# 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 ...
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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1 vote
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $I^0 = \{1\}$. $\newcommand{\im}{\operatorname{im}}$...
1 vote
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### 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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### 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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### 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 ...
1 vote
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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### 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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### 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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### 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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### 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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