# Questions tagged [higher-homotopy-groups]

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### Whitehead theorem for maps between CW-complexes

I was wondering : if $f,g : X \rightarrow Y$ are continuous maps between CW-complexes, if they induce the same morphisms on homotopy groups, does that imply that $f$ and $g$ are homotopic? It would ...
1answer
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### Higher homotopy of a wedge of $3$-sphere and Poincare homology sphere.

I've been preparing for a qualifying exam in topology. I'm struggling with a recurring question to do with computing higher homotopy groups of the wedge of spaces. Example: Let $P$ be the Poincare ...
1answer
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### Role of $0$ in Hatcher's proof of higher homotopy groups distributivity

In Hatcher 4.1, page 341 in my edition Hatcher defines the following. Let $f \in \pi_n(X,x_0)$ and a path $\gamma$ from $x_0$ to $x_1$. Then, $\gamma f$ if defined "by shrinking the domain of $f$ to a ...
1answer
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### $\pi_n(S^n \vee S^n)$ what am I doing wrong?

I have come to contradiction trying to compute $\pi_n(S^n \vee S^n)$ (n > 1). First of all we notice that the composition $S^n \to S^n \vee S^n \to S^n$ is identity map (the first arrow is embedding ...
1answer
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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 ...
0answers
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### Another multiplication in homotopy groups of simplicial loop space

So, I'm trying to fill up the details in some of the propositions of Goerss and Jardine's book, "Simplicial Homotopy Theory". On Lemma 7.6, I've tried to do something similar to what we do in the case ...
1answer
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### Do trivial homotopy groups imply existence of boundary preserving homotopies?

Let $N$ be a smooth $d$-dimensional connected orientable manifold $N$ which have the following property: For every smooth $d$-dimensional manifold $M$ with non-empty boundary, and for every smooth ...
0answers
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### Higher homotopy groups equality question.

I'm self learning Algebraic Topology from Rotman's Algebraic Topology and I've come across this: How are these two expressions in the red box equal? I understand how $\Sigma ^nS^0 = S^n$, but I don'...
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### Homotopy groups O(N) and SO(N): $\pi_m(O(N))$ v.s. $\pi_m(SO(N))$ ($m$=1~10 and $N$=2~11 )

I have known the data of $\pi_m(SO(N))$ from this Table: I wonder whether there are some useful information that I can relate $\pi_m(SO(N))$ and $\pi_m(O(N))$? Here is the difficulty somehow posted ...
1answer
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### Homotopy groups U(N) and SU(N): $\pi_m(U(N))=\pi_m(SU(N))$

Am I correct that homotopy groups of $U(N)$ and $SU(N)$ are the same, $$\pi_m(U(N))=\pi_m(SU(N)), \text{ for } m \geq 2$$ except that $$\pi_1(U(N))=\mathbb{Z}, \;\;\pi_1(SU(N))=0,$$ Hence the ...
0answers
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