos
Numerade Educator

Problem 16

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.

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Problem 16

Show that the fundamental group of the surface of infinite genus shown below is free on an infinite number of generators.
FIGURE CANT COPY

Nick Johnson
Nick Johnson
Numerade Educator
02:16

Problem 16

Using the technique in the proof of Proposition $1.14,$ show that if a space $X$ is obtained from a path-connected subspace $A$ by attaching a cell $e^{n}$ with $n \geq 2,$ then the inclusion $A \hookrightarrow X$ induces a surjection on $\pi_{1}$.

Uma Kumari
Uma Kumari
Numerade Educator
00:59

Problem 17

Given a group $G$ and a normal subgroup $N,$ show that there exists a normal covering space $\tilde{X} \rightarrow X$ with $\pi_{1}(X) \approx G, \pi_{1}(\tilde{X}) \approx N,$ and deck transformation group $G(\tilde{X}) \approx G / N$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:22

Problem 17

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

Angelo Rendina
Angelo Rendina
Numerade Educator
01:21

Problem 17

Modify the proof of Proposition 1.14 to give an elementary proof that $\pi_{1}\left(S^{1}\right)$ is cyclic, generated by the standard loop winding once around the circle. [The more difficult part of the calculation of $\pi_{1}\left(S^{1}\right)$ is therefore the fact that no iterate of this loop is nullhomotopic.]

Jay Patel
Jay Patel
Numerade Educator
02:03

Problem 18

Suppose $f_{t}: X \rightarrow X$ is a homotopy such that $f_{0}$ and $f_{1}$ are each the identity map. Use Lemma 1.19 to show that for any $x_{0} \in X,$ the loop $f_{t}\left(x_{0}\right)$ represents an element of the center of $\pi_{1}\left(X, x_{0}\right) .$ [One can interpret the result as saying that a loop represents an element of the center of $\left.\pi_{1}(X) \text { if it extends to a loop of maps } X \rightarrow X .\right\rfloor$

Nick Johnson
Nick Johnson
Numerade Educator
00:48

Problem 18

In this problem we use the notions of suspension, reduced suspension, cone, and mapping cone defined in Chapter 0. Let $X$ be the subspace of $\mathbb{R}$ consisting of the sequence $1,1 / 2,1 / 3,1 / 4, \cdots$ together with its limit point $0 .$
(a) For the suspension $S X,$ show that $\pi_{1}(S X)$ is free on a countably infinite set of generators, and deduce that $\pi_{1}(S X)$ is countable. In contrast to this, the reduced suspension $\Sigma X,$ obtained from $S X$ by collapsing the segment $\{0\} \times I$ to a point, is the shrinking wedge of circles in Example $1.25,$ with an uncountable fundamental group.
(b) Let $C$ be the mapping cone of the quotient map $S X \rightarrow \Sigma X .$ Show that $\pi_{1}(C)$ is uncountable by constructing a homomorphism from $\pi_{1}(C)$ onto $\Pi_{\infty} \mathbb{Z} / \oplus_{\infty} \mathbb{Z} .$ Note that $C$ is the reduced suspension of the cone $C X .$ Thus the reduced suspension of a contractible space need not be contractible, unlike the unreduced suspension.

Nick Johnson
Nick Johnson
Numerade Educator
01:36

Problem 18

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 $X=S^{1} \vee S^{1} \vee S^{1}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
10:24

Problem 19

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

Chris Trentman
Chris Trentman
Numerade Educator
05:51

Problem 19

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.

Mengchun Cai
Mengchun Cai
Numerade Educator
03:45

Problem 20

Let $X$ be the subspace of $\mathbb{R}^{2}$ that is the union of the circles $C_{n}$ of radius $n$ and center $(n, 0)$ for $n=1,2, \cdots$. Show that $\pi_{1}(X)$ is the free group $*_{n} \pi_{1}\left(C_{n}\right),$ the same as for the infinite wedge sum $V_{\infty} S^{1} .$ Show that $X$ and $V_{\infty} S^{1}$ are in fact homotopy equivalent, but not homeomorphic.
21. Show that the join $X * Y$ of two nonempty spaces $X$ and $Y$ is simply-connected
if $X$ is path-connected.

Narayan Hari
Narayan Hari
Numerade Educator
03:11

Problem 20

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

James Kiss
James Kiss
Numerade Educator
08:26

Problem 21

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 $\mathrm{RP}^{2}$.

Cameron Bunney
Cameron Bunney
Numerade Educator
01:58

Problem 22

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 $X_{1} / G_{1} \times X_{2} / G_{2}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 23

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.

Raj Bala
Raj Bala
Numerade Educator
01:51

Problem 24

Given a covering space action of a group $G$ on a path-connected, locally pathconnected space $X$, then each subgroup $H \subset G$ determines a composition of covering spaces $X \rightarrow X / H \rightarrow X / G$. Show:
(a) Every path-connected covering space between $X$ and $X / G$ is isomorphic to $X / H$ for some subgroup $H \subset G$.
(b) Two such covering spaces $X / H_{1}$ and $X / H_{2}$ of $X / G$ are isomorphic iff $H_{1}$ and
$H_{2}$ are conjugate subgroups of $G$
(c) The covering space $X / H \rightarrow X / G$ is normal iff $H$ is a normal subgroup of $G,$ in which case the group of deck transformations of this cover is $G / H$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:38

Problem 25

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.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:36

Problem 26

For a covering space $p: \tilde{X} \rightarrow X$ with $X$ connected, locally path-connected, and semilocally simply-connected, show:
(a) The components of $\tilde{X}$ are in one-to-one correspondence with the orbits of the action of $\pi_{1}\left(X, x_{0}\right)$ on the fiber $p^{-1}\left(x_{0}\right)$
(b) Under the Galois correspondence between connected covering spaces of $X$ and subgroups of $\pi_{1}\left(X, x_{0}\right),$ the subgroup corresponding to the component of $\tilde{X}$ containing a given lift $\tilde{x}_{0}$ of $x_{0}$ is the stabilizer of $\tilde{x}_{0},$ the subgroup consisting of elements whose action on the fiber leaves $\tilde{x}_{0}$ fixed.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
18:07

Problem 27

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?

Donald Albin
Donald Albin
Numerade Educator
03:14

Problem 28

Generalize the proof of Theorem 1.7 to show that for a covering space action of a group $G$ on a simply-connected space $Y, \pi_{1}(Y / G)$ is isomorphic to $G .$ IIf $Y$ is locally path-connected, this is a special case of part (b) of Proposition 1.40.]

Chris Trentman
Chris Trentman
Numerade Educator
01:36

Problem 29

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).

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:23

Problem 30

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

Lucas Finney
Lucas Finney
Numerade Educator
02:34

Problem 31

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.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator

Problem 32

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.

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04:33

Problem 33

In Example 1.44 let $d$ be the greatest common divisor of $m$ and $n$, and let $m^{\prime}=m / d$ and $n^{\prime}=n / d .$ Show that the graph $T_{m, n} / K$ consists of $m^{\prime}$ vertices labeled $a, n^{\prime}$ vertices labeled $b,$ together with $d$ edges joining each $a$ vertex to each $b$ vertex. Deduce that the subgroup $K \subset G_{m, n}$ is free on $\ell m^{\prime} n^{\prime}-m^{\prime}-n^{\prime}+1$ generators.

Brian Lin
Brian Lin
Numerade Educator