# Questions tagged [higher-homotopy-groups]

136 questions

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### Confusion about free homotopy, based homotopy and homotopy groups

Unfortunately, this becomes a very general post: I have some questions concerning the homotopy invariance of homotopy groups. I start from what should be clear:
If $f,g:(X,x_0)\to (Y,y_0)$ are based homotopic, then $\pi_k(f)=\pi_k(g)$, which is…

FKranhold

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### 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 some simple counter-example to show $\pi_2(X, A)$…

Evaristo

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### 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 can we show that the class $[f]\in \pi_n(S^n)$…

E. Kevin

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### 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 $\pi_n (G/H)$ using the long exact sequence.
In general, if I…

Stratiev

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### Proof of a property of Whitehead product

Let $\alpha\in \pi_n(X)$ and $\beta\in\pi_k(X)$. Let $[\alpha,\beta]\in \pi_{n+k-1}(X)$ be the Whitehead product of $\alpha$ and $\beta$. I am having trouble understanding the following property of the Whitehead…

J. Wang

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### Space of Non-Surjective maps between Spheres

Let $\text{Map}_{ns}(S^n,S^n)$ denote the space of all continuous, non-surjective maps from $S^n$ to $S^n$ carrying the compact-open topology. There is the following fibration:$$
\text{Map}_{ns}((S^n,N),(S^n,N))\to \text{Map}_{ns}(S^n,S^n)\to…

ThorbenK

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### How does the Hopf map generate $\pi_3(S^2)$?

I have been studying the Hopf fibration which is an example of a map from $S^3$ to $S^2$. It is a member of $\pi_3(S^2)$ and shows that this group is non-trivial. It can be shown using a long exact sequence applied to the Hopf fibration that…

Jack Holmes

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### Explicit isomorphism $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_1(\mathbb{RP}^{n-1})$

From covering space theory we know that $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_{n+1}(\mathbb{S}^n)$.
From wikipedia I can notice that $\pi_{n+1}(\mathbb{S}^n) \cong \pi_1(\mathbb{RP}^{n-1})$.*
My question is: is there an explicit isomorphism…

CNS709

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### $\mathbf{B}A$ as a 2-group in a long fiber sequence

I am trying to digest the following statement about 2-group:
From nlab Observation 4.2:
"Let $A \to \hat G$ be the inclusion of a subgroup, exhibiting a central extension $A \to \hat G \to G$ with $G := \hat G/A$. Then this short exact sequence of…

wonderich

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### What are the homotopy groups of the space of matrices with rank bigger than $k$?

Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$.
What are the homotopy groups $\pi_n(H_{>k})$?
In particular, I would like to know whether or not they are finitely generated for $n \ge 2$?
(The…

Asaf Shachar

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### Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$?

Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$? I can't find any reference about who did it first.

Luis

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### Higher homotopy groups in terms of the fundamental groupoid

Let $X$ be a topological space. Then we can construct the following structure. Let an $n$-morphism be a map $I^n\to X$. We can view $n+1$ morphisms exactly as homotopies between $n$-morphisms.
Let $f,g$ and $h$ be $n$-morphisms and $H$ and $G$ be…

Chetan Vuppulury

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### 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
$X_p := \pi^{-1}(B^p)$.
By construction the pair…

user7391733

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### 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 such that $f_*$ is an isomorphism for
$i

jacopoburelli

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### 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 $S^1\vee S^1$, but for all $n$ and all $k$ I couldn't prove…

user949426