# Questions tagged [higher-homotopy-groups]

136 questions

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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 homotopy group is defined as…

GRJ

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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 $I^0 = \{1\}$.
$\newcommand{\im}{\operatorname{im}}$Exactness at $\pi_1(X,A,x_0)$ : $[f]\in\pi_1(X,x_0,x_0$)…

one potato two potato

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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). Is it true that $X$ and $Y$ are homotopy…

Aivazian Arshak

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### Can the fundamental group $\pi_1(X)$ of topological space $X$ be non-abelian? or be continuous?

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 of
$\mathbb{Z}$ and $\mathbb{Z}/p$ for some…

Марина Marina S

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### 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 continuos map $f: (X,A) \longmapsto (Y \star)$ into…

jacopoburelli

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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 induced homomorphism on the singular homology…

B.Hueber

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### 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 $f$ still induces a surjection on $\pi_k$ since the
…

Zest

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### Relative homotopy groups $\pi_k (S^n, S^1)$

I'm an undergraduate student currently studying Algebraic topology.
I've been struggling to find all relative homotopy groups $\pi_k (S^n, S^1)$ for $n\geq 3$, $k\leq n$.
Here are my thoughts: If either $S^n$ or $S^1$ were contractible, this would…

Andrea

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### $f:X\to Y$ extends to a map $Z\to Y$ iff $f_*[g] = 0$

Let $f \colon(X,x_0)\to (Y,y_0)$ and $g \colon(S^n,s_0)\to (X,x_0)$ be base
point preserving maps.
Let $Z$ be the space that arises from $X$ by attaching an $(n+1)$-disk via $g$.
I want to prove that $f_*[g]=0 \in \pi_n(Y,y_0)$ implies that…

Zest

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### Long exact sequence of the Klein bottle as a $S^1$-fiber bundle

If we look at he Klein bottle $K$ as a $S^1$-fiber bundle over $S^1$, we can apply the long exact sequence in Homotopy for fibers.…

Timmathy

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### Boundary map between homotopy groups

I would like to understand the boundary map bellow
$\dots \to \pi_{n}(B, b_{0}) \stackrel{\partial}{\to} \pi_{n-1}(F, x_{0}) \to \dots$
where $p\colon E \to B$ has the homotopy lifting property with respect to disks $D^{k}$ for all $k > 0$, $b_{0}…

Gregory math

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### Left homotopy groups

What are $\delta_0$ and $\delta_1$ in the diagram of the definition $2.1$
in the notion of homotopy here?

user122424

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### Contractibility of CW complex without Whitehead

Suppose I have a CW complex $X$ with skeleta $(X_n)_{n\ge 0}$ such that $\pi_k(X)=0$ for all $k\ge 0$. I want to conclude that $X$ is contractible without invoking Whitehead's theorem.
It would be enough to see that the identity $X$ is…

FKranhold

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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't see how the next relationship is…

Oliver G

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### The cartesian product of contractible spaces is contractible

Let $X_i$, $i\in I$ be contractible spaces. Is the Cartesian product $\prod_iX_i$ contractible, too?

Flavius Aetius

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