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Algebraic Topology

Allen Hatcher

Chapter 2

Homology - all with Video Answers

Educators

Chapter Questions

25:17

Problem 1

Prove the Brouwer fixed point theorem for maps $f: D^{n} \rightarrow D^{n}$ by applying degree theory to the map $S^{n} \rightarrow S^{n}$ that sends both the northern and southern hemispheres of $S^{n}$ to the southern hemisphere via $f .$ IThis was Brouwer's original proof.

Steven Swee
Steven Swee
Numerade Educator
00:59

Problem 1

If $T_{n}(X, A)$ denotes the torsion subgroup of $H_{n}(X, A ; \mathbb{Z}),$ show that the functors $(X, A) \mapsto T_{n}(X, A),$ with the obvious induced homomorphisms $T_{n}(X, A) \rightarrow T_{n}(Y, B)$ and boundary maps $T_{n}(X, A) \rightarrow T_{n-1}(A),$ do not define a homology theory. Do the same for the 'mod torsion' functor $M T_{n}(X, A)=H_{n}(X, A ; \mathbb{Z}) / T_{n}(X, A).$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:47

Problem 1

What is the minimum number of edges in simplicial complex structures $K$ and $L$ on $S^{1}$ such that there is a simplicial map $K \rightarrow L$ of degree $n ?$

Aadit Sharma
Aadit Sharma
Numerade Educator
02:16

Problem 1

Compute $H_{i}\left(S^{n}-X\right)$ when $X$ is a subspace of $S^{n}$ homeomorphic to $S^{k} \vee S^{\ell}$ or to $S^{k} \mathrm{II} S^{\ell}$

Uma Kumari
Uma Kumari
Numerade Educator
01:20

Problem 1

What familiar space is the quotient $\Delta$ -complex of a 2 -simplex $\left[v_{0}, v_{1}, v_{2}\right]$ obtained by identifying the edges $\left[v_{0}, v_{1}\right]$ and $\left[v_{1}, v_{2}\right],$ preserving the ordering of vertices?

Vikash Ranjan
Vikash Ranjan
Numerade Educator
03:34

Problem 2

Given a map $f: S^{2 n} \rightarrow S^{2 n}$, show that there is some point $x \in S^{2 n}$ with either $f(x)=x$ or $f(x)=-x .$ Deduce that every map $\mathbb{R P}^{2 n} \rightarrow \mathbb{R} P^{2 n}$ has a fixed point. Construct maps $\mathrm{RP}^{2 n-1} \rightarrow \mathrm{RP}^{2 n-1}$ without fixed points from linear transformations $\mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}$ without eigenvectors.

Arun Bana
Arun Bana
Numerade Educator
02:01

Problem 2

Use the Lefschetz fixed point theorem to show that a map $S^{n} \rightarrow S^{n}$ has a fixed point unless its degree is equal to the degree of the antipodal map $x \mapsto-x$.

Amy Jiang
Amy Jiang
Numerade Educator
42:21

Problem 2

Show that the $\Delta$ -complex obtained from $\Delta^{3}$ by performing the edge identifications $\left[v_{0}, v_{1}\right] \sim\left[v_{1}, v_{3}\right]$ and $\left[v_{0}, v_{2}\right] \sim\left[v_{2}, v_{3}\right]$ deformation retracts onto a Klein bottle. Find other pairs of identifications of edges that produce $\Delta$ -complexes deformation retracting onto a torus, a 2-sphere, and $\mathbb{R P}^{2}$.

Chris Trentman
Chris Trentman
Numerade Educator
03:56

Problem 2

Show that $\tilde{H}_{i}\left(S^{n}-X\right) \approx \tilde{H}_{n-i-1}(X)$ when $X$ is homeomorphic to a finite connected graph. First do the case that the graph is a tree.

Victoria Dollar
Victoria Dollar
Numerade Educator
01:19

Problem 2

Define a candidate for a reduced homology theory on CW complexes by $\tilde{h}_{n}(X)=$ $\prod_{i} \tilde{H}_{i}(X) / \oplus_{i} \tilde{H}_{i}(X) .$ Thus $\tilde{h}_{n}(X)$ is independent of $n$ and is zero if $X$ is finitedimensional, but is not identically zero, for example for $X=V_{i} S^{i} .$ Show that the axioms for a homology theory are satisfied except that the wedge axiom fails.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
20:07

Problem 3

3. Show that if $h$ is a reduced homology theory, then $h_{n}(p o i n t)=0$ for all $n$. Deduce that there are suspension isomorphisms $\tilde{h}_{n}(X) \approx \tilde{h}_{n+1}(S X)$ for all $n.$

Mahnoor Amin
Mahnoor Amin
Numerade Educator
03:58

Problem 3

Let $(D, S) \subset\left(D^{n}, S^{n-1}\right)$ be a pair of subspaces homeomorphic to $\left(D^{k}, S^{k-1}\right),$ with $D \cap S^{n-1}=S .$ Show the inclusion $S^{n-1}-S \hookrightarrow D^{n}-D$ induces an isomorphism on homology. IGlue two copies of $\left(D^{n}, D\right)$ together along $\left(S^{n-1}, S\right)$ and examine the Mayer-Vietoris sequence for the complement of the resulting $k$ -sphere in $S^{n}$, decomposed into two copies of $D^{n}-D . .$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:09

Problem 3

Construct a $\Delta$ -complex structure on $\mathbb{R P}^{n}$ as a quotient of a $\Delta$ -complex structure on $S^{n}$ having vertices the two vectors of length 1 along each coordinate axis in $\mathbb{R}^{n+1}$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:09

Problem 3

Let $f: S^{n} \rightarrow S^{n}$ be a map of degree zero. Show that there exist points $x, y \in S^{n}$ with $f(x)=x$ and $f(y)=-y .$ Use this to show that if $F$ is a continuous vector field defined on the unit ball $D^{n}$ in $\mathbb{R}^{n}$ such that $F(x) \neq 0$ for all $x,$ then there exists a point on $\partial D^{n}$ where $F$ points radially outward and another point on $\partial D^{n}$ where $F$ points radially inward.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
03:41

Problem 3

Verify that the formula $f\left(z_{1}, \cdots, z_{2 k}\right)=\left(\bar{z}_{2},-\bar{z}_{1}, \bar{z}_{4},-\bar{z}_{3}, \cdots, \bar{z}_{2 k},-\bar{z}_{2 k-1}\right)$ defines
a map $f: \mathbb{C}^{2 k} \rightarrow \mathbb{C}^{2 k}$ inducing a quotient map $\mathrm{CP}^{2 k-1} \rightarrow \mathrm{CP}^{2 k-1}$ without fixed points.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:26

Problem 4

Show that the wedge axiom for homology theories follows from the other axioms in the case of finite wedge sums.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:38

Problem 4

In the unit sphere $S^{p+q-1} \subset \mathbb{R}^{p+q}$ let $S^{p-1}$ and $S^{q-1}$ be the subspheres consisting of points whose last $q$ and first $p$ coordinates are zero, respectively. (a) Show that $S^{p+q-1}-S^{p-1}$ deformation retracts onto $S^{q-1},$ and is in fact homeomorphic to $S^{q-1} \times \mathbb{R}^{p}$ (b) Show that $S^{p-1}$ and $S^{q-1}$ are not the boundaries of any pair of disjointly embedded disks $D^{p}$ and $D^{q}$ in $D^{p+q} .$ [The preceding exercise may be useful.]

Adam Dehollander
Adam Dehollander
Numerade Educator
03:01

Problem 4

If $X$ is a finite simplicial complex and $f: X \rightarrow X$ is a simplicial homeomorphism, show that the Lefschetz number $\tau(f)$ equals the Euler characteristic of the set of fixed points of $f .$ In particular, $\tau(f)$ is the number of fixed points if the fixed points are isolated. IHint: Barycentrically subdivide $X$ to make the fixed point set a subcomplex.]

Regina Hays
Regina Hays
Numerade Educator
04:10

Problem 4

Compute the simplicial homology groups of the triangular parachute obtained from
$\Delta^{2}$ by identifying its three vertices to a single point.

Gideon Idumah
Gideon Idumah
Numerade Educator
10:00

Problem 4

Construct a surjective map $S^{n} \rightarrow S^{n}$ of degree zero, for each $n \geq 1$.

Mengchun Cai
Mengchun Cai
Numerade Educator
03:56

Problem 5

Compute the simplicial homology groups of the Klein bottle using the $\Delta$ -complex structure described at the beginning of this section.

VS
Vivek Singh
Numerade Educator
04:13

Problem 5

Let $S$ be an embedded $k$ -sphere in $S^{n}$ for which there exists a disk $D^{n} \subset S^{n}$ intersecting $S$ in the disk $D^{k} \subset D^{n}$ defined by the first $k$ coordinates of $D^{n} .$ Let $D^{n-k} \subset D^{n}$ be the disk defined by the last $n-k$ coordinates, with boundary sphere $S^{n-k-1} .$ Show that the inclusion $S^{n-k-1} \hookrightarrow S^{n}-S$ induces an isomorphism on homology groups.

Uma Kumari
Uma Kumari
Numerade Educator
03:39

Problem 5

Show that any two reflections of $S^{n}$ across different $n$ -dimensional hyperplanes are homotopic, in fact homotopic through reflections. [The linear algebra formula for
a reflection in terms of inner products may be helpful.]

Angelo Rendina
Angelo Rendina
Numerade Educator
00:34

Problem 5

Let $M$ be a closed orientable surface embedded in $\mathbb{R}^{3}$ in such a way that reflection across a plane $P$ defines a homeomorphism $r: M \rightarrow M$ fixing $M \cap P,$ a collection of circles. Is it possible to homotope $r$ to have no fixed points?

Ashley High
Ashley High
Numerade Educator
View

Problem 6

Modify the construction of the Alexander horned sphere to produce an embedding $S^{2} \hookrightarrow \mathbb{R}^{3}$ for which neither component of $\mathbb{R}^{3}-S^{2}$ is simply-connected.

Victor Salazar
Victor Salazar
Numerade Educator
01:40

Problem 6

Show that every map $S^{n} \rightarrow S^{n}$ can be homotoped to have a fixed point if $n>0$.

Angelo Rendina
Angelo Rendina
Numerade Educator
03:15

Problem 6

Do an even-genus analog of Example $2 \mathrm{C} .4$ by replacing the central torus by a sphere letting $f$ be a homeomorphism that restricts to the antipodal map on this sphere.

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
01:25

Problem 6

Compute the simplicial homology groups of the $\Delta$ -complex obtained from $n+1$ 2 -simplices $\Delta_{0}^{2}, \cdots, \Delta_{n}^{2}$ by identifying all three edges of $\Delta_{0}^{2}$ to a single edge, and for $i>0$ identifying the edges $\left[v_{0}, v_{1}\right]$ and $\left[v_{1}, v_{2}\right]$ of $\Delta_{i}^{2}$ to a single edge and the edge $\left[v_{0}, v_{2}\right]$ to the edge $\left[v_{0}, v_{1}\right]$ of $\Delta_{i-1}^{2}$.

Lucía Guerrero
Lucía Guerrero
Numerade Educator
00:45

Problem 7

Find a way of identifying pairs of faces of $\Delta^{3}$ to produce a $\Delta$ -complex structure on $S^{3}$ having a single 3 -simplex, and compute the simplicial homology groups of this $\Delta$ -complex.

Aadit Sharma
Aadit Sharma
Numerade Educator
01:29

Problem 7

For an invertible linear transformation $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ show that the induced map on $H_{n}\left(\mathbb{R}^{n}, \mathbb{R}^{n}-\{0\}\right) \approx \tilde{H}_{n-1}\left(\mathbb{R}^{n}-\{0\}\right) \approx \mathbb{Z}$ is $\mathbb{1}$ or $-\mathbb{1}$ according to whether the
determinant of $f$ is positive or negative. IUse Gaussian elimination to show that the matrix of $f$ can be joined by a path of invertible matrices to a diagonal matrix with $\left.\pm 1^{\prime} s \text { on the diagonal. }\right]$

Angelo Rendina
Angelo Rendina
Numerade Educator
00:55

Problem 7

Verify that the Lefschetz fixed point theorem holds also when $\tau(f)$ is defined using homology with coefficients in a field $F$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:27

Problem 7

Analyze what happens when the number of handles in the basic building block for the Alexander horned sphere is doubled, as in the figure at the right.

Jake Rempel
Jake Rempel
Numerade Educator
07:21

Problem 8

Construct a 3 -dimensional $\Delta$ -complex $X$ from $n$ tetrahedra $T_{1}, \cdots, T_{n}$ by the following two steps. First arrange the tetrahedra in a cyclic pattern as in the figure, so that each $T_{i}$ shares a common vertical face with its two neighbors $T_{i-1}$ and $T_{i+1},$ subscripts being taken mod $n .$ Then identify the bottom face of $T_{i}$ with the top face of $T_{i+1}$ for each $i .$ Show the simplicial homology groups of $X$ in dimensions 0,1,2,3 are $Z, Z_{n}, 0, Z,$ respectively. IThe space $X$ is an example of a lens space, see Example 2.43 for the general case.

Chris Trentman
Chris Trentman
Numerade Educator
01:51

Problem 8

Let $X$ be homotopy equivalent to a finite simplicial complex and let $Y$ be homotopy equivalent to a finite or countably infinite simplicial complex. Using the simplicial approximation theorem, show that there are at most countably many homotopy classes of maps $X \rightarrow Y$.

Brittany Knowlton
Brittany Knowlton
Numerade Educator
03:23

Problem 8

A polynomial $f(z)$ with complex coefficients, viewed as a map $\mathrm{C} \rightarrow \mathrm{C}$, can always be extended to a continuous map of one-point compactifications $\hat{f}: S^{2} \rightarrow S^{2}$. Show that the degree of $\hat{f}$ equals the degree of $f$ as a polynomial. Show also that the local degree of $\hat{f}$ at a root of $f$ is the multiplicity of the root.

Nick Johnson
Nick Johnson
Numerade Educator
05:40

Problem 8

Show that $\mathbb{R}^{2 n+1}$ is not a division algebra over $\mathbb{R}$ if $n>0$ by showing that if it were, then for nonzero $a \in \mathbb{R}^{2 n+1}$ the map $S^{2 n} \rightarrow S^{2 n}, x \mapsto a x /|a x|$ would be homotopic to $x \mapsto-a x /|a x|$ but these maps have different degrees.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:27

Problem 9

Show that there are only countably many homotopy types of finite CW complexes.

Angelo Rendina
Angelo Rendina
Numerade Educator
01:51

Problem 9

Make the transfer sequence explicit in the case of a trivial covering $\tilde{X} \rightarrow X,$ where $\tilde{X}=X \times S^{0}$

Wendi Zhao
Wendi Zhao
Numerade Educator
14:50

Problem 9

Compute the homology groups of the $\Delta$ -complex $X$ obtained from $\Delta^{n}$ by identifying all faces of the same dimension. Thus $X$ has a single $k$ -simplex for each $k \leq n$.

Chris Trentman
Chris Trentman
Numerade Educator
01:50

Problem 9

Compute the homology groups of the following 2-complexes:
(a) The quotient of $S^{2}$ obtained by identifying north and south poles to a point.
(b) $S^{1} \times\left(S^{1} \vee S^{1}\right)$
(c) The space obtained from $D^{2}$ by first deleting the interiors of two disjoint subdisks
in the interior of $D^{2}$ and then identifying all three resulting boundary circles together via homeomorphisms preserving clockwise oricntations of these circles.
(d) The quotient space of $S^{1} \times S^{1}$ obtained by identifying points in the circle $S^{1} \times\left\{x_{0}\right\}$ that differ by $2 \pi / m$ rotation and identifying points in the circle $\left\{x_{0}\right\} \times S^{1}$ that differ by $2 \pi / n$ rotation.

Carson Merrill
Carson Merrill
Numerade Educator
01:48

Problem 10

Use the transfer sequence for the covering $S^{\infty} \rightarrow \mathbb{R} P^{\infty}$ to compute $H_{n}\left(\mathbb{R} P^{\infty} ; \mathbb{Z}_{2}\right)$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
04:24

Problem 10

(a) Show the quotient space of a finite collection of disjoint 2 -simplices obtained by identifying pairs of edges is always a surface, locally homeomorphic to $\mathbb{R}^{2}$.
(b) Show the edges can always be oriented so as to define a $\Delta$ -complex structure on the quotient surface. IThis is more difficult.]

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
03:47

Problem 10

Let $X$ be the quotient space of $S^{2}$ under the identifications $x \sim-x$ for $x$ in the equator $S^{1} .$ Compute the homology groups $H_{i}(X) .$ Do the same for $S^{3}$ with antipodal points of the equatorial $S^{2} \subset S^{3}$ identified.

Regina Hays
Regina Hays
Numerade Educator
01:40

Problem 11

Show that if $A$ is a retract of $X$ then the map $H_{n}(A) \rightarrow H_{n}(X)$ induced by the inclusion $A \subset X$ is injective.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:39

Problem 11

In an exercise for $\$ 1.2$ we described a 3 -dimensional CW complex obtained from the cube $I^{3}$ by identifying opposite faces via a one-quarter twist. Compute the homology groups of this complex.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
00:52

Problem 11

Use the transfer sequence for the covering $X \times S^{\infty} \rightarrow X \times \mathbb{R} \mathrm{P}^{\infty}$ to produce isomorphisms $H_{n}\left(X \times \mathbb{R P}^{\infty} ; \mathbb{Z}_{2}\right) \approx \oplus_{i \leq n} H_{i}\left(X ; \mathbb{Z}_{2}\right)$ for all $n$

Doruk Isik
Doruk Isik
Numerade Educator
16:59

Problem 12

Show that chain homotopy of chain maps is an equivalence relation.

Chris Trentman
Chris Trentman
Numerade Educator
07:44

Problem 12

Show that the quotient map $S^{1} \times S^{1} \rightarrow S^{2}$ collapsing the subspace $S^{1} \vee S^{1}$ to a point is not nullhomotopic by showing that it induces an isomorphism on $H_{2} .$ On the other hand, show via covering spaces that any map $S^{2} \rightarrow S^{1} \times S^{1}$ is nullhomotopic.

Anthony Ramos
Anthony Ramos
Numerade Educator
08:25

Problem 13

Verify that $f \simeq g$ implies $f_{*}=g_{*}$ for induced homomorphisms of reduced homology groups.

Ely Crowder
Ely Crowder
Numerade Educator
02:43

Problem 13

Let $X$ be the 2 -complex obtained from $S^{1}$ with its usual cell structure by attaching two 2-cells by maps of degrees 2 and $3,$ respectively.
(a) Compute the homology groups of all the subcomplexes $A \subset X$ and the corresponding quotient complexes $X / A$
(b) Show that $X \approx S^{2},$ and that the only subcomplex $A \subset X$ with $X / A=S^{2}$ is the trivial subcomplex consisting of the 0-cell alone.

Aman Gupta
Aman Gupta
Numerade Educator
04:03

Problem 14

A map $f: S^{n} \rightarrow S^{n}$ satisfying $f(x)=f(-x)$ for all $x$ is called an even map. Show that an even map $S^{n} \rightarrow S^{n}$ must have even degree, and that the degree must in fact be zero when $n$ is even. When $n$ is odd, show there exist even maps of any given even degree. IHints: If $f$ is even, it factors as a composition $S^{n} \rightarrow \mathbb{R} P^{n} \rightarrow S^{n} .$ Using the calculation of $H_{n}\left(\mathbb{R} P^{n}\right)$ in the text, show that the induced map $H_{n}\left(S^{n}\right) \rightarrow H_{n}\left(\mathbb{R P}^{n}\right)$ sends a generator to twice a generator when $n$ is odd. It may be helpful to show that the quotient map $\mathbb{R P}^{n} \rightarrow \mathbb{R} \mathrm{P}^{n} / \mathbb{R} \mathrm{P}^{n-1}$ induces an isomorphism on $H_{n}$ when $n$ is odd.]

Nick Johnson
Nick Johnson
Numerade Educator
03:16

Problem 14

Determine whether there exists a short exact sequence $0 \rightarrow \mathbb{Z}_{4} \rightarrow \mathbb{Z}_{8} \oplus \mathbb{Z}_{2} \rightarrow \mathbb{Z}_{4} \rightarrow 0$
More generally, determine which abelian groups $A$ fit into a short exact sequence $0 \rightarrow \mathbb{Z}_{p^{m}} \rightarrow A \rightarrow \mathbb{Z}_{p^{m}} \rightarrow 0$ with $p$ prime. What about the case of short exact sequences $0 \rightarrow \mathbb{Z} \rightarrow A \rightarrow \mathbb{Z}_{n} \rightarrow 0 ?$

Anas Venkitta
Anas Venkitta
Numerade Educator
07:44

Problem 15

Show that if $X$ is a CW complex then $H_{n}\left(X^{n}\right)$ is free by identifying it with the kernel of the cellular boundary $\operatorname{map} H_{n}\left(X^{n}, X^{n-1}\right) \rightarrow H_{n-1}\left(X^{n-1}, X^{n-2}\right)$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:17

Problem 15

For an exact sequence $A \rightarrow B \rightarrow C \rightarrow D \rightarrow E$ show that $C=0$ iff the map $A \rightarrow B$ is surjective and $D \rightarrow E$ is injective. Hence for a pair of spaces $(X, A),$ the inclusion $A \hookrightarrow X$ induces isomorphisms on all homology groups iff $H_{n}(X, A)=0$ for all $n$.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
03:37

Problem 16

Let $\Delta^{n}=\left[v_{0}, \cdots, v_{n}\right]$ have its natural $\Delta$ -complex structure with $k$ -simplices $\left[v_{i_{0}}, \cdots, v_{i_{k}}\right]$ for $i_{0}<\cdots<i_{k} .$ Compute the ranks of the simplicial (or cellular) chain groups $\Delta_{i}\left(\Delta^{n}\right)$ and the subgroups of cycles and boundaries. IHint: Pascal's triangle.] Apply this, using also the previous problem, to show that the $k$ -skeleton of $\Delta^{n}$ has homology groups $\tilde{\boldsymbol{H}}_{t}\left(\left(\Delta^{n}\right)^{k}\right)$ equal to 0 for $i<k,$ and free of rank $\left(\begin{array}{l}n \\ k+1\end{array}\right)$ for $i=k$.

Shiksha Dutta
Shiksha Dutta
Numerade Educator
00:53

Problem 16

(a) Show that $H_{0}(X, A)=0$ iff $A$ meets each path-component of $X .$
(b) Show that $H_{1}(X, A)=0$ iff $H_{1}(A) \rightarrow H_{1}(X)$ is surjective and each path-component of $X$ contains at most one path-component of $A$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
08:50

Problem 17

Show the isomorphism between cellular and singular homology is natural in the following sense: $A$ map $f: X \rightarrow Y$ that is cellular $-$ satisfying $f\left(X^{n}\right) \subset Y^{n}$ for
all $n-$ induces a chain map $f_{*}$ between the cellular chain complexes of $X$ and
$Y,$ and the map $f_{*}: H_{n}^{C W}(X) \rightarrow H_{n}^{C W}(Y)$ induced by this chain map corresponds to $f_{*}: H_{n}(X) \rightarrow H_{n}(Y)$ under the isomorphism $H_{n}^{C W} \approx H_{n}$.

Ely Crowder
Ely Crowder
Numerade Educator
00:40

Problem 17

(a) Compute the homology groups $H_{n}(X, A)$ when $X$ is $S^{2}$ or $S^{1} \times S^{1}$ and $A$ is a finite set of points in $X .$
(b) Compute the groups $H_{n}(X, A)$ and $H_{n}(X, B)$ where $X$ is a closed orientable surface of genus two and $A$ and $B$ are the circles shown.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
07:44

Problem 18

Show that for the subspace $\mathbb{Q} \subset \mathbb{R},$ the relative homology group $H_{1}(\mathbb{R}, \mathbb{Q})$ is free abelian and find a basis.

Anthony Ramos
Anthony Ramos
Numerade Educator
05:34

Problem 18

For a CW pair $(X, A)$ show there is a relative cellular chain complex formed by the groups $H_{i}\left(X^{i}, X^{i-1} \cup A^{i}\right),$ having homology groups isomorphic to $H_{n}(X, A)$.

Chris Trentman
Chris Trentman
Numerade Educator
01:15

Problem 19

Compute $H_{i}\left(\mathrm{RP}^{n} / \mathrm{RP}^{m}\right)$ for $m<n$ by cellular homology, using the standard CW structure on $\mathbb{R P}^{n}$ with $\mathbb{R P}^{m}$ as its $m$ -skeleton.

Adriano Chikande
Adriano Chikande
Numerade Educator
06:39

Problem 19

Compute the homology groups of the subspace of $I \times I$ consisting of the four boundary edges plus all points in the interior whose first coordinate is rational.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:38

Problem 20

For finite CW complexes $X$ and $Y,$ show that $\chi(X \times Y)=\chi(X) \chi(Y)$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
16:33

Problem 20

Show that $\tilde{H}_{n}(X) \approx \tilde{H}_{n+1}(S X)$ for all $n,$ where $S X$ is the suspension of $X .$ More generally, thinking of $S X$ as the union of two cones $C X$ with their bases identified, compute the reduced homology groups of the union of $n$ cones $C X$ with their bases identified.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
07:04

Problem 21

If a finite CW complex $X$ is the union of subcomplexes $A$ and $B$, show that $x(X)=x(A)+x(B)-x(A \cap B)$.

Willis James
Willis James
Numerade Educator
04:42

Problem 21

Making the preceding problem more concrete, construct explicit chain maps $s: C_{n}(X) \rightarrow C_{n+1}(S X)$ inducing isomorphisms $\tilde{H}_{n}(X) \rightarrow \tilde{H}_{n+1}(S X)$.

Chris Trentman
Chris Trentman
Numerade Educator
02:39

Problem 22

Prove by induction on dimension the following facts about the homology of a finite-dimensional CW complex $X,$ using the observation that $X^{n} / X^{n-1}$ is a wedge sum of $n$ -spheres:
(a) If $X$ has dimension $n$ then $H_{i}(X)=0$ for $i>n$ and $H_{n}(X)$ is free.
(b) $H_{n}(X)$ is free with basis in bijective correspondence with the $n$ -cells if there are no cells of dimension $n-1$ or $n+1$
(c) If $X$ has $k$ n-cells, then $H_{n}(X)$ is generated by at most $k$ elements.

Chris Trentman
Chris Trentman
Numerade Educator
02:38

Problem 22

For $X$ a finite $\mathrm{CW}$ complex and $p: \tilde{X} \rightarrow X$ an $n$ -sheeted covering space, show that $\chi(\tilde{X})=n \chi(X)$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:23

Problem 23

Show that the second barycentric subdivision of a $\Delta$ -complex is a simplicial complex. Namely, show that the first barycentric subdivision produces a $\Delta$ -complex with the property that each simplex has all its vertices distinct, then show that for a \Delta-complex with this property, barycentric subdivision produces a simplicial complex.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:43

Problem 23

Show that if the closed orientable surface $M_{g}$ of genus $g$ is a covering space
of $M_{h},$ then $g=n(h-1)+1$ for some $n,$ namely, $n$ is the number of sheets in the covering. [Conversely, if $g=n(h-1)+1$ then there is an $n$ -sheeted covering $\left.M_{g} \rightarrow M_{h}, \text { as we saw in Example } 1.41 .\right]$.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
10:46

Problem 24

Suppose we build $S^{2}$ from a finite collection of polygons by identifying edges in pairs. Show that in the resulting CW structure on $S^{2}$ the 1 -skeleton cannot be either of the two graphs shown, with five and six vertices. IThis is one step in a proof that neither of these graphs embeds in $\left.\mathbb{R}^{2} .\right]$

Victoria Dollar
Victoria Dollar
Numerade Educator
01:03

Problem 24

Show that each $n$ -simplex in the barycentric subdivision of $\Delta^{n}$ is defined by $n$ inequalities $t_{i_{0}} \leq t_{i_{1}} \leq \cdots \leq t_{i_{n}}$ in its barycentric coordinates, where $\left(i_{0}, \cdots, i_{n}\right)$ is a permutation of $(0, \cdots, n)$.

Amy Jiang
Amy Jiang
Numerade Educator
02:06

Problem 25

Show that for each $n \in \mathbb{Z}$ there is a unique function $\varphi$ assigning an integer to each finite CW complex, such that (a) $\varphi(X)=\varphi(Y)$ if $X$ and $Y$ are homeomorphic, (b) $\varphi(X)=\varphi(A)+\varphi(X / A)$ if $A$ is a subcomplex of $X,$ and $(c) \varphi\left(S^{0}\right)=n .$ For such
a function $\varphi,$ show that $\varphi(X)=\varphi(Y)$ if $X \simeq Y$.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
02:00

Problem 25

Find an explicit, noninductive formula for the barycentric subdivision operator $S: C_{n}(X) \rightarrow C_{n}(X)$.

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:05

Problem 26

Show that $H_{1}(X, A)$ is not isomorphic to $\tilde{H}_{1}(X / A)$ if $X=[0,1]$ and $A$ is the sequence $1,1 / 2,1 / 3, \cdots$ together with its limit $0 .$ ISee Example 1.25 .1

Anthony Ramos
Anthony Ramos
Numerade Educator
03:45

Problem 26

For a pair $(X, A),$ let $X \cup C A$ be $X$ with a cone on $A$ attached.
(a) Show that $X$ is a retract of $X \cup C A$ iff $A$ is contractible in $X:$ There is a homotopy $f_{t}: A \rightarrow X$ with $f_{0}$ the inclusion $A \sqcup X$ and $f_{1}$ a constant map.
(b) Show that if $A$ is contractible in $X$ then $H_{n}(X, A) \approx \tilde{H}_{n}(X) \oplus \tilde{H}_{n-1}(A),$ using the fact that $(X \cup C A) / X$ is the suspension $S A$ of $A$.

Narayan Hari
Narayan Hari
Numerade Educator
14:32

Problem 27

Let $f:(X, A) \rightarrow(Y, B)$ be a map such that both $f: X \rightarrow Y$ and the restriction $f: A \rightarrow B$ are homotopy equivalences.
(a) Show that $f_{*}: H_{n}(X, A) \rightarrow H_{n}(Y, B)$ is an isomorphism for all $n$
(b) For the case of the inclusion $f:\left(D^{n}, S^{n-1}\right) \hookrightarrow\left(D^{n}, D^{n}-\{0\}\right),$ show that $f$ is not a homotopy equivalence of pairs - there is no $g:\left(D^{n}, D^{n}-\{0\}\right) \rightarrow\left(D^{n}, S^{n-1}\right)$ such that $f g$ and $g f$ are homotopic to the identity through maps of pairs.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:01

Problem 27

The short exact sequences $0 \rightarrow C_{n}(A) \rightarrow C_{n}(X) \rightarrow C_{n}(X, A) \rightarrow 0$ always split, but why does this not always yield splittings $H_{n}(X) \approx H_{n}(A) \oplus H_{n}(X, A) ?$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
12:31

Problem 28

Let $X$ be the cone on the 1 -skeleton of $\Delta^{3},$ the union of all line segments joining points in the six edges of $\Delta^{3}$ to the barycenter of $\Delta^{3}$. Compute the local homology groups $H_{n}(X, X-[x])$ for all $x \in X .$ Define $\partial X$ to be the subspace of points $x$ such that $H_{n}(X, X-\{x\})=0$ for all $n,$ and compute the local homology groups $H_{n}(\partial X, \partial X-\{x\}) .$ Use these calculations to determine which subsets $A \subset X$ have the property that $f(A) \subset A$ for all homeomorphisms $f: X \rightarrow X$.

Carlos Pinilla
Carlos Pinilla
Numerade Educator
11:31

Problem 28

(a) Use the Mayer-Vietoris sequence to compute the homology groups of 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.
(b) Do the same for the space obtained by attaching a Mobius band to $\mathbb{R P}^{2}$ via a homeomorphism of its boundary circle to the standard $\mathbb{R} P^{1} \subset \mathbb{R} P^{2}$.

Chris Trentman
Chris Trentman
Numerade Educator
01:55

Problem 29

The surface $M_{g}$ of genus $g$, embedded in $\mathbb{R}^{3}$ in the standard way, bounds a compact region $R .$ Two copies of $R,$ glued together by the identity map between their boundary surfaces $M_{g},$ form a closed 3 -manifold $X .$ Compute the homology groups of $X$ via the Mayer-Vietoris sequence for this decomposition of $X$ into two copies of $R .$ Also compute the relative groups $H_{i}\left(R, M_{g}\right)$

Elliott Walker
Elliott Walker
Numerade Educator
03:26

Problem 29

Show that $S^{1} \times S^{1}$ and $S^{1} \vee S^{1} \vee S^{2}$ have isomorphic homology groups in all dimensions, but their universal covering spaces do not.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:01

Problem 30

For the mapping torus $T_{f}$ of a map $f: X \rightarrow X,$ we constructed in Example 2.48 a long exact sequence $\cdots \rightarrow H_{n}(X) \stackrel{1-f_{*}}{\longrightarrow} H_{n}(X) \rightarrow H_{n}\left(T_{f}\right) \rightarrow H_{n-1}(X) \rightarrow \cdots$ Use this to compute the homology of the mapping tori of the following maps:
(a) A reflection $S^{2} \rightarrow S^{2}$
(b) A map $S^{2} \rightarrow S^{2}$ of degree 2
(c) The map $S^{1} \times S^{1} \rightarrow S^{1} \times S^{1}$ that is the identity on one factor and a reflection on the other.
(d) The map $S^{1} \times S^{1} \rightarrow S^{1} \times S^{1}$ that is a reflection on each factor.
(e) The map $S^{1} \times S^{1} \rightarrow S^{1} \times S^{1}$ that interchanges the two factors and then reflects one of the factors.

Anthony Ramos
Anthony Ramos
Numerade Educator
14:32

Problem 30

In each of the following commutative diagrams assume that all maps but one are isomorphisms. Show that the remaining map must be an isomorphism as well. (FIGURE CANNOT COPY)

Anthony Ramos
Anthony Ramos
Numerade Educator
04:04

Problem 31

Using the notation of the five-lemma, give an example where the maps $\alpha, \beta, \delta$ and $\varepsilon$ are zero but $y$ is nonzero. This can be done with short exact sequences in which all the groups are either $\mathbb{Z}$ or $0 .$

Carson Merrill
Carson Merrill
Numerade Educator
07:01

Problem 31

Use the Mayer-Vietoris sequence to show there are isomorphisms $\tilde{H}_{n}(X \vee Y) \approx$ $\tilde{H}_{n}(X) \oplus \tilde{H}_{n}(Y)$ if the basepoints of $X$ and $Y$ that are identified in $X \vee Y$ are defor mation retracts of neighborhoods $U \subset X$ and $V \subset Y$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:35

Problem 32

For $S X$ the suspension of $X,$ show by a Mayer-Vietoris sequence that there are isomorphisms $\tilde{H}_{n}(S X) \approx \tilde{H}_{n-1}(X)$ for all $n$.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:16

Problem 33

Suppose the space $X$ is the union of open sets $A_{1}, \cdots, A_{n}$ such that each intersection $A_{i_{1}} \cap \cdots \cap A_{i_{k}}$ is either empty or has trivial reduced homology groups. Show that $\tilde{H}_{i}(X)=0$ for $i \geq n-1,$ and give an example showing this inequality is best possible, for each $n$.

Uma Kumari
Uma Kumari
Numerade Educator
05:22

Problem 34

Derive the long exact sequence of a pair $(X, A)$ from the Mayer-Vietoris sequence applied to $X \cup C A,$ where $C A$ is the cone on $A$. IWe showed after the proof of Proposition 2.22 that $H_{n}(X, A) \approx \tilde{H}_{n}(X \cup C A)$ for all $n .1$.

Angelo Rendina
Angelo Rendina
Numerade Educator
02:39

Problem 35

Use the Mayer-Vietoris sequence to show that a nonorientable closed surface, or more generally a finite simplicial complex $X$ for which $H_{1}(X)$ contains torsion, cannot be embedded as a subspace of $\mathrm{R}^{3}$ in such a way as to have a neighborhood homeomorphic to the mapping cylinder of some map from a closed orientable surface
to $X .$ IThis assumption on a neighborhood is in fact not needed if one deduces the result from Alexander duality in $\$ 3.3 .\rfloor$.

Chris Trentman
Chris Trentman
Numerade Educator
01:05

Problem 36

36. Show that $H_{i}\left(X \times S^{n}\right) \approx H_{i}(X) \oplus H_{i-n}(X)$ for all $i$ and $n,$ where $H_{i}=0$ for $i<0$ by definition. Namely, show $H_{i}\left(X \times S^{n}\right) \approx H_{i}(X) \oplus H_{i}\left(X \times S^{n}, X \times\left[x_{0}\right]\right)$ and $H_{i}\left(X \times S^{n}, X \times\left\{x_{0}\right\}\right) \approx H_{i-1}\left(X \times S^{n-1}, X \times\left\{x_{0}\right\}\right) .$ IFor the latter isomorphism the relative Mayer-Vietoris sequence yields an easy proof.|

Anthony Ramos
Anthony Ramos
Numerade Educator
02:08

Problem 37

Give an elementary derivation for the Mayer-Vietoris sequence in simplicial homology for a $\Delta$ -complex $X$ decomposed as the union of subcomplexes $A$ and $B$.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:21

Problem 38

Show that a commutative diagram
CAN'T COPY THE FIGURE
with the two sequences across the top and bottom exact, gives rise to an exact sequence $\cdots \rightarrow E_{n+1} \rightarrow B_{n} \rightarrow C_{n} \oplus D_{n} \rightarrow E_{n} \rightarrow B_{n-1} \rightarrow \cdots$ where the maps are obtained from those in the previous diagram in the obvious way, except that $B_{n} \rightarrow C_{n} \oplus D_{n}$ has a minus sign in one coordinate.

Jay Patel
Jay Patel
Numerade Educator
01:01

Problem 39

Use the preceding exercise to derive relative Mayer-Vietoris sequences for CW pairs $(X, Y)=(A \cup B, C \cup D)$ with $A=B$ or $C=D$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
08:25

Problem 40

From the long exact sequence of homology groups associated to the short exact sequence of chain complexes $0 \rightarrow C_{i}(X) \stackrel{n}{\longrightarrow} C_{i}(X) \rightarrow C_{i}\left(X ; \mathbb{Z}_{n}\right) \rightarrow 0$ deduce
immediately that there are short exact sequences
$$
0 \rightarrow H_{i}(X) / n H_{i}(X) \rightarrow H_{i}\left(X ; Z_{n}\right) \rightarrow n-\text {Torsion}\left(H_{i-1}(X)\right) \rightarrow 0
$$
where $n$ -Torsion(G) is the kernel of the map $G \stackrel{n}{\longrightarrow} G, g \mapsto n g .$ Use this to show that $\tilde{H}_{i}\left(X ; \mathbb{Z}_{p}\right)=0$ for all $i$ and all primes $p$ iff $\tilde{H}_{i}(X)$ is a vector space over $\mathbb{Q}$ for all $i$.

Ely Crowder
Ely Crowder
Numerade Educator
06:05

Problem 41

For $X$ a finite $\mathrm{CW}$ complex and $F$ a field, show that the Fuler characteristic $\chi(X)$ can also be computed by the formula $x(X)=\Sigma_{n}(-1)^{n} \operatorname{dim} H_{n}(X ; F),$ the alternating sum of the dimensions of the vector spaces $H_{n}(X ; F)$.

Jacob Fry
Jacob Fry
Numerade Educator
08:25

Problem 42

Let $X$ be a finite connected graph having no vertex that is the endpoint of just one edge, and suppose that $H_{1}(X ; \mathbb{Z})$ is free abelian of rank $n>1,$ so the group of automorphisms of $H_{1}(X ; \mathbb{Z})$ is $G L_{n}(Z),$ the group of invertible $n \times n$ matrices with integer entries whose inverse matrix also has integer entries. Show that if $G$ is a finite group of homeomorphisms of $X$, then the homomorphism $G \rightarrow G L_{n}(Z)$ assigning to $g: X \rightarrow X$ the induced homomorphism $g_{*}: H_{1}(X ; Z) \rightarrow H_{1}(X ; Z)$ is injective. Show the same result holds if the coefficient group $\mathbb{Z}$ is replaced by $\mathbb{Z}_{m}$ with $m>2 .$ What goes wrong when $m=2 ?$

Ely Crowder
Ely Crowder
Numerade Educator
08:25

Problem 43

(a) Show that a chain complex of free abelian groups $C_{n}$ splits as a direct sum of subcomplexes $0 \rightarrow L_{n+1} \rightarrow K_{n} \rightarrow 0$ with at most two nonzero terms. IShow the short exact sequence $0 \rightarrow \operatorname{Ker} \partial \rightarrow C_{n} \rightarrow \operatorname{Im} \partial \rightarrow 0$ splits and take $K_{n}=$ Ker $\partial .$ (b) In case the groups $C_{n}$ are finitely generated, show there is a further splitting into summands $0 \rightarrow \mathbb{Z} \rightarrow 0$ and $0 \rightarrow \mathbb{Z} \stackrel{m}{\longrightarrow} \mathbb{Z} \rightarrow 0$. [Reduce the matrix of the boundary map $L_{n+1} \rightarrow K_{n}$ to echelon form by elementary row and column operations.
(c) Deduce that if $X$ is a CW complex with finitely many cells in each dimension, then $H_{n}(X ; G)$ is the direct sum of the following groups:
- a copy of $G$ for each $\mathbb{Z}$ summand of $H_{n}(X)$
- a copy of $G / m G$ for each $\mathbb{Z}_{m}$ summand of $H_{n}(X)$
- a copy of the kemel of $G \stackrel{m}{\longrightarrow} G$ for each $\mathbb{Z}_{m}$ summand of $H_{n-1}(X)$

Ely Crowder
Ely Crowder
Numerade Educator