Questions tagged [higher-homotopy-groups]

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1answer
54 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 ...
2
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0answers
41 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 ...
5
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1answer
106 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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1answer
50 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 $I^0 = \{1\}$. $\newcommand{\im}{\operatorname{im}}$...
1
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1answer
32 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 ...
2
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2answers
66 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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1answer
41 views

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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0answers
22 views

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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0answers
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 ...
2
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0answers
30 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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31 views

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,...
3
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1answer
134 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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1answer
94 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 ...
2
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1answer
53 views

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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0answers
42 views

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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1answer
25 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 ...
3
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1answer
58 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 $S^1\vee S^1$, ...
4
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1answer
102 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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0answers
138 views

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 ...
6
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1answer
120 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 $...
2
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1answer
64 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 ...
2
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2answers
71 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 $U(n)\to U(n+1) \to S^{2n+1}$ ...
1
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1answer
96 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 \...
3
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2answers
107 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\...
2
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1answer
71 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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0answers
36 views

Action of fundamental group on relative homotopy groups, and relative free homotopy classes?

It is well-known that for a path-connected topological space $X$, its fundamental group $\pi_1(X;x_0)$ acts on homotopy groups $\pi_n(X;x_0)$, such that there is a bijection between the quotient space ...
2
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1answer
87 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\{\...
2
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1answer
50 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 ...
1
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4answers
118 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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0answers
67 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 ...
1
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1answer
63 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(...
6
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1answer
171 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 ...
1
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1answer
65 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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0answers
33 views

Reference for first appearance of Samelson product

Samelson noticed that the commutator in any based loop space induced the Pontrjagin product on the homology of that loop space. What is the original reference where the name Samelson product is ...
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1answer
172 views

How torsion arise in homotopy groups of spheres?

The example I have in mind of a torsion element in fundamental group is what happens on the projective plane: there is a cell $D^2$, but its boundary doubly covers a copy of $S^1$. The latter has then ...
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1answer
53 views

Reference requested for homotopy theory theorem

I came across this post: Homotopy groups of compact topological manifold which states exactly the result I need for a theorem I'm working on. However, I would need a reference, since the audience ...
2
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0answers
74 views

Quotient map for the two formulations of relative homotopy groups

In Hatcher Chapter 4, he defines the relative homotopy group $\pi_n(X,A, x_0)$ as maps $(I^n, \partial I^n, J^{n-1})\to (X, A, x_0)$ up to homotopy through such maps. $J^{n-1}$ is defined as the ...
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0answers
404 views

Whitehead product and a homotopy group of a wedge sum

Note : this question has been crossposted on the mathematics Overflow. Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-complex. My goal is to prove ...
8
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1answer
227 views

What does fundamental (homotopy) groups measure?

As I read an algebraic topology book, I felt I knew exactly what the fundamental group is geometrically! I thought it counts the number of independent cycles. (my definition of dependence cycles (that ...
2
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2answers
60 views

element of a homotopy group

I am having trouble understanding the idea of an element of a homotopy group. This is the definition from wikipedia: In the n-sphere $S^{n}$ we choose a base point $a$. For a space $X$ with base point ...
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0answers
87 views

Hatcher's proof of the Freudenthal suspension theorem, definition of the suspension map

In Corollary 4.24 (Freudenthal Susepension Theroem) of Hatcher - Algebraic Topology, Hatcher says that the suspension map $S:\pi_i(X)\to \pi_{i+1}(SX)$ is same as the map $\pi_i(X)\cong \pi_{i+1}(C_+X,...
2
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1answer
54 views

If $A\subset X$ is a deformation retract, then do we have $\pi_k(X,A)=0$?

Let $A$ be a deformation retract of the topological space $X$; the example in my mind is for example: $X=(\mathbb C^*)^n$ and $A=(S^1)^n$. Notice that if we consider the homology instead, then we know ...
5
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2answers
287 views

homotopy groups of product space

I'm trying to prove that the sequence below is exact $1 \to \pi_{n}(X) \to \pi_{n}(X \times Y) \to \pi_{n}(Y)$ Such that $i_{*}:\pi_{n}(X) \to \pi_{n}(X \times Y)$ and $\pi_{y*}:\pi_{n}(X \times Y) \...
1
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1answer
76 views

Homotopy groups in large dimensions

Let $X$ be an $(n-1)$-connected CW complex of dimension $n$ and $\{\varphi_i : \mathbb{S}^n \to X \mid i \in I\}$ a generating set of $\pi_n(X)$. If $\dot{X}$ denotes the complex obtained from $X$ by ...
3
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1answer
256 views

Does the functor $\pi_n\colon \mathsf{Top}_* \to \mathsf{Grp}$ preserve products?

One of the very first propositions about the fundamental group in Hatcher's book [Hat01] states that the fundamental group functor preserves finite products (it is not hard to see that the isomorphism ...
2
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0answers
82 views

weak equivalence of the geometric realisation of a total singular complex and a topological space (from P.May concise course in Algebraic Topology)

In P.May's book "A concise course in Algebraic Topology", chapter 16, He establishes a weak equivalence between $\Gamma X = |S_*(X)|$ and $X$, where $X$ is a topological space, $S_*(X)$ is ...
4
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2answers
196 views

Homotopy groups of quotient groups.

I'd like to ask how to compute homotopy groups of quotient groups, whose homotopy groups I already know. I found this answer, but I don't understand how to derive the homotopy group of $\pi_n (G/H)$ ...
1
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0answers
43 views

Does a deformation retraction of $X$ onto a subspace $A\subset X$ induce an isomorphism $\pi_n(X) \to \pi_n(A)$?

Let's say we have a topological space $X$ and a subspace $A\subset X$. Assume $A$ is a deformation retraction of $X$. Does that imply that the induced homomorphism of the deformation retraction is an ...
1
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0answers
35 views

CW approximation: why does $S^{n+1}\to X$ qualify as an attaching map for attaching $S^{n+1}$ to another space $Z$?

So i am still trying to understand the general proof of the CW approximation. At one point in the proof we have the inductively build CW complex $Z^{n+1}$ together with a map $f:Z^{n+1} \to X$ such ...
0
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1answer
32 views

Question regarding surjectivity of induced homormophism in an old version of Hatcher's proof of Prop. 4.13

So I am currently trying to understand the given proof of Hatcher's proof of proposition 4.13. It's this particular part (in the middle of the screenshot) I don't understand: The extended $f$ ...