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Algebraic Topology

Allen Hatcher

Chapter 3

Cohomology - all with Video Answers

Educators

Chapter Questions

01:07

Problem 1

Show that $\operatorname{Ext}(H, G)$ is a contravariant functor of $H$ for fixed $G,$ and a covariant functor of $G$ for fixed $H.$

Carson Merrill
Carson Merrill
Numerade Educator
03:57

Problem 1

Show that there exist nonorientable 1 -dimensional manifolds if the Hausdorff condition is dropped from the definition of a manifold.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:43

Problem 1

Assuming as known the cup product structure on the torus $S^{1} \times S^{1},$ compute the cup product structure in $H^{*}\left(M_{g}\right)$ for $M_{g}$ the closed orientable surface of genus $g$ by using the quotient map from $M_{g}$ to a wedge sum of $g$ tori, shown below.

Uma Kumari
Uma Kumari
Numerade Educator
01:07

Problem 1

Compute the groups $H_{i}\left(\mathbb{R P}^{m} \times \mathbb{R} P^{n} ; G\right)$ and $H^{i}\left(\mathbb{R} P^{m} \times \mathbb{R} P^{n} ; G\right)$ for $G=\mathbb{Z}$ and $\mathbb{Z}_{2}$
via the cellular chain and cochain complexes. Isee Example 3B.4.]

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:40

Problem 1

Given maps $f_{i}: X_{i} \rightarrow X_{i+1}$ for integers $i<0,$ show that the 'reverse mapping telescope' obtained by glueing together the mapping cylinders of the $f_{i}$ 's in the obvious way deformation retracts onto $X_{0} .$ Similarly, if maps $f_{i}: X_{i} \rightarrow X_{i+1}$ are given for all $i \in \mathbb{Z},$ show that the resulting 'double mapping telescope' deformation retracts onto any of the ordinary mapping telescopes contained in it, the union of the mapping cylinders of the $f_{i}$ 's for $i$ greater than a given number $n$.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:58

Problem 1

Suppose that $X$ is a CW complex with basepoint $e \in X$ a 0-cell. Show that $X$ is an H-space if there is a map $\mu: X \times X \rightarrow X$ such that the maps $X \rightarrow X, x \mapsto \mu(x, e)$ and $x \mapsto \mu(e, x),$ are homotopic to the identity. Isometimes this is taken as the definition of an H-space, rather than the more restrictive condition in the definition we have given. With the same hypotheses, show also that $\mu$ can be homotoped so that $e$ is a strict two-sided identity.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:36

Problem 1

Show that a topological group that has a CW structure is an orientable manifold. I Consider the homeomorphisms $x \mapsto x g$ for a fixed group element $g .1$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:36

Problem 1

Compute $H_{*}\left(S^{1} ; E\right)$ and $H^{*}\left(S^{1} ; E\right)$ for $E \rightarrow S^{1}$ the nontrivial bundle with fiber $\mathbb{Z}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
06:48

Problem 1

Use the universal coefficient theorem to show that if $H_{*}(X ; \mathbb{Z})$ is finitely generated, so the Euler characteristic $x(X)=\sum_{n}(-1)^{n} \operatorname{rank} H_{n}(X ; \mathbb{Z})$ is defined, then for any coefficient field $F$ we have $\chi(X)=\sum_{n}(-1)^{n} \operatorname{dim} H_{n}(X ; F)$

Chris Trentman
Chris Trentman
Numerade Educator
01:30

Problem 1

Show that $H^{*}\left(K\left(\mathbb{Z}_{m}, 1\right) ; \mathbb{Z}_{k}\right)$ is isomorphic as a ring to $H^{*}\left(K\left(\mathbb{Z}_{m}, 1\right) ; \mathbb{Z}_{m}\right) \otimes \mathbb{Z}_{k}$ if $k$
divides $m .$ In particular, if $m / k$ is even, this is $\Lambda_{z_{k}}[x] \otimes \mathbb{Z}_{k}[y]$

Abid Hussain
Abid Hussain
Numerade Educator
00:36

Problem 1

Compute $H_{*}\left(S^{1} ; E\right)$ and $H^{*}\left(S^{1} ; E\right)$ for $E \rightarrow S^{1}$ the nontrivial bundle with fiber $\mathbb{Z}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:07

Problem 2

Show that the maps $G \stackrel{n}{\longrightarrow} G$ and $H \stackrel{n}{\rightarrow} H$ multiplying each element by the integer $n$ induce multiplication by $n$ in $\operatorname{Ext}(H, G).$

Carson Merrill
Carson Merrill
Numerade Educator
03:26

Problem 2

Show that deleting a point from a manifold of dimension greater than 1 does not affect orientability of the manifold.

Anthony Ramos
Anthony Ramos
Numerade Educator
09:31

Problem 2

Using the cup product $H^{k}(X, A ; R) \times H^{\ell}(X, B ; R) \rightarrow H^{k+\ell}(X, A \cup B ; R)$, show that if $X$ is the union of contractible open subsets $A$ and $B$, then all cup products of positive-dimensional classes in $H^{*}(X ; R)$ are zero. This applies in particular if $X$ is a suspension. Generalize to the situation that $X$ is the union of $n$ contractible open subsets, to show that all $n$ -fold cup products of positive-dimensional classes are zero.

Chris Trentman
Chris Trentman
Numerade Educator
08:12

Problem 2

Let $C$ and $C^{\prime}$ be chain complexes, and let $I$ be the chain complex consisting of
$\mathbb{Z}$ in dimension 1 and $\mathbb{Z} \times \mathbb{Z}$ in dimension $0,$ with the boundary map taking a generator $e$ in dimension 1 to the difference $v_{1}-v_{0}$ of generators $v_{i}$ of the two $\mathbb{Z}$ 's in dimension 0. Show that a chain map $f: I \otimes C \rightarrow C^{\prime}$ is precisely the same as a chain homotopy between the two chain maps $f_{i}: C \rightarrow C^{\prime}, c \mapsto f\left(v_{i} \otimes c\right), i=0,1 .$ [The chain homotopy is $h(c)=f(e \otimes c) .1$

Yuou Sun
Yuou Sun
Numerade Educator
01:36

Problem 2

Show that $\lim _{x}^{1} G_{i}=0$ if the sequence $\cdots \longrightarrow G_{2} \stackrel{\alpha_{2}}{\longrightarrow} G_{1} \stackrel{\alpha_{1}}{\longrightarrow} G_{0}$ satisfies the Mittag-Leffler condition that for each $i$ the images of the maps $G_{i+n} \rightarrow G_{i}$ are independent of $n$ for sufficiently large $n$.

Nick Johnson
Nick Johnson
Numerade Educator
03:27

Problem 2

Show that a retract of an H-space is an H-space if it contains the identity element.

Runpeng Li
Runpeng Li
Numerade Educator
01:01

Problem 2

Using the CW structure on $S O(n),$ show that $\pi_{1} S O(n) \approx \mathbb{Z}_{2}$ for $n \geq 3$. Find a loop representing a generator, and describe how twice this loop is nullhomotopic.

Raj Bala
Raj Bala
Numerade Educator
02:04

Problem 2

Compute the homology groups with local coefficients $H_{n}\left(M ; M_{Z}\right)$ for a closed nonorientable surface $M$.

WZ
Wen Zheng
Numerade Educator
02:59

Problem 2

In this problem we will derive one half of the classification of lens spaces up to homotopy equivalence, by showing that if $L_{m}\left(\ell_{1}, \cdots, \ell_{n}\right) \simeq L_{m}\left(\ell_{1}^{\prime}, \cdots, \ell_{n}^{\prime}\right)$ then
$\ell_{1} \cdots \ell_{n} \equiv \pm \ell_{1}^{\prime} \cdots \ell_{n}^{\prime} k^{n} \bmod m$ for some integer $k .$ The converse is Exercise 29
for \S 4.2
(a) Let $L=L_{m}\left(\ell_{1}, \cdots, \ell_{n}\right)$ and let $\mathbb{Z}_{m}^{*}$ be the multiplicative group of invertible elements of $\mathbb{Z}_{m}$. Define $t \in \mathbb{Z}_{m}^{*}$ by the equation $x y^{n-1}=t z$ where $x$ is a generator of $H^{1}\left(L ; \mathbb{Z}_{m}\right), y=\beta(x),$ and $z \in H^{2 n-1}\left(L ; \mathbb{Z}_{m}\right)$ is the image of a generator of $H^{2 n-1}(L ; \mathbb{Z}) .$ Show that the image $\tau(L)$ of $t$ in the quotient group $\mathbb{Z}_{m}^{*} / \pm\left(\mathbb{Z}_{m}^{*}\right)^{n}$ depends only on the homotopy type of $L$.
(b) Given nonzero integers $k_{1}, \cdots, k_{n},$ define a map $\tilde{f}: S^{2 n-1} \rightarrow S^{2 n-1}$ sending the unit vector $\left(r_{1} e^{i \theta_{1}}, \cdots, r_{n} e^{i \theta_{n}}\right)$ in $\mathbb{C}^{n}$ to $\left(r_{1} e^{i k_{1} \theta_{1}}, \cdots, r_{n} e^{i k_{n} \theta_{n}}\right) .$ Show:
(i) $\tilde{f}$ has degree $k_{1} \cdots k_{n}$
(iii) $\tilde{f}$ induces a quotient map $f: L \rightarrow L^{\prime}$ for $L^{\prime}=L_{m}\left(\ell_{1}^{\prime}, \cdots, \ell_{n}^{\prime}\right)$ provided that $k_{j} \ell_{j} \equiv \ell_{j}^{\prime} \bmod m$ for each $j$
(iii) $f$ induces an isomorphism on $\pi_{1}$, hence on $H^{1}\left(-; \mathbb{Z}_{m}\right)$.
(iv) $f$ has degree $k_{1} \cdots k_{n},$ i.e., $f_{*}$ is multiplication by $k_{1} \cdots k_{n}$ on $H_{2 n-1}(-; \mathbb{Z})$
(c) Using the $f$ in (b), show that $\tau(L)=k_{1} \cdots k_{n} \tau\left(L^{\prime}\right)$
(d) Deduce that if $L_{m}\left(\ell_{1}, \cdots, \ell_{n}\right) \simeq L_{m}\left(\ell_{1}^{\prime}, \cdots, \ell_{n}^{\prime}\right),$ then $\ell_{1} \cdots \ell_{n} \equiv \pm \ell_{1}^{\prime} \cdots \ell_{n}^{\prime} k^{n}$
mod $m$ for some integer $k$

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
02:41

Problem 2

Show that $\operatorname{Tor}(A, \mathbb{Q} / \mathbb{Z})$ is isomorphic to the torsion subgroup of $A$. Deduce that $A$ is torsionfree iff $\operatorname{Tor}(A, B)=0$ for all $B$

Nick Johnson
Nick Johnson
Numerade Educator
02:04

Problem 2

Compute the homology groups with local coefficients $H_{n}\left(M ; M_{Z}\right)$ for a closed nonorientable surface $M$.

WZ
Wen Zheng
Numerade Educator
01:34

Problem 3

Regarding $\mathbb{Z}_{2}$ as a module over the ring $\mathbb{Z}_{4},$ construct a resolution of $\mathbb{Z}_{2}$ by free modules over $\mathbb{Z}_{4}$ and use this to show that $\operatorname{Ext}_{z_{4}}^{n}\left(\mathbb{Z}_{2}, \mathbb{Z}_{2}\right)$ is nonzero for all $n.$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
05:55

Problem 3

Show that every covering space of an orientable manifold is an orientable manifold.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:13

Problem 3

Show that the splitting in the topological KĂ¼nneth formula cannot be natural by considering the map $f \times \mathbb{1}: M\left(\mathbb{Z}_{m}, n\right) \times M\left(\mathbb{Z}_{m}, n\right) \rightarrow S^{n+1} \times M\left(\mathbb{Z}_{m}, n\right)$ where $f$ collapses
the $n$ -skeleton of $M\left(\mathbb{Z}_{m}, \boldsymbol{n}\right)=S^{n} \cup e^{n+1}$ to a point.

Gideon Idumah
Gideon Idumah
Numerade Educator
06:50

Problem 3

(a) Using the cup product structure, show there is no map $\mathbb{R P}^{n} \rightarrow \mathbb{R} P^{m}$ inducing a nontrivial map $H^{1}\left(\mathbb{R P}^{m} ; \mathbb{Z}_{2}\right) \rightarrow H^{1}\left(\mathbb{R P}^{n} ; \mathbb{Z}_{2}\right)$ if $n>m$. What is the corresponding result for maps $\mathbb{C} \mathrm{P}^{n} \rightarrow \mathrm{CP}^{m} ?$
(b) Prove the Borsuk-Ulam theorem by the following argument. Suppose on the contrary that $f: S^{n} \rightarrow \mathbb{R}^{n}$ satisfies $f(x) \neq f(-x)$ for all $x$. Then define $g: S^{n} \rightarrow S^{n-1}$ by $g(x)=(f(x)-f(-x)) /|f(x)-f(-x)|,$ so $g(-x)=-g(x)$ and $g$ induces a map $\mathbb{R P}^{n} \rightarrow \mathbb{R P}^{n-1} .$ Show that part (a) applies to this map.

Donald Albin
Donald Albin
Numerade Educator
01:28

Problem 3

Show that $\operatorname{Ext}(A, \mathbb{Q})=0$ for all $A .$ Iconsider the homology with $\mathbb{Q}$ coefficients of a Moore space $M(A, n) .]$

Harshita Goel
Harshita Goel
Numerade Educator
02:05

Problem 3

Show that if $X$ is an H-space such that the set of path-components of $X$ is a group with respect to the multiplication induced by the H-space structure, then all the path-components are homotopy equivalent.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:54

Problem 3

Compute the Pontryagin ring structure in $H_{*}(S O(5) ; \mathbb{Z})$.

Lottie Adams
Lottie Adams
Numerade Educator
08:25

Problem 3

Let $\mathcal{B}(X ; G)$ be the set of isomorphism classes of bundles of groups $E \rightarrow X$ with fiber $G,$ and let $E_{0} \rightarrow B$ Aut $(G)$ be the bundle corresponding to the 'identity' action $\rho: \operatorname{Aut}(G) \rightarrow \operatorname{Aut}(G) .$ Show that the map $[X, B \text { Aut }(G)] \rightarrow \mathcal{B}(X, G),[f] \mapsto f^{*}\left(E_{0}\right),$ is a bijection if $X$ is a CW complex, where $[X, Y]$ denotes the set of homotopy classes of maps $X \rightarrow Y$

Ely Crowder
Ely Crowder
Numerade Educator
08:33

Problem 3

Let $X$ be the smash product of $k$ copies of a Moore space $M\left(\mathbb{Z}_{p}, n\right)$ with $p$ prime. Compute the Bockstein homomorphisms in $H^{*}\left(X ; \mathbb{Z}_{p}\right)$ and use this to describe $H^{*}(X ; \mathbb{Z})$

Carrie Frizzell
Carrie Frizzell
Numerade Educator
01:30

Problem 3

Show that if $\tilde{H}^{n}(X ; \mathbb{Q})$ and $\tilde{H}^{n}\left(X ; \mathbb{Z}_{p}\right)$ are zero for all $n$ and all primes $p,$ then $\tilde{H}_{n}(X ; \mathbb{Z})=0$ for all $n,$ and hence $\tilde{H}^{n}(X ; G)=0$ for all $G$ and $n$.

Ameer Said
Ameer Said
Numerade Educator
08:25

Problem 3

Let $\mathcal{B}(X ; G)$ be the set of isomorphism classes of bundles of groups $E \rightarrow X$ with fiber $G,$ and let $E_{0} \rightarrow B$ Aut $(G)$ be the bundle corresponding to the 'identity' action $\rho: \operatorname{Aut}(G) \rightarrow \operatorname{Aut}(G) .$ Show that the map $[X, B \operatorname{Aut}(G)] \rightarrow \mathcal{B}(X, G),[f] \mapsto f^{*}\left(E_{0}\right),$ is
a bijection if $X$ is a CW complex, where $[X, Y]$ denotes the set of homotopy classes
of maps $X \rightarrow Y$

Ely Crowder
Ely Crowder
Numerade Educator
01:35

Problem 4

What happens if one defines homology groups $h_{n}(X ; G)$ as the homology groups of the chain complex $\cdots \rightarrow \operatorname{Hom}\left(G, C_{n}(X)\right) \rightarrow \operatorname{Hom}\left(G, C_{n-1}(X)\right) \rightarrow \cdots ?$ More specifically, what are the groups $h_{n}(X ; G)$ when $G=\mathbb{Z}, \mathbb{Z}_{m},$ and $\mathbb{Q} ?$

Evey Z
Evey Z
Numerade Educator
01:07

Problem 4

Given a covering space action of a group $G$ on an orientable manifold $M$ by orientation-preserving homeomorphisms, show that $M / G$ is also orientable.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:46

Problem 4

Show that the cross product of fundamental classes for closed $R$-orientable manifolds $M$ and $N$ is a fundamental class for $M \times N$.

Nick Johnson
Nick Johnson
Numerade Educator
02:01

Problem 4

Apply the Lefschetz fixed point theorem to show that every map $f: \mathbb{C} \mathrm{P}^{n} \rightarrow \mathbb{C} \mathrm{P}^{n}$ has a fixed point if $n$ is even, using the fact that $f^{*}: H^{*}\left(\mathbb{C P}^{n} ; \mathbb{Z}\right) \rightarrow H^{*}\left(\mathbb{C P}^{n} ; \mathbb{Z}\right)$ is a ring homomorphism. When $n$ is odd show there is a fixed point unless $f^{*}(\alpha)=-\alpha,$ for $\alpha$ a generator of $H^{2}\left(\mathrm{CP}^{n} ; \mathbb{Z}\right) .$ ISee Exercise 2 in $\$ 2 .$ C for an example of a map without fixed points in this exceptional case.

Amy Jiang
Amy Jiang
Numerade Educator
01:40

Problem 4

An abelian group $G$ is defined to be divisible if the map $G \stackrel{n}{\longrightarrow} G, g \mapsto n g,$ is surjective for all $n>1 .$ Show that a group is divisible iff it is a quotient of a direct sum of $\mathbb{Q}$ 's. Deduce from the previous problem that if $G$ is divisible then $\operatorname{Ext}(A, G)=0$ for all $A$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:23

Problem 4

Show that an H-space or topological group structure on a path-connected, locally path-connected space can be lifted to such a structure on its universal cover. IFor the group $S O(n)$ considered in the next section, the universal cover for $n>2$ is a 2-sheeted cover, a group called $\operatorname{spin}(n) .]$

Bryan Lynn
Bryan Lynn
Numerade Educator
07:44

Problem 4

Show that if finite connected CW complexes $X$ and $Y$ are homotopy equivalent, then their universal covers $\tilde{X}$ and $\tilde{Y}$ are proper homotopy equivalent.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:05

Problem 4

Using the cup product structure in $H^{*}(S O(5) ; \mathbb{Z}),$ show that $S O(5)$ is not homotopy equivalent to the product of any two CW complexes with nontrivial cohomology.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:52

Problem 4

Show that $\otimes$ and Tor commute with direct limits:
$\left(\lim _{\rightarrow} A_{\alpha}\right) \otimes B=\lim _{\rightarrow}\left(A_{\alpha} \otimes B\right)$ and
Tor $\left({\lim_\longrightarrow } A_{\alpha}, B\right)=\lim _{\longrightarrow} \operatorname{Tor}\left(A_{\alpha}, B\right)$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
07:44

Problem 4

Show that if finite connected CW complexes $X$ and $Y$ are homotopy equivalent, then their universal covers $\tilde{X}$ and $\tilde{Y}$ are proper homotopy equivalent.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:25

Problem 5

Regarding a cochain $\varphi \in C^{1}(X ; G)$ as a function from paths in $X$ to $G,$ show that if $\varphi$ is a cocycle, then
(a) $\varphi(f \cdot g)=\varphi(f)+\varphi(g)$
(b) $\varphi$ takes the value 0 on constant paths,
(c) $\varphi(f)=\varphi(g)$ if $f \simeq g$
(d) $\varphi$ is a coboundary iff $\varphi(f)$ depends only on the endpoints of $f,$ for all $f$
[In particular, (a) and (c) give a map $H^{1}(X ; G) \rightarrow \operatorname{Hom}\left(\pi_{1}(X), G\right),$ which the universal coefficient theorem says is an isomorphism if $X$ is path-connected.]

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:19

Problem 5

Show that $M \times N$ is orientable iff $M$ and $N$ are both orientable.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
07:42

Problem 5

Show that
$$
\begin{array}{ll}
H_{n}(X \times Y ; R) \times H^{j}(Y ; R) \rightarrow H_{n-j}(Y ; R), & \left(e^{i} \times e^{j}, \varphi\right) \mapsto \varphi\left(e^{j}\right) e^{i} \\
H^{n}(X \times Y ; R) \times H_{j}(Y ; R) \rightarrow H^{n-j}(Y ; R), & \left(\varphi, e^{j}\right) \mapsto\left(e^{i} \mapsto \varphi\left(e^{i} \times e^{j}\right)\right)
\end{array}
$$
can be defined via the indicated cellular formulas. [These 'products' are in some ways more like division than multiplication, and this is reflected in the common notation $a / b$ for them, or $a \backslash b$ when the order of the factors is reversed. The first of the two slant products is related to cap product in the same way that the cohomology cross product is related to cup product.]

Jayashree Behera
Jayashree Behera
Numerade Educator
01:05

Problem 5

Show the ring $H^{*}\left(\mathbb{R P}^{\infty} ; \mathbb{Z}_{m}\right)$ is isomorphic to $\mathbb{Z}_{m}[\alpha, \beta] /\left(2 \alpha, 2 \beta, \alpha^{2}\right)$ if $m>2$ where $|\alpha|=1$ and $|\beta|=2 .$ IAdapt the proof of Theorem 3.12 to $\mathbb{Z}_{m}$ coefficients.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:22

Problem 5

Show that if $(X, e)$ is an H-space then $\pi_{1}(X, e)$ is abelian. [Compare the usual composition $f \cdot g$ of loops with the product $\mu(f(t), g(t))$ coming from the H-space multiplication $\mu .]$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:05

Problem 5

Show that $\operatorname{Ext}(A, \mathbb{Z})$ is isomorphic to the cokernel of Hom $(A, \mathbb{Q}) \rightarrow \operatorname{Hom}(A, \mathbb{Q} / \mathbb{Z})$ the map induced by the quotient map $\mathbb{Q} \rightarrow \mathbb{Q} / \mathbb{Z}$. Use this to get another proof that $\operatorname{Ext}\left(\mathbb{Z}_{\left.p^{\infty}, \mathbb{Z}\right)} \approx \hat{\mathbb{Z}}_{p} \text { for } p$ prime. \right.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:34

Problem 5

If $X$ is a finite nonsimply-connected graph, show that $H^{n}\left(X ; \mathbb{Z}\left[\boldsymbol{\pi}_{1} X\right]\right)$ is zero unless $n=1,$ when it is the direct sum of a countably infinite number of $\mathbb{Z}^{\prime}$ 's. IUse Proposition 3H.5 and compute $H_{c}^{n}(\tilde{X})$ as $\lim _{\longrightarrow} H^{n}\left(\tilde{X}, \tilde{X}-T_{i}\right)$ for a suitable sequence of finite subtrees $\left.T_{1} \subset T_{2} \subset \cdots \text { of } \tilde{X} \text { with } \cup_{i} T_{i}=\tilde{X} .\right]$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
11:16

Problem 5

From the fact that $\operatorname{Tor}(A, B)=0$ if $A$ is free, deduce that $\operatorname{Tor}(A, B)=0$ if $A$ is torsionfree by applying the previous problem to the directed system of finitely generated subgroups $A_{\alpha}$ of $A .$

Ely Crowder
Ely Crowder
Numerade Educator
02:49

Problem 5

If $X$ is a finite nonsimply-connected graph, show that $H^{n}\left(X ; \mathbb{Z}\left[\pi_{1} X\right]\right)$ is zero unless $n=1,$ when it is the direct sum of a countably infinite number of $\mathbb{Z}$ 's. [Use Proposition 3H.5 and compute $H_{c}^{n}(\tilde{X})$ as $\lim _{\text {roposition }} H^{n}\left(\tilde{X}, \tilde{X}-T_{i}\right)$ for a suitable sequence of finite subtrees $\left.T_{1} \subset T_{2} \subset \cdots \text { of } \tilde{X} \text { with } \cup_{i} T_{i}=\tilde{X} .\right]$

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:11

Problem 6

(a) Directly from the definitions, compute the simplicial cohomology groups of $S^{1} \times S^{1}$ with $\mathbb{Z}$ and $\mathbb{Z}_{2}$ coefficients, using the $\Delta$ -complex structure given in $\$ 2.1.$
(b) Do the same for $\mathbb{R P}^{2}$ and the Klein bottle.

James Kiss
James Kiss
Numerade Educator
06:33

Problem 6

Given two disjoint connected $n$ -manifolds $M_{1}$ and $M_{2}$, a connected $n$ -manifold $M_{1} \geqslant M_{2},$ their connected sum, can be constructed by deleting the interiors of closed $n$ -balls $B_{1} \subset M_{1}$ and $B_{2} \subset M_{2}$ and identifying the resulting boundary spheres $\partial B_{1}$ and $\partial B_{2}$ via some homeomorphism between them. (Assume that each $B_{i}$ embeds nicely in a larger ball in $M_{i} .$ )
(a) Show that if $M_{1}$ and $M_{2}$ are closed then there are isomorphisms $H_{i}\left(M_{1} \neq M_{2} ; \mathbb{Z}\right) \approx$ $H_{i}\left(M_{1} ; \mathbb{Z}\right) \oplus H_{i}\left(M_{2} ; \mathbb{Z}\right)$ for $0<i<n,$ with one exception: If both $M_{1}$ and $M_{2}$ are nonorientable, then $H_{n-1}\left(M_{1} \# M_{2} ; \mathbb{Z}\right)$ is obtained from $H_{n-1}\left(M_{1} ; \mathbb{Z}\right) \oplus H_{n-1}\left(M_{2} ; \mathbb{Z}\right)$ by
replacing one of the two $\mathbb{Z}_{2}$ summands by a $\mathbb{Z}$ summand. IEuler characteristics may help in the exceptional case.
(b) Show that $x\left(M_{1} \neq M_{2}\right)=x\left(M_{1}\right)+x\left(M_{2}\right)-x\left(S^{n}\right)$ if $M_{1}$ and $M_{2}$ are closed.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:46

Problem 6

Use cup products to compute the map $H^{*}\left(\mathrm{CP}^{n} ; \mathbb{Z}\right) \rightarrow H^{*}\left(\mathrm{CP}^{n} ; \mathbb{Z}\right)$ induced by the map $\mathrm{CP}^{n} \rightarrow \mathrm{CP}^{n}$ that is a quotient of the map $\mathrm{C}^{n+1} \rightarrow \mathrm{C}^{n+1}$ raising each coordinate to the $d^{\text {th}}$ power, $\left(z_{0}, \cdots, z_{n}\right) \mapsto\left(z_{0}^{d}, \cdots, z_{n}^{d}\right),$ for a fixed integer $d>0 .$ First do the case $n=1.1.$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
03:27

Problem 6

Show that $S^{n}$ is an H-space iff the attaching map of the $2 n$ -cell of $J_{2}\left(S^{n}\right)$ is homotopically trivial.

Runpeng Li
Runpeng Li
Numerade Educator
07:46

Problem 6

$$\text { Show that } \operatorname{Ext}\left(\mathbb{Z}_{p^{\infty}}, \mathbb{Z}_{p}\right) \approx \mathbb{Z}_{p}$$

Shafiq Rehman
Shafiq Rehman
Numerade Educator
02:01

Problem 6

Show that homology groups $H_{n}^{\ell f}(X ; G)$ can be defined using locally finite chains, which are formal sums $\Sigma_{\sigma} g_{\sigma} \sigma$ of singular simplices $\sigma: \Delta^{n} \rightarrow X$ with coefficients $g_{\sigma} \in G,$ such that each $x \in X$ has a neighborhood meeting the images of only finitely many $\sigma$ 's with $g_{\sigma} \neq 0 .$ Develop this homology theory far enough to show that for a locally compact CW complex $X, H_{n}^{\ell f}(X ; G)$ can be computed using infinite cellular chains $\sum_{\alpha} g_{\alpha} e_{\alpha}^{n}$.

Christopher Callahan
Christopher Callahan
Numerade Educator
01:57

Problem 6

Show that $\operatorname{Tor}(A, B)$ is always a torsion group, and that $\operatorname{Tor}(A, B)$ contains an element of order $n$ iff both $A$ and $B$ contain elements of order $n$.

Wendi Zhao
Wendi Zhao
Numerade Educator
02:01

Problem 6

Show that homology groups $H_{n}^{\ell f}(X ; G)$ can be defined using locally finite chains, which are formal sums $\sum_{\sigma} g_{\sigma} \sigma$ of singular simplices $\sigma: \Delta^{n} \rightarrow X$ with coefficients $g_{\sigma} \in G,$ such that each $x \in X$ has a neighborhood meeting the images of only finitely many $\sigma$ 's with $g_{\sigma} \neq 0 .$ Develop this homology theory far enough to show that for a locally compact $\mathrm{CW}$ complex $X, H_{n}^{\ell f}(X ; G)$ can be computed using infinite cellular chains $\sum_{\alpha} g_{\alpha} e_{\alpha}^{n}$

Christopher Callahan
Christopher Callahan
Numerade Educator
01:05

Problem 7

Show that the functors $h^{n}(X)=\operatorname{Hom}\left(H_{n}(X), \mathbb{Z}\right)$ do not define a cohomology theory on the category of CW complexes.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:53

Problem 7

Use cup products to show that $\mathbb{R P}^{3}$ is not homotopy equivalent to $\mathbb{R P}^{2} \vee S^{3}$.

Clarissa Noh
Clarissa Noh
Numerade Educator

Problem 7

For a map $f: M \rightarrow N$ between connected closed orientable $n$ -manifolds with fundamental classes $[M]$ and $[N],$ the degree of $f$ is defined to be the integer $d$ such that $f_{*}([M])=d[N],$ so the sign of the degree depends on the choice of fundamental classes. Show that for any connected closed orientable $n$ -manifold $M$ there is a degree $1 \operatorname{map} M \rightarrow S^{n}.$

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01:18

Problem 7

What are the primitive elements of the Hopf algebra $\mathbb{Z}_{p}[x]$ for $p$ prime?

James Chok
James Chok
Numerade Educator
08:25

Problem 7

Show that for a short exact sequence of abelian groups $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0,$ a Moore space $M(\mathcal{C}, \boldsymbol{n})$ can be realized as a quotient $M(B, n) / M(A, n) .$ Applying the long exact sequence of cohomology for the pair $(M(B, n), M(A, n))$ with any coefficient group $G,$ deduce an exact sequence $0 \rightarrow \operatorname{Hom}(C, G) \rightarrow \operatorname{Hom}(B, G) \rightarrow \operatorname{Hom}(A, G) \rightarrow \operatorname{Ext}(C, G) \rightarrow \operatorname{Ext}(B, G) \rightarrow \operatorname{Ext}(A, G) \rightarrow 0$

Ely Crowder
Ely Crowder
Numerade Educator
01:05

Problem 8

Many basic homology arguments work just as well for cohomology even though maps go in the opposite direction. Verify this in the following cases:
(a) Compute $H^{i}\left(S^{n} ; G\right)$ by induction on $n$ in two ways: using the long exact sequence
of a pair, and using the Mayer-Vietoris sequence.
(b) Show that if $A$ is a closed subspace of $X$ that is a deformation retract of some neighborhood, then the quotient map $X \rightarrow X / A$ induces isomorphisms $H^{n}(X, A ; G) \approx$ $\widetilde{H}^{n}(X / A ; G)$ for all $n$
(c) Show that if $A$ is a retract of $X$ then $H^{n}(X ; G) \approx H^{n}(A ; G) \oplus H^{n}(X, A ; G)$

Anthony Ramos
Anthony Ramos
Numerade Educator
07:44

Problem 8

Let $X$ be $\mathrm{CP}^{2}$ with a cell $e^{3}$ attached by a map $S^{2} \rightarrow \mathbb{C} \mathrm{P}^{1} \subset \mathbb{C} \mathrm{P}^{2}$ of degree $p,$ and let $Y=M\left(\mathbb{Z}_{p}, 2\right) \vee S^{4} .$ Thus $X$ and $Y$ have the same 3-skeleton but differ in the way their 4 -cells are attached. Show that $X$ and $Y$ have isomorphic cohomology rings with $\mathbb{Z}$ coefficients but not with $\mathbb{Z}_{p}$ coefficients.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:22

Problem 8

For a map $f: M \rightarrow N$ between connected closed orientable $n$ -manifolds, suppose there is a ball $B \subset N$ such that $f^{-1}(B)$ is the disjoint union of of balls $B_{i}$ each mapped homeomorphically by $f$ onto $B$. Show the degree of $f$ is $\Sigma_{i} \varepsilon_{i}$ where $\varepsilon_{i}$ is +1 or -1 according to whether $f: B_{i} \rightarrow B$ preserves or reverses local orientations induced from given fundamental classes $[M]$ and $[N].$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
View

Problem 8

Show that the tensor product of two Hopf algebras is a Hopf algebra.

Victor Salazar
Victor Salazar
Numerade Educator
08:25

Problem 8

Show that for a Moore space $M(G, n)$ the Bockstein long exact sequence in cohomology associated to the short exact sequence of coefficient groups $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ reduces to an exact sequence $0 \rightarrow \operatorname{Hom}(G, A) \rightarrow \operatorname{Hom}(G, B) \rightarrow \operatorname{Hom}(G, C) \rightarrow \operatorname{Ext}(G, A) \rightarrow \operatorname{Ext}(G, B) \rightarrow \operatorname{Ext}(G, C) \rightarrow 0$

Ely Crowder
Ely Crowder
Numerade Educator
03:29

Problem 9

Show that if $f: S^{n} \rightarrow S^{n}$ has degree $d$ then $f^{*}: H^{n}\left(S^{n} ; G\right) \rightarrow H^{n}\left(S^{n} ; G\right)$ is multiplication by $d .$

Will Erickson
Will Erickson
Numerade Educator
03:13

Problem 9

Show that if $H_{n}(X ; \mathbb{Z})$ is finitely generated and free for each $n,$ then $H^{*}\left(X ; \mathbb{Z}_{p}\right)$ and $H^{*}(X ; \mathbb{Z}) \otimes \mathbb{Z}_{p}$ are isomorphic as rings, so in particular the ring structure with $\mathbb{Z}$ coefficients determines the ring structure with $\mathbb{Z}_{p}$ coefficients.

Gideon Idumah
Gideon Idumah
Numerade Educator
28:40

Problem 9

Show that a $p$ -sheeted covering space projection $M \rightarrow N$ has degree $\pm p,$ when $M$ and $N$ are connected closed orientable manifolds.

Donald Albin
Donald Albin
Numerade Educator
04:05

Problem 9

Apply the theorems of Hopf and Borel to show that for a finite CW complex H-space $X$ with $\tilde{H}_{*}(X ; \mathbb{Z}) \neq 0,$ the Euler characteristic $\chi(X)$ is 0.

Harshita Goel
Harshita Goel
Numerade Educator
06:02

Problem 9

For an abelian group $A$ let $p: A \rightarrow A$ be multiplication by $p,$ and let $p A=\operatorname{Ker} p$ $p A=\operatorname{Im} p,$ and $A_{p}=$ Coker $p$ as in the proof of Proposition 3F.12. Show that the sixterm exact sequences involving Hom(- $, \mathbb{Z}$ ) and $\operatorname{Ext}(-, \mathbb{Z})$ associated to the short exact sequences $0 \rightarrow_{p} A \rightarrow A \rightarrow p A \rightarrow 0$ and $0 \rightarrow p A \rightarrow A \rightarrow A_{p} \rightarrow 0$ can be spliced together
to yield the exact sequence across the top of the following diagram where the map labeled " $p$ ' is multiplication by $p .$ Use this to show:
(a) $\operatorname{Ext}(A, \mathbb{Z})$ is divisible iff $A$ is torsionfree.
(b) $\operatorname{Ext}(A, \mathbb{Z})$ is torsionfree if $A$ is divisible, and the converse holds if $\operatorname{Hom}(A, \mathbb{Z})=0$
(FIGURE CAN'T COPY)

Ely Crowder
Ely Crowder
Numerade Educator
00:55

Problem 10

For the lens space $L_{m}\left(\ell_{1}, \cdots, \ell_{n}\right)$ defined in Example $2.43,$ compute the cohomology groups using the cellular cochain complex and taking coefficients in $\mathbb{Z}, \mathbb{Q}, \mathbb{Z}_{m}$ and $\mathbb{Z}_{p}$ for $p$ prime. Verify that the answers agree with those given by the universal coefficient theorem.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
05:34

Problem 10

Show that the cross product map $H^{*}(X ; \mathbb{Z}) \otimes H^{*}(Y ; \mathbb{Z}) \rightarrow H^{*}(X \times Y ; \mathbb{Z})$ is not an isomorphism if $X$ and $Y$ are infinite discrete sets. [This shows the necessity of the hypothesis of finite generation in Theorem 3.16.1.

Chris Trentman
Chris Trentman
Numerade Educator
04:58

Problem 10

Let $X$ be a path-connected H-space with $H^{*}(X ; R)$ free and finitely generated in each dimension. For maps $f, g: X \rightarrow X,$ the product $f g: X \rightarrow X$ is defined by $(f g)(x)=f(x) g(x),$ using the H-space product.
(a) Show that $(f g)^{*}(\alpha)=f^{*}(\alpha)+g^{*}(\alpha)$ for primitive elements $\alpha \in H^{*}(X ; R)$
(b) Deduce that the $k^{t h}$ -power map $x \mapsto x^{k}$ induces the map $\alpha \mapsto k \alpha$ on primitive elements $\alpha .$ In particular the quaternionic $k^{t h}$ -power map $S^{3} \rightarrow S^{3}$ has degree $k$
(c) Show that every polynomial $a_{n} x^{n} b_{n}+\cdots+a_{1} x b_{1}+a_{0}$ with coefficients in $\mathbb{H}$ has a root in $\mathbb{H}$ if $n>0 .$ Isee Theorem 1.8 .1

Anthony Ramos
Anthony Ramos
Numerade Educator
01:21

Problem 10

10. Show that for a degree 1 map $f: M \rightarrow N$ of connected closed orientable manifolds, the induced map $f_{\star}: \pi_{1} M \rightarrow \pi_{1} N$ is surjective, hence also $f_{*}: H_{1}(M) \rightarrow H_{1}(N)$. II ift
$f$ to the covering space $\tilde{N} \rightarrow N$ corresponding to the subgroup $\operatorname{Im} f_{\star} \subset \pi_{1} N,$ then consider the two cases that this covering is finite-sheeted or infinite-sheeted.]

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:07

Problem 11

Let $X$ be a Moore space $M\left(\mathbb{Z}_{m}, n\right)$ obtained from $S^{n}$ by attaching a cell $e^{n+1}$ by a map of degree $m$
(a) Show that the quotient map $X \rightarrow X / S^{n}=S^{n+1}$ induces the trivial map on $\tilde{H}_{i}(-; \mathbb{Z})$ for all $i,$ but not on $H^{n+1}(-; \mathbb{Z}) .$ Deduce that the splitting in the universal coefficient theorem for cohomology cannot be natural.
(b) Show that the inclusion $S^{n} \hookrightarrow X$ induces the trivial map on $\tilde{H}^{i}(-; \mathbb{Z})$ for all $i,$ but not on $H_{n}(-; \mathbb{Z}).$

Anthony Ramos
Anthony Ramos
Numerade Educator
06:50

Problem 11

Using cup products, show that every map $s^{k+\ell} \rightarrow S^{k} \times S^{\ell}$ induces the trivial homomorphism $H_{k+\ell}\left(S^{k+\ell}\right) \rightarrow H_{k+\ell}\left(S^{k} \times S^{\ell}\right),$ assuming $k>0$ and $\ell>0.$

Donald Albin
Donald Albin
Numerade Educator
01:38

Problem 11

If $M_{g}$ denotes the closed orientable surface of genus $g,$ show that degree 1 maps $M_{g} \rightarrow M_{h}$ exist iff $g \geq h.$

Gregory Higby
Gregory Higby
Numerade Educator
08:26

Problem 11

If $T^{n}$ is the $n$ -dimensional torus, the product of $n$ circles, show that the Pontryagin ring $H_{*}\left(T^{n} ; \mathbb{Z}\right)$ is the exterior algebra $\Lambda_{Z}\left[x_{1}, \cdots, x_{n}\right]$ with $\left|x_{i}\right|=1$.

Cameron Bunney
Cameron Bunney
Numerade Educator
02:07

Problem 12

Show $H^{k}\left(X, X^{n} ; G\right)=0$ if $X$ is a CW complex and $k \leq n,$ by using the cohomology version of the second proof of the corresponding result for homology in Lemma 2.34.

Narayan Hari
Narayan Hari
Numerade Educator
03:58

Problem 12

Show that the spaces $\left(S^{1} \times \mathrm{CP}^{\infty}\right) /\left(S^{1} \times\left\{x_{0}\right\}\right)$ and $S^{3} \times \mathrm{CP}^{\infty}$ have isomorphic cohomology rings with $\mathbb{Z}$ or any other coefficients. IAn exercise for $\$ 4 . \mathrm{L}$ is to show these two spaces are not homotopy equivalent.]

Anthony Ramos
Anthony Ramos
Numerade Educator
01:39

Problem 12

As an algebraic application of the preceding problem, show that in a free group $F$ with basis $x_{1}, \cdots, x_{2 k},$ the product of commutators $\left[x_{1}, x_{2}\right] \cdots\left[x_{2 k-1}, x_{2 k}\right]$ is not equal to a product of fewer than $k$ commutators $\left[v_{i}, w_{i}\right]$ of elements $v_{i}, w_{i} \in F$ IRecall that the 2 -cell of $M_{k}$ is attached by the product $\left[x_{1}, x_{2}\right] \cdots\left[x_{2 k-1}, x_{2 k}\right] .$ From a relation $\left[x_{1}, x_{2}\right] \cdots\left[x_{2 k-1}, x_{2 k}\right]=\left[v_{1}, w_{1}\right] \cdots\left[v_{j}, w_{j}\right]$ in $F,$ construct a degree 1 $\operatorname{map} M_{j} \rightarrow M_{k} .1$

Nick Johnson
Nick Johnson
Numerade Educator
04:07

Problem 12

Compute the Pontryagin product structure in $H_{*}\left(L ; \mathbb{Z}_{p}\right)$ where $L$ is an infinitedimensional lens space $S^{\infty} / \mathbb{Z}_{p},$ for $p$ an odd prime, using the coproduct in $H^{*}\left(L ; \mathbb{Z}_{p}\right)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
07:44

Problem 13

13. Let $\langle X, Y\rangle$ denote the set of basepoint-preserving homotopy classes of basepointpreserving maps $X \rightarrow Y .$ Using Proposition $1 \mathrm{B} .9,$ show that if $X$ is a connected CW complex and $G$ is an abelian group, then the map $\langle X, K(G, 1)\rangle \rightarrow H^{1}(X ; G)$ sending a map $f: X \rightarrow K(G, 1)$ to the induced homomorphism $f_{*}: H_{1}(X) \rightarrow H_{1}(K(G, 1)) \approx G$ is a bijection, where we identify $H^{1}(X ; G)$ with Hom $\left(H_{1}(X), G\right)$ via the universal coefficient theorem.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:56

Problem 13

Describe $H^{*}\left(\mathrm{CP}^{\infty} / \mathrm{CP}^{1} ; \mathbb{Z}\right)$ as a ring with finitely many multiplicative generators. How does this ring compare with $H^{*}\left(S^{6} \times 00 \mathrm{P}^{\infty} ; \mathbb{Z}\right) ?$

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
06:04

Problem 13

Verify that the Hopf algebras $\Lambda_{R}[\alpha]$ and $\mathbb{Z}_{p}[\alpha] /\left(\alpha^{p}\right)$ are self-dual.

Mengchun Cai
Mengchun Cai
Numerade Educator
04:30

Problem 13

Let $M_{h}^{\prime} \subset M_{g}$ be a compact subsurface of genus $h$ with one boundary circle, so $M_{h}^{\prime}$ is homeomorphic to $M_{h}$ with an open disk removed. Show there is no retraction $M_{g} \rightarrow M_{h}^{\prime}$ if $h>g / 2 .$ IApply the previous problem, using the fact that $M_{g}-M_{h}^{\prime}$ has genus $g-h .1.$

Gurjyot Kaur
Gurjyot Kaur
Numerade Educator

Problem 14

Let $q: \mathbb{R} P^{\infty} \rightarrow C P^{\infty}$ be the natural quotient map obtained by regarding both spaces as quotients of $S^{\infty},$ modulo multiplication by real scalars in one case and complex scalars in the other. Show that the induced map $q^{*}: H^{*}\left(\mathrm{CP}^{\infty} ; \mathbb{Z}\right) \rightarrow H^{*}\left(\mathbb{R} \mathrm{P}^{\infty} ; \mathbb{Z}\right)$ is sur-
jective in even dimensions by showing first by a geometric argument that the restriction $q: \mathbb{R}^{2} \rightarrow \mathbb{C}^{1}$ induces a surjection on $H^{2}$ and then appealing to cup product structures. Next, form a quotient space $X$ of $\mathbb{R P}^{\infty}$ II $\mathbb{C}^{n}$ by identifying each point $x \in \mathbb{R}^{2 n}$ with $q(x) \in \mathbb{C} \mathrm{P}^{n} .$ Show there are ring isomorphisms $H^{*}(X ; \mathbb{Z}) \approx \mathbb{Z}[\alpha] /\left(2 \alpha^{n+1}\right)$ and $H^{*}\left(X ; \mathbb{Z}_{2}\right) \approx \mathbb{Z}_{2}[\alpha, \beta] /\left(\beta^{2}-\alpha^{n+1}\right),$ where $|\alpha|=2$ and $|\beta|=2 n+1 .$ Make a similar construction and analysis for the quotient map $q: \mathbb{C} \mathrm{P}^{\infty} \rightarrow \mathbb{H P}^{\infty}$.

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07:44

Problem 14

Show that the coproduct in the Hopf algebra $H_{*}(X ; R)$ dual to $H^{*}(X ; R)$ is induced by the diagonal map $X \rightarrow X \times X, x \mapsto(x, x)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:39

Problem 14

14. Let $X$ be the subspace of $\mathbb{R}^{2}$ consisting of the circles of radius $1 / n$ and center $(1 / n, 0)$ for $n=1,2, \cdots$
(a) If $f_{n}: I \rightarrow X$ is the loop based at the origin winding once around the $n^{t h}$ circle, show that the infinite product of commutators $\left[f_{1}, f_{2}\right]\left[f_{3}, f_{4}\right] \cdots$ defines a loop in $X$ that is nontrivial in $H_{1}(X) .$ IUse Exercise 12.1.
(b) If we view $X$ as the wedge sum of the sub spaces $A$ and $B$ consisting of the odd-numbered and even-numbered circles, respectively, use the same loop to show that the map $H_{1}(X) \rightarrow H_{1}(A) \oplus H_{1}(B)$ induced by the retractions of $X$ onto $A$ and $B$ is not an isomorphism.

Chris Trentman
Chris Trentman
Numerade Educator
00:38

Problem 15

For a fixed coefficient field $F,$ define the Poincaré series of a space $X$ to be the formal power series $p(t)=\Sigma_{i} a_{i} t^{i}$ where $a_{i}$ is the dimension of $H^{i}(X ; F)$ as a vector space over $F$, assuming this dimension is finite for all $i .$ Show that $p(X \times Y)= p(X) p(Y) .$ Compute the Poincaré series for $S^{n}, \mathbb{R P}^{n}, \mathbb{R P}^{\infty}, \mathbb{C P}^{n}, \mathbb{C P}^{\infty},$ and the spaces in the preceding three exercises.

Frank Lin
Frank Lin
Numerade Educator
07:44

Problem 15

Suppose that $X$ is a path-connected H-space such that $H^{*}(X ; \mathbb{Z})$ is free and finitely generated in each dimension, and $H^{*}(X ; \mathbb{Q})$ is a polynomial ring $\mathbb{Q}[\alpha]$. Show that the Pontryagin ring $H_{*}(X ; \mathbb{Z})$ is commutative and associative, with a structure uniquely determined by the ring $H^{*}(X ; \mathbb{Z})$.

Anthony Ramos
Anthony Ramos
Numerade Educator
08:25

Problem 15

For an $n$ -manifold $M$ and a compact subspace $A \subset M,$ show that $H_{n}(M, M-A ; R)$ is isomorphic to the group $\Gamma_{R}(A)$ of sections of the covering space $M_{R} \rightarrow M$ over $A$ that is, maps $A \rightarrow M_{R}$ whose composition with $M_{R} \rightarrow M$ is the identity.

Ely Crowder
Ely Crowder
Numerade Educator
06:36

Problem 16

Show that if $X$ and $Y$ are finite CW complexes such that $H^{*}(X ; \mathbb{Z})$ and $H^{*}(Y ; \mathbb{Z})$ contain no elements of order a power of a given prime $p,$ then the same is true for $X \times Y .$ IApply Theorem 3.16 with coefficients in various fields.

WM
William Mead
Numerade Educator
01:12

Problem 16

Classify algebraically the Hopf algebras $A$ over $\mathbb{Z}$ such that $A^{n}$ is free for each $n$ and $A \otimes \mathbb{Q} \approx \mathbb{Q}[\alpha] .$ In particular, determine which Hopf algebras $A \otimes \mathbb{Z}_{p}$ arise from such $A$ 's.

Prem Bijarniya
Prem Bijarniya
Numerade Educator
06:45

Problem 16

Show that $(\alpha \sim \varphi) \sim \psi=\alpha \sim(\varphi \smile \psi)$ for all $\alpha \in C_{k}(X ; R), \varphi \in C^{\ell}(X ; R),$ and $\psi \in C^{m}(X ; R) .$ Deduce that cap product makes $H_{*}(X ; R)$ a right $H^{*}(X ; R)$ -module.

Sandip Ranjan
Sandip Ranjan
Numerade Educator
View

Problem 17

Show that $H^{*}\left(J\left(S^{n}\right) ; \mathbb{Z}\right)$ for $n$ odd is the tensor product of an exterior algebra on an $n$ -dimensional generator with a divided polynomial algebra on a $2 n$ -dimensional generator.

Victor Salazar
Victor Salazar
Numerade Educator
01:20

Problem 17

Show that a direct limit of exact sequences is exact. More generally, show that homology commutes with direct limits: If $\left\{C_{\alpha}, f_{\alpha \beta}\right\}$ is a directed system of chain complexes, with the maps $f_{\alpha \beta}: C_{\alpha} \rightarrow C_{\beta}$ chain maps, then $H_{n}\left(\underline{\lim } C_{\alpha}\right)=\lim _{x \rightarrow a} H_{n}\left(C_{\alpha}\right).$

Sarah Wharton
Sarah Wharton
Numerade Educator
01:00

Problem 18

For the closed orientable surface $M$ of genus $g \geq 1,$ show that for each nonzero $\alpha \in H^{1}(M ; \mathbb{Z})$ there exists $\beta \in H^{1}(M ; \mathbb{Z})$ with $\alpha \beta \neq 0 .$ Deduce that $M$ is not homotopy equivalent to a wedge sum $X \vee Y$ of $C W$ complexes with nontrivial reduced homology. Do the same for closed nonorientable surfaces using cohomology with $\mathbb{Z}_{2}$ coefficients.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:40

Problem 18

Show that a direct limit $\varliminf \underline{\lim G_{\alpha}}$ of torsionfree abelian groups $G_{\alpha}$ is torsionfree. More generally, show that any finitely generated subgroup of $\lim _{x \rightarrow \alpha}$ is realized as a subgroup of some $G_{\alpha}.$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:55

Problem 19

19. Show that a direct limit of countable abelian groups over a countable indexing set is countable. Apply this to show that if $X$ is an open set in $\mathbb{R}^{n}$ then $H_{i}(X ; \mathbb{Z})$ is countable for all $i.$

Angelo Rendina
Angelo Rendina
Numerade Educator
02:59

Problem 20

Show that $H_{c}^{0}(X ; G)=0$ if $X$ is path-connected and noncompact.

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
04:01

Problem 21

For a space $X,$ let $X^{+}$ be the one-point compactification. If the added point, denoted $\infty,$ has a neighborhood in $X^{+}$ that is a cone with $\infty$ the cone point, show that the evident map $H_{c}^{n}(X ; G) \rightarrow H^{n}\left(X^{+}, \infty ; G\right)$ is an isomorphism for all $n$. [Question:
Does this result hold when $X=\mathbb{Z} \times \mathbb{R}\rceil.$

Anthony Ramos
Anthony Ramos
Numerade Educator
03:29

Problem 22

Show that $H_{c}^{n}(X \times \mathbb{R} ; G) \approx H_{c}^{n-1}(X ; G)$ for all $n.$

Will Erickson
Will Erickson
Numerade Educator
01:58

Problem 23

Show that for a locally compact $\Delta$ -complex $X$ the simplicial and singular cohomology groups $H_{c}^{i}(X ; G)$ are isomorphic. This can be done by showing that $\Delta_{c}^{i}(X ; G)$ is the union of its subgroups $\Delta^{i}(X, A ; G)$ as $A$ ranges over subcomplexes of $X$ that contain all but finitely many simplices, and likewise $C_{c}^{i}(X ; G)$ is the union of its subgroups $C^{i}(X, A ; G)$ for the same family of subcomplexes $A.$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:12

Problem 24

Let $M$ be a closed connected 3 -manifold, and write $H_{1}(M ; \mathbb{Z})$ as $\mathbb{Z}^{r} \oplus F$, the direct sum of a free abelian group of rank $r$ and a finite group $F .$ Show that $H_{2}(M ; Z)$ is $Z^{r}$ if $M$ is orientable and $Z^{r-1} \oplus Z_{2}$ if $M$ is nonorientable. In particular, $r \geq 1$ when $M$ is nonorientable. Using Exercise $6,$ construct examples showing there are no other restrictions on the homology groups of closed 3 -manifolds. IIn the nonorientable case consider the manifold $N$ obtained from $S^{2} \times I$ by identifying $S^{2} \times\{0\}$ with $S^{2} \times\{1\}$ via a reflection of $s^{2} .1$

Yuou Sun
Yuou Sun
Numerade Educator
07:44

Problem 25

Show that if a closed orientable manifold $M$ of dimension $2 k$ has $H_{k-1}(M ; Z)$ torsionfree, then $H_{k}(M ; \mathbb{Z})$ is also torsionfree.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:22

Problem 26

Compute the cup product structure in $H^{*}\left(S^{2} \times S^{8} \neq S^{4} \times S^{6} ; Z\right),$ and in particular show that the only nontrivial cup products are those dictated by Poincaré duality. Isee Exercise $6 .$ The result has an evident generalization to connected sums of $S^{i} \times S^{n-i}$ 's for fixed $n \text { and varying } i .\rfloor.$

Victor Salazar
Victor Salazar
Numerade Educator
03:19

Problem 27

Show that after a suitable change of basis, a skew-symmetric nonsingular bilinear form over $\mathbb{Z}$ can be represented by a matrix consisting of $2 \times 2$ blocks $\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$ along the diagonal and zeros elsewhere. [For the matrix of a bilinear form, the following operation can be realized by a change of basis: Add an integer multiple of the $i^{t h}$ row to the $j^{t h}$ row and add the same integer multiple of the $i^{t h}$ column to the $j^{t h}$ column. Use this to fix up each column in turn. Note that a skew-symmetric matrix must have zeros on the diagonal.]

Srilakshmi E K
Srilakshmi E K
Numerade Educator
02:16

Problem 28

28. Show that a nonsingular symmetric or skew-symmetric bilinear pairing over a field $F,$ of the form $F^{n} \times F^{n} \rightarrow F,$ cannot be identically zero when restricted to all pairs of vectors $v, w$ in a $k$ -dimensional subspace $V \subset F^{n}$ if $k>n / 2.$

Nick Johnson
Nick Johnson
Numerade Educator
00:32

Problem 29

Use the preceding problem to show that if the closed orientable surface $M_{g}$ of genus $g$ retracts onto a graph $X \subset M_{g},$ then $H_{1}(X)$ has rank at most $g .$ Deduce an alternative proof of Exercise 13 from this, and construct a retraction of $M_{g}$ onto a wedge sum of $k$ circles for each $k \leq g$.

Ashley High
Ashley High
Numerade Educator
07:10

Problem 30

Show that the boundary of an $R$ -orientable manifold is also $R$ -orientable.

Bob Monday
Bob Monday
Numerade Educator
03:18

Problem 31

Show that if $M$ is a compact $R$ -orientable $n$ -manifold, then the boundary map $H_{n}(M, \partial M ; R) \rightarrow H_{n-1}(\partial M ; R)$ sends a fundamental class for $(M, \partial M)$ to a fundamental class for $\partial M.$

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
01:13

Problem 32

Show that a compact manifold does not retract onto its boundary.

James Chok
James Chok
Numerade Educator
07:44

Problem 33

Show that if $M$ is a compact contractible $n$ -manifold then $\partial M$ is a homology $(n-1)-$ sphere, that is, $H_{i}(\partial M ; \mathbb{Z}) \approx H_{i}\left(S^{n-1} ; \mathbb{Z}\right)$ for all $i.$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:01

Problem 34

For a compact manifold $M$ verify that the following diagram relating Poincaré duality for $M$ and $\partial M$ is commutative, up to sign at least:
$H^{k-1}(\partial M ; R) \longrightarrow H^{k}(M, \partial M ; R) \longrightarrow H^{k}(M ; R) \longrightarrow H^{k}(\partial M ; R)$
$$
|0 M| \sim \quad|| M |
$$
$|\operatorname{lom}| \sim$
$H_{n-k}(\partial M ; R) \longrightarrow H_{n-k}(M ; R) \longrightarrow H_{n-k}(M, \partial M ; R) \longrightarrow H_{n-k-1}(\partial M ; R)$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:07

Problem 35

If $M$ is a noncompact $R$ -orientable $n$ -manifold with boundary $\partial M$ having a collar neighborhood in $M,$ show that there are Poincaré duality isomorphisms $H_{c}^{k}(M ; R) \approx$ $H_{n-k}(M, \partial M ; R)$ for all $k,$ using the five-lemma and the following diagram:
(equation can't copy)

Narayan Hari
Narayan Hari
Numerade Educator