# Questions tagged [higher-homotopy-groups]

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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 ...
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### 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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### 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 ...
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### 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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### 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 ...
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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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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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 $\pi_n (G/H)$ ...
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### 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 ...
### 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 ...
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 $f$ ...