Questions tagged [higher-homotopy-groups]

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Does the proof that the homotopy groups are abelian in Switzer's book only work for hausdorff spaces?

The proof that the homotopy groups of a pointed space, $\pi_n(X,x_0)$, are abelian for $n\geq 2$, given by Robert P. Switzer in his book Algebraic Topology - Homology and Homotopy, relies on the ...
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2 votes
2 answers
62 views

Loop-Suspension adjunction unit is stable equivalence

Let $X$ be a sequential spectrum, then the unit of the $(\Sigma,\Omega)$-adjunction yields a map $\eta : X \to \Omega \Sigma X$. The authors of the book Foundations of Stable Homotopy Theory (p. 48) ...
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The homotopy fiber $F$ of the map $p_2 : S^2 \to \mathbb{C} P^\infty$ is weakly homotopy equivalent to $S^3$

I'm stuck with the following question: Consider the Postnikov tower $S^2 \to \ldots \to P_3S^2 \to P_2S^2 \to P_1S^2$ of the $2$-sphere. Show that $P_2S^2$ and $\mathbb{C} P^\infty$ are weakly ...
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Construction of Eilenberg-Maclane Space

I have just learned about the construction of the Eilenberg-Maclane space $K(A,n)$ ($n \geq 1$). I am using Hatcher's text. He briefly describes the construction on p.365. I will recap the ...
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2 votes
1 answer
72 views

Order of the suspension of the Hopf map

I am working on the following problem from an old qualifying exam in algebraic topology: The Hopf map $\eta:S^3 \rightarrow S^2$ can be defined as $\eta(z_0,z_1) = z_0/z_1$, with $S^3$ the unit ...
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1 vote
1 answer
33 views

Homotopy groups of infinite dimension lens space

Let $n > 1$. We define the infinite dimensional lens space $L$ as follows. Let $S^{\infty}$ be the unit sphere in the infinite dimensional complex vector space $C^{\infty}$, and let $\mathbb{Z}/n = ...
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1 answer
56 views

Homotopy groups of wedge of spaces [duplicate]

Let $X$ and $Y$ be topological spaces with $\pi_{i}(X) = \pi_{i}(Y) = 0$ for $i \geqslant n$ Is it true that $\pi_{i}(X\vee Y) = 0$ for $i \geqslant n$? If no, is it true for $CW$ complexes?
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1 answer
53 views

Equivalence between two homotopy groups definition

Homotopy groups of a (pointed) topological space can be defined in multiple ways. In particular, I'm interested in proving rigourosly that the following two definitions are equivalent: $\pi_n(X,x_0)=[...
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4 votes
1 answer
69 views

Examples of spaces with non-abelian $\pi_2(X, A)$

It is well known that $\pi_n(X)$ are abelian for all $n\geq2$, but this only follows in relative homotopy groups for $n\geq3$. I am writing some notes on higher homotopy groups, and was searching for ...
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$H_n(X)/h(\pi(X))\cong H_n(K(\pi_1(X),1))$, Hatcher section 4.2 ex. 25

Let $X$ be a connected CW-complex, with $\pi_i(X)=0$ for $1<i<n$. I want to show that $H_n( K(\pi_1(X),1) ) \cong H_n(X)/h(\pi_n(X))$, where $h$ denotes the Hurewicz map. This is exercise 25, ...
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74 views

Homotopy Lifting Property in Hatcher's Spectral Sequences

Let $\pi: X \to B$ be a fibration with $B$ path-connected CW complex filtered by $p$-skeleta $B_0 \subset B^1 \subset ... \subset B^p \subset ... B^{\dim(B)}=B$. This induces a filtration on $X$ via $...
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1 answer
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A pair $(X,A)$ is $n$-connected iff the inclusion $A\rightarrow X$ is an $n$-equivalence.

This book says followings: Definition. A continuous map $e:A\rightarrow Z$ is an n-equivalence if, for all $y\in Y$, the induced map $e_{*}:\pi_q(Y,y)\rightarrow \pi_q(Z,e(y))$ is an injection for $q&...
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Explicit description of $\pi_1(M)$-action on the relative homotopy groups $\pi_k(\Omega_0M,M)$

Let $M$ be a path-connected space (say, a connected manifold) and denote by $\Omega_0 M$ the based loop space component of the constant loop $x_0$. If $c \colon M \to \Omega_0 M$ denotes the obvious ...
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2 answers
70 views

If we have a weak homotopy equivalence $f:X \rightarrow Y$, does there necessarily exist a weak homotopy equivalence $g:Y \rightarrow X$?

Suppose $X$ and $Y$ are topological spaces. Is it true that, if we have a weak homotopy equivalence $f:X \rightarrow Y$, then there exists a weak homotopy equivalence $g:Y \rightarrow X$? I can't ...
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1 answer
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Why is this CW complex connected?

I am working on Exercise 4.1.11 in Hatcher's Algebraic Topology text, which is as follows: Show that a CW complex $X$ is contractible if it is the union of an increasing sequence of subcomplexes $X_1 ...
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Prove the theorem : $𝜋_𝑛(𝐵)≃𝜋_𝑛(𝐸)⊕𝜋_{𝑛−1}(𝐹)$ [duplicate]

Theorem: For a fiber bundle 𝐹→𝐸→𝐵 such that the inclusion 𝐹→𝐸 is homotopic to a constant map the long exact sequence in homotopy breaks up into split short exact sequences $𝜋_𝑛(𝐵)≃𝜋_𝑛(𝐸)⊕𝜋...
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1 answer
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Homotopy group of pairs: equivalent descriptions

I'm reading May's A concise Course in Algebraic Topology and I have a question about the definition of the homotopy group of a pair. Given a pair $(X,A)$ of pointed topological spaces, the relative ...
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1 answer
171 views

What is $\pi_3(S^2\times S^1)$?

What is the third homotopy group of $S^2\times S^1$? Here $S^n$ means the $n$-dimensional sphere. I am a physicist and did not take courses on too mathematical topics. I just need a fact of this in a ...
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1 vote
1 answer
91 views

A map between connected $n$-dimensional CW complexes is a homotopy equivalence

Show that a map between connected $n$-dimensional CW complexes is a homotopy equivalence if it induces an isomorphism on $\pi_i$ for $i\leq n$. [Pass to universal covers an use homology]. I'm in the ...
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2 votes
0 answers
57 views

Action of $\pi_1(\Bbb RP^n)$ on $\pi_n(\Bbb RP^n)$

The following is one of Hatcher AT exercise problem. Show the action of $\pi_1(\Bbb RP^n)$ on $\pi_n(\Bbb RP^n)\simeq\Bbb Z$ is trivial for $n$ odd and nontrivial for $n$ even. Using the fact that ...
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5 votes
1 answer
141 views

Does there exist a smooth compact manifold whose homotopy groups are not finitely generated?

Does there exist a smooth compact manifold whose homotopy groups are not finitely generated? I found a counter-example for topological manifolds, but I did not understand whether it is possible to ...
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0 votes
1 answer
100 views

Long exact sequence of homotopy group

$\pi_1(X,x_0)\xrightarrow{j_*}\pi_1(X,A,x_0)\xrightarrow{\partial}\pi_0(A,x_0)\xrightarrow{i_*}\pi_0(X,x_0)$ is exact. Here, I interpreted $I^0 = \{1\}$. $\newcommand{\im}{\operatorname{im}}$...
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1 vote
1 answer
41 views

Why does the contractibility of $CX$ imply that $\pi_n(CX,X,x_0)\cong\pi_{n-1}(X,x_0)$?

My Algebraic Topology notes say that because the cone of $X$ is contractible (which is easy enough to see, $H([x,t],s)=[x,t(1-s)]$ does the job) the Long Exact Sequence of Relative Homotopy Groups ...
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2 votes
2 answers
112 views

Definition of weak homotopy equivalence

Recall a weak homotopy equivalence is a continuous map $f:X_1 \rightarrow X_2$ that induces a bijection $f_*:[Y,X_1] \rightarrow [Y,X_2]$ for any CW-complex $Y$, where $[Y,X]$ denotes the homotopy ...
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1 answer
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If higher homotopy groups are trivial, then the fundamental group is a complete invariant?

Let $X$ and $Y$ be two n-manifolds all of whose higher homotopy groups are trivial, and the first homotopy groups are isomorphic (but the existence of a mapping inducing an isomorphism is not assumed)....
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Showing an element of $\pi_n(S^n)$ is a sum of maps acting locally, invertibly near one point

By simplicial approximation, a continuous map $f:S^n \to S^n$ is homotopy equivalant to a simplicial map. Such a map must send vertices to vertices, from which the full map is determined by linearly ...
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1 vote
0 answers
68 views

Covering ${\Bbb R}^{3}$ by disoint circles.

Theorem. It is impossible to make ${\Bbb R}^{3} = \underset{\lambda \in \Lambda}{\bigcup} {(S_1)}_{\lambda}$, where each $(S_1)_{\lambda}$ never meets other $(S_1)_{\lambda'}$. That is any two circles ...
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2 votes
0 answers
33 views

Is an homotopy class represented by the homotopy class of inclusion as subspace?

The following question is related to this other question. In the question I have to show the exactness of $$\pi_1(A,x_0) \overset{i_*}{\longmapsto} \pi_1(X,x_0)\overset{j_*}{\longmapsto} \pi_1(X,A,x_0)...
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1 vote
0 answers
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Understanding Theorem 6.3.1 Tom Dieck

I'd like to understand a part of the Theorem on Tammo Tom Dieck p.$130$. The theorem is the following: I do understand that $T$ is homeomorphic to $J^n$ with the exchange of the last two coordinates,...
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3 votes
1 answer
162 views

Understanding $CW$ approximation theorem.

The statement is the following: Approximation theorem: Let $A$ be a $CW$ and $k \in \mathbb{Z} \cup \left\lbrace-1\right\rbrace$. Let $Y$ be a topological space with $f: A \longmapsto Y$ continuous ...
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0 votes
1 answer
123 views

Can the fundamental group $\pi_1(X)$ of topological space $X$ be non-abelian? or be continuous? [closed]

We are familiar with the homotopy group of spheres $S^n$: https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres. There we learn that $\pi_d(S^n)$ must be abelian and discrete. They are direct sum ...
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2 votes
1 answer
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Homotopy groups of projective Lie groups PO(N), PSO(N), and PSpin(N)

Previously, we have learned from Homotopy groups O(N) and SO(N): $\pi_m(O(N))$ v.s. $\pi_m(SO(N))$ that: $\pi_m(SO(N))$: a table consisting of the groups $\pi_m(SO(N))$ for $1 \leq m \leq 15$ and $2 \...
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2 votes
0 answers
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Question regarding the proof of Theorem 6.1.2 Tom Dieck

I'd like to understand the proof of the exact sequence of the pair in homotopy. I'm reffering to the one I found here: https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf (p.123) The implication ...
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0 votes
1 answer
26 views

Bijection $[((X,A)),((Y \star))]\simeq[X/A,Y]^0$

I'd like to prove the following, to well define homotopy classes and essentialy work with the most useful space, (i.e being able to switch between $(I,\partial I)$ and $\mathbb{S}^1$): Lemma : A ...
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3 votes
1 answer
72 views

The n-sphere is not a deformation retract of the bouquet of k n-spheres

I'm trying to show that $S^n$ is not a deformation retract of $\bigvee^k S^n$ as a generalization of a proof (or an attempt) I made showing that $S^1$ is not a deformation retract of $S^1\vee S^1$, ...
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4 votes
1 answer
140 views

Reflection of $S^n$ is the inverse of the identity map in $\pi_n(S^n)$

Regard $S^n$ as a subspace of $\Bbb R^{n+1}$, and consider the reflection $f:S^n\to S^n$, $(x_1,\dots,x_{n+1})\mapsto (-x_1,x_2,\dots,x_{n+1})$. This defines an element of the group $\pi_n(S^n)$. How ...
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7 votes
0 answers
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Five lemma for homotopy exact sequences of triple

Suppose we have topological spaces $B\subset A\subset X$ and $B'\subset A'\subset X'$, and a continuous map $f:X\to X'$ with $f(A)\subset A'$, $f(B)\subset B'$. Consider the homotopy long exact ...
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6 votes
1 answer
133 views

Homotopy cardinality of fiber bundles

Consider the Homotopy cardinality (or $\infty$-groupoid cardinality) $\chi(X):=\sum_{[x]\in\pi_0(X)}\prod_{i\geq0}|\pi_i(X,x)|^{(-1)^{i+1}}$ associated to a space $X$. Suppose we have a fibre bundle $...
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2 votes
1 answer
74 views

What can be said about the homotopy groups of $(\widetilde{K}\times X)/G$

Let $G$ be a group and $X$ a simply connected $G$-space. For a $K(G,1)$ space $K$ with universal cover $\widetilde{K}\rightarrow K$ we have that $G$ acts on $\widetilde{K}$ via the unique homotopy ...
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2 votes
2 answers
87 views

How to show that the homotopy group $\pi_4(U(3))$ of a unitary group is finite

I have dealt with the stable unitary groups and calculated the groups for $\pi_i(U(n))$ when $i=1,2,3$ and I know that $\pi_4(U(3))\cong \pi_4(U(4))$ and the fibration $U(n)\to U(n+1) \to S^{2n+1}$ ...
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1 vote
1 answer
122 views

Homotopy group of $O(p,q)$

I am interested in knowing the homotopy group of $O(p,q)$ as the orthogonal group of indefinite quadratic form over the reals. Here $O(p,q)$ is defined as $$ O(p,q) ={O}(p,q; \mathbb{R}) = \left\{Q \...
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3 votes
2 answers
131 views

Defining higher homotopy in terms of embeddings $S^n \hookrightarrow X$?

I was looking around on https://en.wikipedia.org/wiki/Homotopy_group, and saw that the definition of $\pi_n(X)$ is the set of homotopy classes of maps that map $S^n \to X$ (with fixed base points $a\...
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  • 5,684
2 votes
1 answer
100 views

How to show that higher Hopf fibrations are not nullhomotopic?

Here Proving that hopf map from $S^3 \to S^2 $ is not null homotopic is an aswer showing Hopf map is not null homotopic. How would one show that the higher fibrations $S^3 \to S^7 \to S^4$ and $S^7 \...
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  • 205
2 votes
1 answer
94 views

A confusion on $\Omega$ and $\Sigma$ functors

We know that for any pointed topological space there are two functors $$ \Omega:\left\{\text{pointed topological spaces}\right\}\longrightarrow \left\{\text{H-groups}\right\} $$ and $$ \Sigma:\left\{\...
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2 votes
1 answer
89 views

Calculating higher homotopy groups of (complements of) knots

There are techniques to calculate the group of a knot, i.e. the fundamental group of its complement in a manifold, but are there techniques to calculate its higher homotopy groups? Can anyone suggest ...
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  • 341
1 vote
4 answers
154 views

Prove that $S^n$ is $(n-1)$-connected.

I heard someone mention that since $S^n$ is $(n-1)$-connected (i.e. $\pi_k (S^n) = 0$ for $k<n$), $\pi_2 (S^3)=0$. However, I can't seem to imagine how this is the case. The person who said it made ...
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0 votes
0 answers
68 views

Show that $(f+g)_{\ast}=f_{\ast}+g_{\ast}$

I have the following problem: Let $X$ be some (path-connected) topological space. I have to show that for two $f,g\in\pi_{n}(X)$ we have that $(f+g)_{\ast}=f_{\ast}+g_{\ast}$, where $\ast$ denotes the ...
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  • 1,542
1 vote
1 answer
103 views

Action of $\pi_1(G)$ on $\pi_n(G)$ is trivial for a topological group $G$, i.e. $G$ is a n-simple space.

My question is a follow up of this question Bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_n/\pi_1$. For a topological group $G$, there is a natural action of $\pi_1(...
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6 votes
1 answer
189 views

Is the splitting $\pi_{k}(X,A)\simeq\pi_{k}(X)\times \pi_{k-1}(A)$ a $\pi_1(A)$-modules isomorphism?

Let $(X,A)$ be a pair of topological spaces with $A\subset X$. Fix a basepoint $x_0$ of $X$ which lies in $A$. Assume that the inclusion $(A,x_0)\to (X,x_0)$ is homotopic to a constant map relatively ...
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1 vote
1 answer
83 views

Isomorphism between homotopy groups of CW-complexes

Let $(Y, y_0)$ and $(Y’, y_0)$ pointed CW-complexes, with $Y’$ obtained from $Y$ by attaching $n+1$-cells. Why is it true that $i_{*}: \pi_{q}(Y, y_0) \to \pi_{q}(Y’, y_0)$ is an isomorphism for $q &...
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