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Algebraic Topology

Allen Hatcher

Chapter 4

Homotopy Theory - all with Video Answers

Educators

Chapter Questions

02:29

Problem 1

Show there is a map $\mathbb{R P}^{\infty} \rightarrow \mathbb{C} \mathbf{P}^{\infty}=K(\mathbb{Z}, 2)$ which induces the trivial map on $\tilde{H}_{*}(-; \mathbb{Z})$ but a nontrivial map on $\tilde{H}^{*}(-; \mathbb{Z}) .$ How is this consistent with the universal coefficient theorem?

Gregory Higby
Gregory Higby
Numerade Educator
02:52

Problem 1

Suppose a sum $f+^{\prime} g$ of maps $f, g:\left(I^{n}, \partial I^{n}\right) \rightarrow\left(X, x_{0}\right)$ is defined using a coordinate of $I^{n}$ other than the first coordinate as in the usual sum $f+g .$ Verify the formula $(f+g)+^{\prime (h+k)=\left(f+^{\prime} h\right)+\left(g+^{\prime} k\right),$ and deduce that $f+^{\prime} k \simeq f+k$ so the two sums agree on $\pi_{n}\left(X, x_{0}\right),$ and also that $g+^{\prime} h \simeq h+g$ so the addition is abelian.

Wendi Zhao
Wendi Zhao
Numerade Educator
05:40

Problem 1

Show directly that if $X$ is a topological group with identity element $x_{0},$ then any two maps $f, g:\left(Z, z_{0}\right) \rightarrow\left(X, x_{0}\right)$ which are homotopic are homotopic through base point preserving maps.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:13

Problem 1

Use homotopy groups to show there is no retraction $$\mathbb{R} P^{n} \rightarrow \mathbb{R} P^{k} \text { if } n>k>0$$.

Gideon Idumah
Gideon Idumah
Numerade Educator
01:23

Problem 1

Show that the Hopf invariant of a composition $S^{2 n-1} \stackrel{f}{\rightarrow} S^{2 n-1} \stackrel{g}{\rightarrow} S^{n}$ is given by $H(g f)=(\operatorname{deg} f) H(g),$ and for a composition $S^{2 n-1} \stackrel{f}{\longrightarrow} S^{n} \stackrel{g}{\longrightarrow} S^{n}$ the Hopf invariant
satisfies $H(g f)=(\operatorname{deg} g)^{2} H(f)$.

Raj Bala
Raj Bala
Numerade Educator
05:40

Problem 1

Show that for a sequence of maps $X_{0} \stackrel{f_{1}}{\longrightarrow} X_{1} \stackrel{f_{2}}{\longrightarrow} \cdots,$ the infinite iterated mapping cylinder $M\left(f_{1}, f_{2}, \cdots\right),$ which is the union of the finite iterated mapping cylinders $M\left(f_{1}, \cdots, f_{n}\right),$ deformation retracts onto the mapping telescope.

Anthony Ramos
Anthony Ramos
Numerade Educator
08:50

Problem 1

By Exercise 35 in 4.2 there is a bundle $S^{2} \rightarrow \mathrm{CP}^{3} \rightarrow S^{4} .$ Let $S^{2} \rightarrow E_{k} \rightarrow S^{4}$ be the pullback of this bundle via a degree $k$ map $S^{4} \rightarrow S^{4}, k>1$. Use the Leray-Hirsch theorem to show that $H^{*}\left(E_{k} ; \mathbb{Z}\right)$ is additively isomorphic to $H^{*}\left(\mathrm{CP}^{3} ; \mathbb{Z}\right)$ but has a different cup product structure in which the square of a generator of $H^{2}\left(E_{k} ; \mathbb{Z}\right)$ is $k$ times a generator of $H^{4}\left(E_{k} ; \mathbb{Z}\right)$.

Ely Crowder
Ely Crowder
Numerade Educator
02:27

Problem 1

Assuming the first two axioms for a homology theory on the CW category, show that the direct limit axiom implies the wedge sum axiom. Show that the converse also holds for countable CW complexes.

Angelo Rendina
Angelo Rendina
Numerade Educator
02:16

Problem 1

Show that if $A \hookrightarrow X$ is a cofibration of compact Hausdorff spaces, then for any space
$Y,$ the map $Y^{X} \rightarrow Y^{A}$ obtained by restriction of functions is a fibration. Iff $A \hookrightarrow X$ is a cofibration, so is $A \times Y \hookrightarrow X \times Y$ for any space $Y .$ ]

Uma Kumari
Uma Kumari
Numerade Educator
04:43

Problem 1

Show that Corollary $4 \mathrm{K} .2$ remains valid when $X$ and $Y$ are $\mathrm{CW}$ complexes and the subspaces $U_{i}$ and $V_{i}$ are subcomplexes rather than open sets.

VU
Viswesh Uppalapati
Numerade Educator
07:44

Problem 1

If a connected CW complex $X$ retracts onto a subcomplex $A,$ show that $\Sigma X \approx$ $\Sigma A \vee \Sigma(X / A) .$ [One approach: Show the map $\Sigma r+\Sigma q: \Sigma X \rightarrow \Sigma A \vee \Sigma(X / A)$ induces an isomorphism on homology, where $r: X \rightarrow A$ is the retraction and $q: X \rightarrow X / A$ is the quotient map.]

Anthony Ramos
Anthony Ramos
Numerade Educator
01:19

Problem 1

Determine all cohomology operations $H^{1}(X ; \mathbb{Z}) \rightarrow H^{n}(X ; \mathbb{Z}), H^{2}(X ; \mathbb{Z}) \rightarrow H^{n}(X ; \mathbb{Z})$ and $H^{1}\left(X ; \mathbb{Z}_{p}\right) \rightarrow H^{n}\left(X ; \mathbb{Z}_{p}\right)$ for $p$ prime.

Gregory Higby
Gregory Higby
Numerade Educator
01:50

Problem 1

Show that $\Omega \Sigma X$ for a nonconnected CW complex $X$ reduces to the connected case by showing that each path-component of $\Omega \Sigma X$ is homotopy equivalent to $\Omega \Sigma\left(V_{\alpha} X_{\alpha}\right)$ where the $X_{\alpha}$ 's are the components of $X .$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:40

Problem 2

Show that the group structure on $S^{1}$ coming from multiplication in $\mathbb{C}$ induces a group structure on $\left\langle X, S^{1}\right\rangle$ such that the bijection $\left\langle X, S^{1}\right\rangle \rightarrow H^{1}(X ; \mathbb{Z})$ of Theorem 4.57
is an isomorphism.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:14

Problem 2

Show that if $s^{k} \rightarrow S^{m} \stackrel{p}{\rightarrow} S^{n}$ is a fiber bundle, then $m=2 n-1, k=n-1,$ and, when $n>1, H(p)=\pm 1 .$ Ishow that $C_{p}$ is a manifold and apply Poincaré duality.

Nick Johnson
Nick Johnson
Numerade Educator
04:19

Problem 2

Show that under the map $\langle X, Y\rangle \rightarrow \operatorname{Hom}\left(\pi_{n}\left(X, x_{0}\right), \pi_{n}\left(Y, y_{0}\right)\right),[f] \mapsto f_{*},$ the action of $\pi_{1}\left(Y, y_{0}\right)$ on $\langle X, Y\rangle$ corresponds to composing with the action on $\pi_{n}\left(Y, y_{0}\right)$ that is, $(y f)_{*}=\beta_{y} f_{*} .$ Deduce a bijection of $[X, K(\pi, 1)]$ with the set of orbits of Hom $\left(\pi_{1}(X), \pi\right)$ under composition with inner automorphisms of $\pi .$ In particular, if $\pi$ is abelian then $[X, K(\pi, 1)]=\langle X, K(\pi, 1)\rangle=\operatorname{Hom}\left(\pi_{1}(X), \pi\right)$.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:00

Problem 2

Show the action of $\pi_{1}\left(\mathbb{R}P^{n}\right)$ on $\pi_{n}\left(\mathbb{R}P^{n}\right) \approx \mathbb{Z}$ is trivial for $n$ odd and nontrivial for $n$ even.

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 2

Show that if $X$ is a complex of spaces in which all the maps are homeomorphisms, then the projection $\Delta X \rightarrow \Gamma$ is a fiber bundle.

Check back soon!
01:56

Problem 2

Apply the Leray-Hirsch theorem to the bundle $S^{1} \rightarrow S^{\infty} / \mathbb{Z}_{p} \rightarrow \mathbb{C} \mathrm{P}^{\infty}$ to compute $H^{*}\left(K\left(\mathbb{Z}_{p}, 1\right) ; \mathbb{Z}_{p}\right)$ from $H^{*}\left(\mathbb{C} \mathrm{P}^{\infty} ; \mathbb{Z}_{p}\right)$.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
07:44

Problem 2

For CW complexes $X$ and $Y$ consider the suspension sequence
$$\langle X, Y\rangle \stackrel{\Sigma}{\longrightarrow}\langle\Sigma X, \Sigma Y\rangle \stackrel{\Sigma}{\longrightarrow}\left\langle\Sigma^{2} X, \Sigma^{2} Y\right\rangle \longrightarrow \cdots $$
Show that if $X$ is a finite complex, these maps eventually become isomorphisms. [Use induction on the number of cells of $X$ and the five-lemma.

Anthony Ramos
Anthony Ramos
Numerade Educator
07:44

Problem 2

Show that a simplicial map $f: K \rightarrow L$ is a homotopy equivalence if $f^{-1}(x)$ is contractible for all $x \in L .$ [Consider the cover of $L$ by open stars of simplices and the cover of $K$ by the preimages of these open stars.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:42

Problem 2

Consider a pushout diagram as at the right, where $B \sqcup_{f} X$ is $B$ with $X$ attached along $A$ via $f .$ Show that if $A \hookrightarrow X$ is a cofibration, so is $B \hookrightarrow B \sqcup_{f} X$.

Linh Vu
Linh Vu
Numerade Educator
02:22

Problem 2

Using the Künneth formula, show that $K\left(\mathbb{Z}_{m} \times \mathbb{Z}_{n}, 1\right) \simeq K\left(\mathbb{Z}_{m}, 1\right) \vee K\left(\mathbb{Z}_{n}, 1\right)$ if $m$ and $n$ are relatively prime. Thus to determine stable splittings of $K\left(\mathbb{Z}_{n}, 1\right)$ it suffices to do the case that $n$ is a prime power, as in Proposition 4E3.

James Chok
James Chok
Numerade Educator
04:58

Problem 2

Use cohomology operations to show that the spaces $\left(S^{1} \times \mathrm{CP}^{\infty}\right) /\left(S^{1} \times\left\{x_{0}\right\}\right)$ and $s^{3} \times \mathrm{CP}^{\infty}$ are not homotopy equivalent.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:42

Problem 2

Show that if $\varphi: X \rightarrow Y$ is a homotopy equivalence, then the induced homomorphisms $\varphi_{*}: \pi_{n}\left(X, x_{0}\right) \rightarrow \pi_{n}\left(Y, \varphi\left(x_{0}\right)\right)$ are isomorphisms for all $n .$ [The case $n=1$
is Proposition 1.18.]

Mengchun Cai
Mengchun Cai
Numerade Educator
03:53

Problem 3

Suppose that a CW complex $X$ contains a subcomplex $S^{1}$ such that the inclusion $S^{1} \hookrightarrow X$ induces an injection $H_{1}\left(S^{1} ; \mathbb{Z}\right) \rightarrow H_{1}(X ; \mathbb{Z})$ with image a direct summand of $H_{1}(X ; \mathbb{Z}) .$ Show that $S^{1}$ is a retract of $X$.

Andrija Isakov
Andrija Isakov
Numerade Educator
02:14

Problem 3

Let $X$ be obtained from a lens space of dimension $2 n+1$ by deleting a point. Compute $\pi_{2 n}(X)$ as a module over $\mathbb{Z}\left[\boldsymbol{\pi}_{1}(X)\right]$

Keshav Singh
Keshav Singh
Numerade Educator
01:40

Problem 3

For a space $X$ let $\operatorname{Aut}(X)$ denote the group of homotopy classes of homotopy equivalences $X \rightarrow X .$ Show that for a CW complex $K(\pi, 1), \operatorname{Aut}(K(\pi, 1))$ is isomorphic to the group of outer automorphisms of $\pi,$ that is, automorphisms modulo inner automorphisms.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:36

Problem 3

What is the nerve of the cover of a simplicial complex by the open stars of its vertices?

Teresa Fuston
Teresa Fuston
Numerade Educator
01:42

Problem 3

Show that for any sequence $z_{1} \rightarrow Z_{2} \rightarrow \cdots,$ the natural map $\lim _{\Omega} \Omega Z_{n} \rightarrow \Omega \lim _{z \rightarrow a} z_{n}$ is
a weak homotopy equivalence, where the direct limits mean mapping telescopes.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:22

Problem 3

Show that $S P_{n}(I)=\Delta^{n}$

Vysakh M
Vysakh M
Numerade Educator
03:57

Problem 3

For fibrations $E_{1} \rightarrow B$ and $E_{2} \rightarrow B$, show that a fiber-preserving map $E_{1} \rightarrow E_{2}$ that is a homotopy equivalence is in fact a fiber homotopy equivalence. IThis is dual to Proposition 0.19.1.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
View

Problem 3

Use the Leray-Hirsch theorem as in Corollary 4D.3 to compute $H^{*}\left(V_{n}\left(\mathbb{C}^{k}\right) ; \mathbb{Z}\right) \approx$ $\Lambda_{z}\left[x_{2 k-2 n+1}, x_{2 k-2 n+3}, \cdots, x_{2 k-1}\right]$ and similarly in the quaternionic case.

Victor Salazar
Victor Salazar
Numerade Educator
04:01

Problem 3

Since there is a fiber bundle $S^{2} \rightarrow \mathrm{CP}^{5} \rightarrow \mathbb{U} \mathrm{P}^{2}$ by Exercise 35 in $\mathrm{S} 4.2,$ one might ask whether there is an analogous bundle $S^{4} \rightarrow \mathbb{D} 0 \mathrm{P}^{5} \rightarrow \mathbb{O P}^{2}$. Use Steenrod powers for the prime 3 to show that such a bundle cannot exist. [The Gysin sequence can be used to determine the map on cohomology induced by the bundle projection $\mathbb{H P}^{5} \rightarrow \mathbb{O P}^{2} .$.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:13

Problem 3

Extending Proposition 4I.3, show that the $(2 k+1)$ -skeleton of the suspension of a lens space with fundamental group of order $p^{n}$ is homotopy equivalent to the wedge sum of the $(2 k+1)$ -skeleta of the spaces $X_{i},$ if these $X_{i}$ 's are chosen to have the minimum number of cells in each dimension, as described in the remarks following the proof.

Gideon Idumah
Gideon Idumah
Numerade Educator
01:47

Problem 3

For an H-space $\left(X, x_{0}\right)$ with multiplication $\mu: X \times X \rightarrow X,$ show that the group operation in $\pi_{n}\left(X, x_{0}\right)$ can also be defined by the rule $(f+g)(x)=\mu(f(x), g(x))$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:07

Problem 4

Given abelian groups $G$ and $H$ and CW complexes $K(G, n)$ and $K(H, n),$ show that the map $\langle K(G, \boldsymbol{n}), K(H, \boldsymbol{n})\rangle \rightarrow \mathrm{Hom}(G, H)$ sending a homotopy class $[f]$ to the induced homomorphism $f_{*}: \pi_{n}(K(G, n)) \rightarrow \pi_{n}(K(H, n))$ is a bijection.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:25

Problem 4

With the notation of the preceding problem, show that Aut(V $s$ s $^{k}$ ) $\approx \mathrm{GL}_{n}$ ( $\mathbb{Z}$ ) for $k>1,$ where $V_{n} S^{k}$ denotes the wedge sum of $n$ copies of $s^{k}$ and $G L_{n}(\mathbb{Z})$ is the group of $n \times n$ matrices with entries in $Z$ having an inverse matrix of the same form. IGL $\left._{n}(\mathbb{Z}) \text { is the automorphism group of } \mathbb{Z}^{n} \approx \boldsymbol{\pi}_{k}\left(\mathrm{V}_{n} S^{k}\right) \approx H_{k}\left(\mathrm{V}_{n} S^{k}\right) .\right]$

Ely Crowder
Ely Crowder
Numerade Educator
03:13

Problem 4

Let $X \subset \mathbb{R}^{n+1}$ be the union of the infinite sequence of spheres $S_{k}^{n}$ of radius $1 / k$ and center $(1 / k, 0, \cdots, 0) .$ Show that $\pi_{i}(X)=0$ for $i<n$ and construct a homomorphism from $\pi_{n}(X)$ onto $\Pi_{k} \pi_{n}\left(S_{k}^{n}\right).$

Gideon Idumah
Gideon Idumah
Numerade Educator
05:34

Problem 4

Show that Proposition $4 \mathrm{G} .2$ and its corollary hold also for $\mathrm{CW}$ complexes and covers by families of sub complexes.

Chris Trentman
Chris Trentman
Numerade Educator
01:45

Problem 4

Show that $S P_{2}\left(S^{1}\right)$ is a Möbius band, and that this is consistent with the description
of $S P_{2}\left(S^{n}\right)$ as a mapping cone given in Example $4 \mathrm{K} .5$

Nilay Maity
Nilay Maity
Numerade Educator
01:48

Problem 4

Define the dual of an iterated mapping cylinder precisely, in terms of maps from $\Delta^{n},$ and use this to give a definition of $\nabla X,$ the dual of $\Delta X,$ for $X$ a complex of spaces.

Narayan Hari
Narayan Hari
Numerade Educator
06:05

Problem 4

For the flag space $F_{n}\left(\mathbb{C}^{n}\right)$ show that $H^{*}\left(F_{n}\left(\mathbb{C}^{n}\right) ; \mathbb{Z}\right) \approx \mathbb{Z}\left[x_{1}, \cdots, x_{n}\right] /\left(\sigma_{1}, \cdots, \sigma_{n}\right)$
where $\sigma_{i}$ is the $i^{t h}$ elementary symmetric polynomial.

Melvin Adkins
Melvin Adkins
Numerade Educator
11:08

Problem 4

Show there is no fiber bundle $S^{7} \rightarrow S^{23} \rightarrow \mathbb{O P}^{2} .$ ICompute the cohomology ring of the mapping cone of the projection $S^{23} \rightarrow$ Op $^{2}$ via Poincaré duality or the Thom isomorphism.]

MR
Melanie Richey
Numerade Educator
01:05

Problem 4

Let $p: \tilde{X} \rightarrow X$ be the universal cover of a path-connected space $X .$ Show that under the isomorphism $\pi_{n}(X) \approx \pi_{n}(\tilde{X}),$ which holds for $n \geq 2,$ the action of $\pi_{1}(X)$ on $\pi_{n}(X)$ corresponds to the action of $\pi_{1}(X)$ on $\pi_{n}(\tilde{X})$ induced by the action of $\pi_{1}(X)$ on $\tilde{X}$ as deck transformations. More precisely, prove a formula like $\gamma p_{*}(\alpha)=p_{*}\left(\beta_{\tilde{\gamma}}\left(\gamma_{*}(\alpha)\right)\right)$ where $\gamma \in \pi_{1}\left(X, x_{0}\right), \alpha \in \pi_{n}\left(\tilde{X}, \tilde{x}_{0}\right),$ and $\gamma_{*}$ denotes the homomorphism induced by the action of $\gamma$ on $\tilde{X}$

Anthony Ramos
Anthony Ramos
Numerade Educator
View

Problem 5

Show that $\left[X, S^{n}\right] \approx H^{n}(X ; \mathbb{Z})$ if $X$ is an $n$ -dimensional CW complex. [Build a $K(\mathbb{Z}, \boldsymbol{n})$ from $S^{n}$ by attaching cells of dimension $\geq n+2.1]$

Victor Salazar
Victor Salazar
Numerade Educator
01:40

Problem 5

This problem involves the spaces constructed in the latter part of this section.
(a) Compute the homology groups of the complex $Z$ in the case $n=3,$ when $Z$ is 2-dimensional.
(b) Letting $\tilde{X}_{n}$ denote the $n$ -dimensional complex $\tilde{X},$ show that $\tilde{X}_{n}$ can be obtained inductively from $\tilde{X}_{n-1}$ as the union of two copies of the mapping torus of the generating deck transformation $\tilde{X}_{n-1} \rightarrow \tilde{X}_{n-1},$ with copies of $\tilde{X}_{n-1}$ in these two mapping tori identified. Thus there is a fiber bundle $\tilde{X}_{n} \rightarrow S^{1} \vee S^{1}$ with fiber $\tilde{X}_{n-1}$
(c) Use part (b) to find a presentation for $\pi_{1}\left(\tilde{X}_{n}\right),$ and show this presentation reduces to a finite presentation if $n>2$ and a presentation with a finite number of generators if $n=2 .$ In the latter case, deduce that $\pi_{1}\left(\tilde{X}_{2}\right)$ has no finite presentation from the fact that $H_{2}\left(\tilde{X}_{2}\right)$ is not finitely generated.

R M
R M
Numerade Educator
04:19

Problem 5

Let $f: S_{\alpha}^{2} \vee S_{\beta}^{2} \rightarrow S_{\alpha}^{2} \vee S_{\beta}^{2}$ be the map which is the identity on the $S_{\alpha}^{2}$ summand and which on the $S_{\beta}^{2}$ summand is the sum of the identity map and a homeomorphism $S_{\beta}^{2} \rightarrow S_{\alpha}^{2} .$ Let $X$ be the mapping torus of $f,$ the quotient space of $\left(S_{\alpha}^{2} \vee S_{\beta}^{2}\right) \times I$ under the identifications $(x, 0) \sim(f(x), 1) .$ The mapping torus of the restriction of $f$ to $S_{\alpha}^{2}$ forms a subspace $A=S^{1} \times S_{\alpha}^{2} \subset X .$ Show that the maps $\pi_{2}(A) \rightarrow \pi_{2}(X) \rightarrow \pi_{2}(X, A)$ form a short exact sequence $0 \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow 0,$ and compute the action of $\pi_{1}(A)$ on these three groups. In particular, show the action of $\pi_{1}(A)$ is trivial on $\pi_{2}(A)$ and $\pi_{2}(X, A)$ but is nontrivial on $\pi_{2}(X).$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
03:45

Problem 5

A map $p: E \rightarrow B$ with $B$ not necessarily path-connected is defined to be a quasifibration if the following equivalent conditions are satisfied:
(i) For all $b \in B$ and $x_{0} \in p^{-1}(b),$ the map $p_{*}: \pi_{i}\left(E, p^{-1}(b), x_{0}\right) \rightarrow \pi_{i}(B, b)$ is an
isomorphism for $i>0$ and $\pi_{0}\left(p^{-1}(b), x_{0}\right) \rightarrow \pi_{0}\left(E, x_{0}\right) \rightarrow \pi_{0}(B, b)$ is exact.
(ii) The inclusion of the fiber $p^{-1}(b)$ into the homotopy fiber $F_{b}$ of $p$ over $b$ is a weak homotopy equivalence for all $b \in B$.
(iii) The restriction of $p$ over each path-component of $B$ is a quasifibration according
to the definition in this section.
Show these three conditions are equivalent, and prove Lemma $4 \mathrm{K} .3$ for quasifibrations over non-pathconnected base spaces.

Narayan Hari
Narayan Hari
Numerade Educator
01:59

Problem 5

Use the Gysin sequence to show that for a fiber bundle $S^{k} \rightarrow S^{m} \stackrel{p}{\longrightarrow} S^{n}$ we must have $k=n-1$ and $m=2 n-1 .$ Then use the Thom isomorphism to show that the Hopf invariant of $p$ must be $\pm 1 .$ IHence $n=1,2,4,8$ by Adams' theorem.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:05

Problem 5

For a pair $(X, A)$ of path-connected spaces, show that $\pi_{1}\left(X, A, x_{0}\right)$ can be identified in a natural way with the set of cosets $\alpha H$ of the subgroup $H \subset \pi_{1}\left(X, x_{0}\right)$ represented by loops in $A$ at $x_{0}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 6

For an abelian group $G,$ Theorem 4.57 gives a map $\mu: K(G, n) \times K(G, n) \rightarrow K(G, n)$ with $\mu^{*}(\alpha)=\alpha \times 1+1 \times \alpha$ where $\alpha$ is a fundamental class for $K(G, n) .$ Show that $\mu$ defines an H-space structure on $K(G, n)$ that is commutative and associative, up to homotopy. Show also that the H-space multiplication $\mu$ is unique up to homotopy.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
22:57

Problem 6

Show that the relative form of the Hurewicz theorem in dimension $n$ implies the absolute form in dimension $n-1$ by considering the pair $(C X, X)$ where $C X$ is the cone on $X.$

Anthony Ramos
Anthony Ramos
Numerade Educator
08:33

Problem 6

Let $X$ be a complex of spaces over a $\Delta$ -complex $\Gamma,$ as defined in $\$ 4 .$ G. Show that the natural projection $\Delta X \rightarrow \Gamma$ is a quasifibration if all the maps in $X$ associated to edges of $\Gamma$ are weak homotopy equivalences.

Carrie Frizzell
Carrie Frizzell
Numerade Educator
07:44

Problem 6

Show that if $M$ is a manifold of dimension $2 n$ for which there exists a fiber bundle $S^{1} \rightarrow S^{2 n+1} \rightarrow M,$ then $M$ is simply-connected and $H^{*}(M ; \mathbb{Z}) \approx H^{*}\left(\mathbb{C P}^{n} ; \mathbb{Z}\right)$ as rings. Conversely, if $M$ is simply-connected and $H^{*}(M ; \mathbb{Z}) \approx H^{*}\left(\mathbb{C P}^{n} ; \mathbb{Z}\right)$ as rings, show there is a bundle $S^{1} \rightarrow E \rightarrow M$ where $E \simeq S^{2 n+1} .$ IWhen $n>1$ there are examples where $M$ is not homeomorphic to $\mathrm{CP}^{n} . \mathrm{J}$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:01

Problem 6

If $p:\left(\tilde{X}, \tilde{A}, \tilde{x}_{0}\right) \rightarrow\left(X, A, x_{0}\right)$ is a covering space with $\tilde{A}=p^{-1}(A),$ show that the $\operatorname{map} p_{*}: \pi_{n}\left(\tilde{X}, \tilde{A}, \tilde{x}_{0}\right) \rightarrow \pi_{n}\left(X, A, x_{0}\right)$ is an isomorphism for all $n>1$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:05

Problem 7

Using an H-space multiplication $\mu$ on $K(G, n),$ define an addition in $\langle X, K(G, n)\rangle$ by $[f]+[g]=[\mu(f, g)]$ and show that under the bijection $H^{n}(X ; G) \approx\langle X, K(G, n)\rangle$ this addition corresponds to the usual addition in cohomology.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:59

Problem 7

Construct a CW complex $X$ with prescribed homotopy groups $\pi_{i}(X)$ and prescribed actions of $\pi_{1}(X)$ on the $\boldsymbol{\pi}_{i}(X)$ 's.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:46

Problem 7

Show that if a disk bundle $D^{n} \rightarrow E \rightarrow B$ has a Thom class with $\mathbb{Z}$ coefficients, then it is orientable.

Bryan Lynn
Bryan Lynn
Numerade Educator
01:12

Problem 7

Extend the results proved near the beginning of this section for the change-ofbasepoint maps $\beta_{\gamma}$ to the case of relative homotopy groups.

Ameer Said
Ameer Said
Numerade Educator
05:40

Problem 8

Show that a map $p: E \rightarrow B$ is a fibration iff the map $\pi: E^{I} \rightarrow E_{p}, \pi(y)=(y(0), p \gamma)$ has a section, that is, a map $s: E_{p} \rightarrow E^{I}$ such that $p s=\mathbb{1} .$

Anthony Ramos
Anthony Ramos
Numerade Educator
View

Problem 8

Show the suspension of an acyclic CW complex is contractible.

Victor Salazar
Victor Salazar
Numerade Educator
06:39

Problem 8

8. If $E$ is the product bundle $B \times D^{n}$ with $B$ a CW complex, show that the Thom space $T(E)$ is homotopy equivalent to the $n$ -fold reduced suspension $\Sigma^{n} B,$ and that the Thom isomorphism specializes to the suspension isomorphism $H^{i}(B ; R) \approx$ $\tilde{H}^{n+i}\left(\Sigma^{n} B ; R\right)$ given by the reduced cross product in 3.2.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:31

Problem 8

Show the sequence $\pi_{1}\left(X, x_{0}\right) \rightarrow \pi_{1}\left(X, A, x_{0}\right) \stackrel{\partial}{\longrightarrow} \pi_{0}\left(A, x_{0}\right) \rightarrow \pi_{0}\left(X, x_{0}\right)$ is exact.

Umar Sohail Qureshi
Umar Sohail Qureshi
Numerade Educator
01:01

Problem 9

Show that a linear projection of a 2 simplex onto one of its edges is a fibration but not a fiber bundle. [Use the preceding problem.]

Raj Bala
Raj Bala
Numerade Educator
16:59

Problem 9

Show that a map between simply-connected CW complexes is a homotopy equivalence if its mapping cone is contractible. Use the preceding exercise to give an example where this fails in the nonsimply-connected case.

Chris Trentman
Chris Trentman
Numerade Educator
08:25

Problem 9

Show that the inclusion $T^{n} \hookrightarrow U(n)$ of the $n$ -torus of diagonal matrices is homotopic to the map $T^{n} \rightarrow U(1) \hookrightarrow U(n)$ sending an $n$ -tuple of unit complex numbers $\left(z_{1}, \cdots, z_{n}\right)$ to the $1 \times 1$ matrix $\left(z_{1} \cdots z_{n}\right) .$ Do the same for the diagonal subgroup of $S p(n) .$ [Hint: Diagonal matrices in $U(n)$ are compositions of scalar multiplication in
$n$ lines in $\mathbb{C}^{n},$ and $\mathbb{C} \mathrm{P}^{n-1}$ is connected.]

Ely Crowder
Ely Crowder
Numerade Educator
04:19

Problem 9

Suppose we define $\pi_{0}\left(X, A, x_{0}\right)$ to be the quotient set $\pi_{0}\left(X, x_{0}\right) / \pi_{0}\left(A, x_{0}\right),$ so that the long exact sequence of homotopy groups for the pair $(X, A)$ extends to $\cdots \rightarrow \pi_{0}\left(X, x_{0}\right) \rightarrow \pi_{0}\left(X, A, x_{0}\right) \rightarrow 0$
(a) Show that with this extension, the five-lemma holds for the map of long exact sequences induced by a map $\left(X, A, x_{0}\right) \rightarrow\left(Y, B, y_{0}\right),$ in the following form: One of the maps between the two sequences is a bijection if the four surrounding maps are bijections for all choices of $x_{0}$
(b) Show that the long exact sequence of a triple $\left(X, A, B, x_{0}\right)$ can be extended only to the term $\boldsymbol{\pi}_{0}\left(A, B, x_{0}\right)$ in general, and that the five-lemma holds for this extension.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:19

Problem 10

Given a fibration $F \rightarrow E \rightarrow B$, use the homotopy lifting property to define an action of $\pi_{1}(E)$ on $\pi_{n}(F),$ a homomorphism $\pi_{1}(E) \rightarrow \operatorname{Aut}\left(\pi_{n}(F)\right),$ such that the composition $\pi_{1}(F) \rightarrow \pi_{1}(E) \rightarrow \operatorname{Aut}\left(\pi_{n}(F)\right)$ is the usual action of $\pi_{1}(F)$ on $\pi_{n}(F) .$ Deduce that if $\boldsymbol{\pi}_{1}(E)=0,$ then the action of $\pi_{1}(F)$ on $\pi_{n}(F)$ is trivial.

Gregory Higby
Gregory Higby
Numerade Educator
02:06

Problem 10

Let the CW complex $X$ be obtained from $S^{1} \vee S^{n}, n \geq 2,$ by attaching a cell $e^{n+1}$ by a map representing the polynomial $p(t) \in \mathbb{Z}\left[t, t^{-1}\right] \approx \pi_{n}\left(S^{1} \vee S^{n}\right),$ so $\pi_{n}(X) \approx \mathbb{Z}\left[t, t^{-1}\right] /(p(t)) .$ Show $\pi_{n}^{\prime}(X)$ is cyclic and compute its order in terms of $p(t) .$ Give examples showing that the group $\pi_{n}(X)$ can be finitely generated or
not, independently of whether $\pi_{n}^{\prime}(X)$ is finite or infinite.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
11:46

Problem 10

Fill in the details of the following argument to show that every $n \times n$ matrix $A$ with entries in $\mathbb{H}$ has an eigenvalue in $\mathbb{H}$. (The usual argument over $\mathbb{C}$ involving roots of the characteristic polynomial does not work due to the lack of a good quaternionic determinant function.) For $t \in[0,1]$ and $\lambda \in S^{3} \subset \mathbb{H},$ consider the matrix $t \lambda I+(1-t) A .$ If $A$ has no eigenvalues, this is invertible for all $t .$ Thus the $\operatorname{map} S^{3} \rightarrow G L_{n}(\mathbb{H}), \lambda \mapsto \lambda I,$ is nullhomotopic. But by the preceding problem and
Exercise $10(\mathrm{b})$ in $\mathrm{S} 3 . \mathrm{C},$ this map represents $n$ times a generator of $\pi_{3} G L_{n}(\mathbb{H})$.

Chris Trentman
Chris Trentman
Numerade Educator
02:33

Problem 10

Show the 'quasi-circle' described in Exercise 7 in $\$ 1.3$ has trivial homotopy groups but is not contractible, hence does not have the homotopy type of a CW complex.

Foster Wisusik
Foster Wisusik
Numerade Educator
14:32

Problem 11

For a space $B,$ let $\mathcal{F}(B)$ be the set of fiber homotopy equivalence classes of fibrations $E \rightarrow B .$ Show that a map $f: B_{1} \rightarrow B_{2}$ induces $f^{*}: \mathcal{F}\left(B_{2}\right) \rightarrow \mathcal{F}\left(B_{1}\right)$ depending only on the homotopy class of $f,$ with $f^{*}$ a bijection if $f$ is a homotopy equivalence.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:52

Problem 11

Let $X$ be a connected CW complex with 1-skeleton $X^{1} .$ Show that $\pi_{2}\left(X, X^{1}\right) \approx$ $\pi_{2}(X) \times K$ where $K$ is the kernel of $\pi_{1}\left(X^{1}\right) \rightarrow \pi_{1}(X),$ a free group. Show also that $\pi_{2}^{\prime}\left(X, X^{1}\right) \approx \boldsymbol{\pi}_{2}^{\prime}(X) \times K^{\prime}$ where $K^{\prime}$ is the quotient of $K$ obtained by factoring out the conjugation action of $\pi_{1}\left(X^{1}\right) .$ Note that $K^{\prime}$ is abelian.

Wendi Zhao
Wendi Zhao
Numerade Educator
View

Problem 11

Show that a CW complex is contractible if it is the union of an increasing sequence of subcomplexes $X_{1} \subset X_{2} \subset \cdots$ such that each inclusion $X_{i} \hookrightarrow X_{i+1}$ is nullhomotopic, a condition sometimes expressed by saying $X_{i}$ is contractible in $X_{i+1} .$ An example is $S^{\infty},$ or more generally the infinite suspension $S^{\infty} X$ of any CW complex $X,$ the union of the iterated suspensions $S^{n} X$.

Victor Salazar
Victor Salazar
Numerade Educator
04:57

Problem 12

Show that for homotopic maps $f, g: A \rightarrow B$ the fibrations $E_{f} \rightarrow B$ and $E_{g} \rightarrow B$ are fiber homotopy equivalent.

Chris Trentman
Chris Trentman
Numerade Educator
07:44

Problem 12

Show that a map $f: X \rightarrow Y$ of connected CW complexes is a homotopy equivalence if it induces an isomorphism on $\pi_{1}$ and if a lift $\tilde{f}: \tilde{X} \rightarrow \tilde{Y}$ to the universal covers induces an isomorphism on homology. IThe latter condition can be restated in terms of homology with local coefficients as saying that $f_{*}: H_{*}\left(X ; \mathbb{Z}\left[\pi_{1} X\right]\right) \rightarrow H_{*}\left(Y ; \mathbb{Z}\left[\pi_{1} Y\right]\right)$
is an isomorphism; see §3.H.]

Anthony Ramos
Anthony Ramos
Numerade Educator
View

Problem 12

Show that an $n$ -connected, $n$ -dimensional CW complex is contractible.

Victor Salazar
Victor Salazar
Numerade Educator
14:32

Problem 13

Given map $f: A \rightarrow B$ and a homotopy equivalence $g: C \rightarrow A,$ show that the fibrations $E_{f} \rightarrow B$ and $E_{f g} \rightarrow B$ are fiber homotopy equivalent. [One approach is to use Corollary 0.21 to reduce to the case of deformation retractions.]

Anthony Ramos
Anthony Ramos
Numerade Educator
01:05

Problem 13

Show that a map between connected $n$ -dimensional CW complexes is a homotopy equivalence if it induces an isomorphism on $\boldsymbol{\pi}_{i}$ for $i \leq n$. [Pass to universal covers and use homology.]

Anthony Ramos
Anthony Ramos
Numerade Educator
View

Problem 13

Use the extension lemma to show that a CW complex retracts onto any contractible subcomplex.

Victor Salazar
Victor Salazar
Numerade Educator
14:32

Problem 14

For a space $B,$ let $\mathcal{M}(B)$ denote the set of equivalence classes of maps $f: A \rightarrow B$ where $f_{1}: A_{1} \rightarrow B$ is equivalent to $f_{2}: A_{2} \rightarrow B$ if there exists a homotopy equivalence $g: A_{1} \rightarrow A_{2}$ such that $f_{1} \simeq f_{2} g .$ Show the natural map $\mathcal{F}(B) \rightarrow \mathcal{M}(B)$ is a bijection. [See Exercises 11 and 13.]

Anthony Ramos
Anthony Ramos
Numerade Educator
01:05

Problem 14

If an $n$ -dimensional CW complex $X$ contains a subcomplex $Y$ homotopy equivalent to $S^{n},$ show that the map $\pi_{n}(Y) \rightarrow \pi_{n}(X)$ induced by inclusion is injective. [Use the Hurewicz homomorphism.]

Anthony Ramos
Anthony Ramos
Numerade Educator
06:39

Problem 14

Use cellular approximation to show that the $n$ -skeletons of homotopy equivalent CW complexes without cells of dimension $n+1$ are also homotopy equivalent.

Anthony Ramos
Anthony Ramos
Numerade Educator
07:44

Problem 15

If the fibration $p: E \rightarrow B$ is a homotopy equivalence, show that $p$ is a fiber homotopy equivalence of $E$ with the trivial fibration $\mathbb{1}: B \rightarrow B$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:02

Problem 15

Show that a closed simply-connected 3-manifold is homotopy equivalent to $S^{3}$. IUse Poincaré duality, and also the fact that closed manifolds are homotopy equivalent to CW complexes, from Corollary A.12 in the Appendix. The stronger statement that a closed simply-connected 3-manifold is homeomorphic to $S^{3}$ is still unproved. This is the Poincaré conjecture, without doubt the most famous open problem in topology. The higher-dimensional analog, that a closed $n$ -manifold homotopy equivalent to $S^{n}$ is homeomorphic to $S^{n}$, has been proved for all $n \geq 4.1.$

Raj Bala
Raj Bala
Numerade Educator
04:33

Problem 15

Show that every map $f: S^{n} \rightarrow S^{n}$ is homotopic to a multiple of the identity map by the following steps.
(a) Use Lemma 4.10 (or simplicial approximation, Theorem $2 \mathrm{C} .1$ ) to reduce to the case that there exists a point $q \in S^{n}$ with $f^{-1}(q)=\left\{p_{1}, \cdots, p_{k}\right\}$ and $f$ is an invertible linear map near each $p_{i}$
(b) For $f$ as in (a), consider the composition $g f$ where $g: S^{n} \rightarrow S^{n}$ collapses the complement of a small ball about $q$ to the basepoint. Use this to reduce (a) further to the case $k=1$.
(c) Finish the argument by showing that an invertible $n \times n$ matrix can be joined by a path of such matrices to either the identity matrix or the matrix of a reflection. (Use Gaussian elimination, for example.)

Lucía Guerrero
Lucía Guerrero
Numerade Educator
07:44

Problem 16

Show that a map $f: X \rightarrow Y$ of connected CW complexes is a homotopy equivalence if it induces an isomorphism on $\pi_{1}$ and its homotopy fiber $F_{f}$ has $\tilde{H}_{*}\left(F_{f} ; \mathbb{Z}\right)=0$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:19

Problem 16

Show that the closed surfaces with infinite fundamental group are $K(\pi, 1)$ 's by showing that their universal covers are contractible, via the Hurewicz theorem and results of §3.3.

John Gehad
John Gehad
Numerade Educator
01:04

Problem 16

Show that a map $f: X \rightarrow Y$ between connected CW complexes factors as a composition $X \rightarrow Z_{n} \rightarrow Y$ where the first map induces isomorphisms on $\pi_{i}$ for $i \leq n$ and the second map induces isomorphisms on $\boldsymbol{\pi}_{i}$ for $i \geq n+1$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:07

Problem 17

Show that $\Omega X$ is an H-space with multiplication the composition of loops.

Carson Merrill
Carson Merrill
Numerade Educator
05:40

Problem 17

Show that the map $\langle X, Y\rangle \rightarrow \operatorname{Hom}\left(\pi_{n}(X), \pi_{n}(Y)\right),[f] \mapsto f_{*},$ is a bijection if $X$ is
an $(n-1)$ -connected CW complex and $Y$ is a path-connected space with $\pi_{i}(Y)=0$ for $i>n .$ Deduce that CW complex $K(G, n)$ 's are uniquely determined, up to homotopy type, by $G$ and $n.$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:35

Problem 17

Show that if $X$ and $Y$ are CW complexes with $X m$-connected and $Y n$ -connected, then $(X \times Y, X \vee Y)$ is $(m+n+1)$ -connected, as is the smash product $X \wedge Y$.

James Chok
James Chok
Numerade Educator
01:57

Problem 18

Show that a fibration sequence $\cdots \rightarrow \Omega B \rightarrow F \rightarrow E \rightarrow B$ induces a long exact sequence $\cdots \rightarrow\langle X, \Omega B\rangle \rightarrow\langle X, F\rangle \rightarrow\langle X, E\rangle \rightarrow\langle X, B\rangle,$ with groups and group homomorphisms except for the last three terms, abelian groups except for the last six terms.

Wendi Zhao
Wendi Zhao
Numerade Educator
05:34

Problem 18

If $X$ and $Y$ are simply-connected CW complexes such that $\tilde{H}_{i}(X)$ and $\tilde{H}_{j}(Y)$ are finite and of relatively prime orders for all pairs ( $i, j),$ show that the inclusion $X \vee Y \hookrightarrow X \times Y$ is a homotopy equivalence and $X \wedge Y$ is contractible. IUse the Künneth formula.]

Chris Trentman
Chris Trentman
Numerade Educator
01:01

Problem 18

Give an example of a weak homotopy equivalence $X \rightarrow Y$ for which there does not exist a weak homotopy equivalence $Y \rightarrow X$.

Robert Daugherty
Robert Daugherty
Numerade Educator
01:19

Problem 19

Given a fibration $F \rightarrow E \stackrel{p}{\longrightarrow} B,$ define a natural action of $\Omega B$ on the homotopy fiber $F_{p}$ and use this to show that exactness at $\langle X, F\rangle$ in the long exact sequence in the preceding problem can be improved to the statement that two elements of $\langle X, F\rangle$ have the same image in $\langle X, E\rangle$ iff they are in the same orbit of the induced action of $\langle X, \Omega B\rangle$ on $\langle X, F\rangle$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
View

Problem 19

If $X$ is a $K(G, 1)$ CW complex, show that $\pi_{n}\left(X^{n}\right)$ is free abelian for $n \geq 2.$

Nick Johnson
Nick Johnson
Numerade Educator
02:44

Problem 19

Consider the equivalence relation $\simeq_{w}$ generated by weak homotopy equivalence:
$X \simeq_{w} Y$ if there are spaces $X=X_{1}, X_{2}, \cdots, X_{n}=Y$ with weak homotopy equivalences $X_{i} \rightarrow X_{i+1}$ or $X_{i} \leftarrow X_{i+1}$ for each $i .$ Show that $X \simeq_{w} Y$ iff $X$ and $Y$ have a common CW approximation.

Ibrahima Barry
Ibrahima Barry
Numerade Educator
10:08

Problem 20

Show that by applying the loopspace functor to a Postnikov tower for $X$ one obtains a Postnikov tower of principal fibrations for $\Omega X$.

Chris Trentman
Chris Trentman
Numerade Educator
00:46

Problem 20

Let $G$ be a group and $X$ a simply-connected space. Show that for the product $K(G, 1) \times X$ the action of $\pi_{1}$ on $\boldsymbol{\pi}_{n}$ is trivial for all $n>1.$

Nick Johnson
Nick Johnson
Numerade Educator
01:56

Problem 20

Show that $[X, Y]$ is finite if $X$ is a finite connected CW complex and $\pi_{i}(Y)$ is finite for $i \leq \operatorname{dim} X$.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
03:58

Problem 21

Show that in the Postnikov tower of an H-space, all the spaces are H-spaces and the maps are H-maps, commuting with the multiplication, up to homotopy.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:20

Problem 21

Given a sequence of CW complexes $K\left(G_{n}, n\right), n=1,2, \cdots,$ let $X_{n}$ be the CW complex formed by the product of the first $n$ of these $K\left(G_{n}, n\right)$ 's. Via the inclusions $X_{n-1} \subset X_{n}$ coming from regarding $X_{n-1}$ as the subcomplex of $X_{n}$ with $n^{t h}$ coordinate equal to a basepoint 0-cell of $K\left(G_{n}, n\right),$ we can then form the union of all the $X_{n}$ 's, a CW complex $X .$ Show $\pi_{n}(X) \approx G_{n}$ for all $n.$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:45

Problem 21

For this problem it is convenient to use the notations $X^{n}$ for the $n^{t h}$ stage in a Postnikov tower for $X$ and $X_{m}$ for an $(m-1)$ -connected covering of $X,$ where $X$ is a connected CW complex. Show that $\left(X^{n}\right)_{m} \simeq\left(X_{m}\right)^{n},$ so the notation $X_{m}^{n}$ is unambiguous. Thus $\boldsymbol{\pi}_{i}\left(X_{m}^{n}\right) \approx \boldsymbol{\pi}_{i}(X)$ for $m \leq i \leq n$ and all other homotopy groups of $X_{m}^{n}$ are zero.
$$X_{m} \longrightarrow X_{m}^{n}$$
$$\downarrow \quad\quad \downarrow$$
$$X \longrightarrow X^{n}$$

Narayan Hari
Narayan Hari
Numerade Educator
03:58

Problem 22

Show that a principal fibration $\Omega C \rightarrow E \stackrel{p}{\rightarrow} B$ is fiber homotopy equivalent to the product $\Omega C \times B$ iff it has a section, a map $s: B \rightarrow E$ with $p s=\mathbb{1} .$

Anthony Ramos
Anthony Ramos
Numerade Educator
03:27

Problem 22

Show that $H_{n+1}(K(G, n) ; \mathbb{Z})=0$ if $n>1 .$ [Build a $K(G, n)$ from a Moore space $M(G, n)$ by attaching cells of dimension $>n+1.1.$

Runpeng Li
Runpeng Li
Numerade Educator
03:26

Problem 22

Show that a path-connected space $X$ is homotopy equivalent to a CW complex with countably many cells iff $\pi_{n}(X)$ is countable for all $n .$ [Use the results on simplicial approximations to maps and spaces in $\S$ $2 . \mathrm{C.}]$

Anthony Ramos
Anthony Ramos
Numerade Educator
07:44

Problem 23

Prove the following uniqueness result for the Quillen plus construction: Given a connected CW complex $X,$ if there is an abelian CW complex $Y$ and a map $X \rightarrow Y$ inducing an isomorphism $H_{*}(X ; \mathbb{Z}) \approx H_{*}(Y ; \mathbb{Z}),$ then such a $Y$ is unique up to homotopy equivalence. [Use Corollary 4.73 with $W$ the mapping cylinder of $X \rightarrow Y .$ ]

Anthony Ramos
Anthony Ramos
Numerade Educator
08:50

Problem 23

Extend the Hurewicz theorem by showing that if $X$ is an $(n-1)$ -connected CW complex, then the Hurewicz homomorphism $h: \pi_{n+1}(X) \rightarrow H_{n+1}(X)$ is surjective when $n>1,$ and when $n=1$ show there is an isomorphism $H_{2}(X) / h\left(\pi_{2}(X)\right) \approx$ $H_{2}\left(K\left(\pi_{1}(X), 1\right)\right) .$ [Build a $K\left(\pi_{n}(X), n\right)$ from $X$ by attaching cells of dimension $n+2$ and greater, and then consider the homology sequence of the pair $(Y, X)$ where $Y$ is $X$ with the $(n+2)$ -cells of $K\left(\pi_{n}(X), n\right)$ attached. Note that the image of the boundary map $H_{n+2}(Y, X) \rightarrow H_{n}(X)$ coincides with the image of $h,$ and $H_{n+1}(Y) \approx$ $\left.H_{n+1}\left(K\left(\boldsymbol{\pi}_{n}(X), \boldsymbol{n}\right)\right) . \text { The previous exercise is needed for the case } n>1 .\right].$

Ely Crowder
Ely Crowder
Numerade Educator
14:32

Problem 23

If $f: X \rightarrow Y$ is a map with $X$ and $Y$ homotopy equivalent to CW complexes, show that the pair $\left(M_{f}, X\right)$ is homotopy equivalent to a CW pair, where $M_{f}$ is the mapping cylinder. Deduce that the mapping cone $C_{f}$ has the homotopy type of a CW complex.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:16

Problem 24

In the situation of the relative lifting problem, suppose one has two different lifts $W \rightarrow X$ that agree on the subspace $A \subset W .$ Show that the obstructions to finding a homotopy rel $A$ between these two lifts lie in the groups $H^{n}\left(W, A ; \pi_{n} F\right)$.

Uma Kumari
Uma Kumari
Numerade Educator
01:07

Problem 24

Show there is a Moore space $M(G, 1)$ with $\pi_{1}(M(G, 1)) \approx G$ iff $H_{2}(K(G, 1) ; \mathbb{Z})=0$ IUse the preceding problem. Build such an $M(G, 1)$ from the 2 -skeleton $K^{2}$ of a $\left.K(G, 1) \text { by attaching } 3 \text { -cells according to a basis for the free group } H_{2}\left(K^{2} ; \mathbb{Z}\right) .\right]$ In particular, there is no $M\left(\mathbb{Z}^{n}, 1\right)$ with fundamental group $\mathbb{Z}^{n},$ free abelian of rank $n$
if $n \geq 2.$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 25

Let $X$ be a CW complex with $\pi_{i}(X)=0$ for $1<i<n$ for some $n \geq 2 .$ Show that $H_{n}(X) / h\left(\pi_{n}(X)\right) \approx H_{n}\left(K\left(\pi_{1}(X), 1\right)\right),$ where $h$ is the Hurewicz homomorphism.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:58

Problem 26

Generalizing the example of $\mathbb{R P}^{2}$ and $S^{2} \times \mathbb{R P}^{\infty},$ show that if $X$ is a connected finite-dimensional CW complex with universal cover $\tilde{X},$ then $X$ and $\tilde{X} \times K\left(\pi_{1}(X), 1\right)$ have isomorphic homotopy groups but are not homotopy equivalent if $\pi_{1}(X)$ contains elements of finite order.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:43

Problem 27

From Lemma 4.39 deduce that the image of the map $\pi_{2}\left(X, x_{0}\right) \rightarrow \pi_{2}\left(X, A, x_{0}\right)$ lies in the center of $\pi_{2}\left(X, A, x_{0}\right).$

Linda Hand
Linda Hand
Numerade Educator
00:59

Problem 28

Show that the group $\mathbb{Z}_{p} \times \mathbb{Z}_{p}$ with $p$ prime cannot act freely on any sphere $S^{n}$ by filling in details of the following argument. Such an action would define a covering space $S^{n} \rightarrow M$ with $M$ a closed manifold. When $n>1,$ build a $K\left(\mathbb{Z}_{p} \times \mathbb{Z}_{p}, 1\right)$ from $M$ by attaching a single $(n+1)$ -cell and then cells of higher dimension. Deduce that $H^{n+1}\left(K\left(\mathbb{Z}_{p} \times \mathbb{Z}_{p}, 1\right) ; \mathbb{Z}_{p}\right)$ is $\mathbb{Z}_{p}$ or $0,$ a contradiction. (The case $n=1$ is more elementary.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:35

Problem 29

Finish the homotopy classification of lens spaces begun in Exercise 2 of §3.E, by showing that two lens spaces $L_{m}\left(\ell_{1}, \cdots, \ell_{n}\right)$ and $L_{m}\left(\ell_{1}^{\prime}, \cdots, \ell_{n}^{\prime}\right)$ are homotopy equivalent if $\ell_{1} \cdots \ell_{n} \equiv \pm k^{n} \ell_{1}^{\prime} \cdots \ell_{n}^{\prime}$ mod $m$ for some integer $k,$ via the following steps:
(a) Reduce to the case $k=1$ by showing that $L_{m}\left(\ell_{1}^{\prime}, \cdots, \ell_{n}^{\prime}\right)=L_{m}\left(k \ell_{1}^{\prime}, \cdots, k \ell_{n}^{\prime}\right)$ if $k$ is relatively prime to $m .$ [Rechoose the generator of the $\mathbb{Z}_{m}$ action on $S^{2 n-1} . .$
(b) Let $f: L \rightarrow L^{\prime}$ be a map constructed as in part (b) of the exercise in §3.E . E. Construct a map $g: L \rightarrow L^{\prime}$ as a composition $L \rightarrow L \vee S^{2 n-1} \rightarrow L \vee S^{2 n-1} \rightarrow L^{\prime}$ where the first map collapses the boundary of a small ball to a point, the second map is the wedge of the identity on $L$ and a map of some degree $d$ on $S^{2 n-1},$ and the third map is $f$ on $L$ and the projection $S^{2 n-1} \rightarrow L^{\prime}$ on $S^{2 n-1} .$ Show that $g$ has degree $k_{1} \cdots k_{n}+d m,$ that is, $g$ induces multiplication by $k_{1} \cdots k_{n}+d m$ on $H_{2 n-1}(-; \mathbb{Z}) .$ [Show first that a lift of $g$ to the universal cover $S^{2 n-1}$ has this degree.
(c) If $\ell_{1} \cdots \ell_{n} \equiv \pm \ell_{1}^{\prime} \cdots \ell_{n}^{\prime} \bmod m,$ choose $d$ so that $k_{1} \cdots k_{n}+d m=\pm 1$ and show this implies that $g$ induces an isomorphism on all homotopy groups, hence is a homotopy equivalence. [For $\pi_{i}$ with $i>1$, consider a lift of $g$ to the universal cover.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:16

Problem 30

Let $E$ be a subspace of $\mathbb{R}^{2}$ obtained by deleting a subspace of $\{0\} \times \mathbb{R} .$ For which such spaces $E$ is the projection $E \rightarrow \mathbb{R},(x, y) \mapsto x,$ a fiber bundle?

Uma Kumari
Uma Kumari
Numerade Educator
04:06

Problem 31

For a fiber bundle $F \rightarrow E \rightarrow B$ such that the inclusion $F \hookrightarrow E$ is homotopic to a constant map, show that the long exact sequence of homotopy groups breaks up into split short exact sequences giving isomorphisms $\pi_{n}(B) \approx \pi_{n}(E) \oplus \pi_{n-1}(F) .$ In particular, for the Hopf bundles $S^{3} \rightarrow S^{7} \rightarrow S^{4}$ and $S^{7} \rightarrow S^{15} \rightarrow S^{8}$ this yields isomorphisms $$\begin{aligned} &\boldsymbol{\pi}_{n}\left(S^{4}\right) \approx \boldsymbol{\pi}_{n}\left(S^{7}\right) \oplus \boldsymbol{\pi}_{n-1}\left(S^{3}\right)\\
&\boldsymbol{\pi}_{n}\left(S^{8}\right) \approx \boldsymbol{\pi}_{n}\left(S^{15}\right) \oplus \boldsymbol{\pi}_{n-1}\left(S^{7}\right)
\end{aligned}$$ Thus $\pi_{7}\left(S^{4}\right)$ and $\pi_{15}\left(S^{8}\right)$ contain $\mathbb{Z}$ summands.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:13

Problem 32

Show that if $S^{k} \rightarrow S^{m} \rightarrow S^{n}$ is a fiber bundle, then $k=n-1$ and $m=2 n-1$ ILook at the long exact sequence of homotopy groups.

Gideon Idumah
Gideon Idumah
Numerade Educator
01:57

Problem 33

Show that if there were fiber bundles $S^{n-1} \rightarrow S^{2 n-1} \rightarrow S^{n}$ for all $n$, then the groups $\pi_{i}\left(S^{n}\right)$ would be finitely generated free abelian groups computable by induction, and nonzero for $i \geq n \geq 2.$

Wendi Zhao
Wendi Zhao
Numerade Educator
04:01

Problem 34

Let $p: s^{3} \rightarrow S^{2}$ be the Hopf bundle and let $q: T^{3} \rightarrow S^{3}$ be the quotient map collapsing the complement of a ball in the 3 -dimensional torus $T^{3}=S^{1} \times S^{1} \times S^{1}$ to a point. Show that $p q: T^{3} \rightarrow S^{2}$ induces the trivial map on $\pi_{*}$ and $\tilde{H}_{*},$ but is not homotopic to a constant map.

Anthony Ramos
Anthony Ramos
Numerade Educator
08:16

Problem 35

35. Show that the fiber bundle $S^{3} \rightarrow S^{4 n+3} \rightarrow \mathbb{H}P^{n}$ gives rise to a quotient fiber bundle $S^{2} \rightarrow \mathbb{C} \mathrm{P}^{2 n+1} \rightarrow \mathbb{H}P^{n}$ by factoring out the action of $S^{1}$ on $S^{4 n+3}$ by complex scalar multiplication.

Cullen Miller
Cullen Miller
Numerade Educator
03:23

Problem 36

For basepoint-preserving maps $f: S^{1} \rightarrow X$ and $g: S^{n} \rightarrow X$ with $n>1,$ show that the Whitehead product $[f, g]$ is $\pm(g-f g),$ where $f g$ denotes the action of $f$ on $g .$

Donald Albin
Donald Albin
Numerade Educator
09:31

Problem 37

Show that all Whitehead products in a path-connected H-space are trivial.

Chris Trentman
Chris Trentman
Numerade Educator
01:49

Problem 38

Show $\pi_{3}\left(S^{1} \vee S^{2}\right)$ is not finitely generated as a module over $\mathbb{Z}\left[\pi_{1}\left(S^{1} \vee S^{2}\right)\right]$ by considering Whitehead products in the universal cover, using the results in Example 4.52. Generalize this to $\pi_{i+j-1}\left(S^{1} \vee S^{i} \vee S^{j}\right)$ for $i, j>1.$

Uma Kumari
Uma Kumari
Numerade Educator
01:40

Problem 39

Show that the indeterminacy of a Toda bracket $\langle f, g, h\rangle$ with $f \in \pi_{i}^{S}, g \in \pi_{j}^{S}$ $h \in \pi_{k}^{s}$ is the subgroup $f \cdot \pi_{j+k+1}^{s}+h \cdot \pi_{i+j+1}^{s}$ of $\pi_{i+j+k+1}^{s}.$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator