# Questions tagged [higher-homotopy-groups]

136 questions

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### Defining higher homotopy in terms of embeddings $S^n \hookrightarrow X$?

I was looking around on https://en.wikipedia.org/wiki/Homotopy_group, and saw that the definition of $\pi_n(X)$ is the set of homotopy classes of maps that map $S^n \to X$ (with fixed base points $a\in S^n$ and $b\in X$). Reading…

D.R.

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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 provided in the proof is natural):
Proposition…

Jakub Opršal

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### Homotopy groups of $S^\infty$

I have seen that it is possible to see $S^\infty$ is contractible, which gives trivial homotopy groups $\pi_k(S^\infty)=0$ for all $k\geq1$.
Are there different proofs to show the homotopy groups are trivial, besides constructing homotopys with the…

Timmathy

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### Show that $S^2$ and $S^3 \times \mathbb CP^\infty$ have isomorphic homotopy groups.

I want to show that $S^2$ and $S^3 \times \mathbb CP^\infty$ have isomorphic homotopy groups in each degree. My first approach was to calculate the homotopy group of $\mathbb CP^\infty$ and use the fact that the homotopy group of a product is the…

samlanader

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### Why doesn't the Homotopy group satisfy excision?

I'm studying higher homotopy groups from the book Algebraic Topology by author Allen Hatcher, there he says that the sequence $A \to X \to X/A$ does not induce an exact sequence of homotopy groups. Why? Is this related to excision?

Gregory math

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### The $n$-dimensional cube modulus its boundary is homeomorphic to an $n$-dimensional sphere

Let ${I}^{n}$ denote the $n$-dimensional cube, $\partial{I}^{n}$ be its boundary and ${S}^{n}$ denote the $n$-dimensional unit sphere.
Now for a pointed space $(X,x_{0})$ the $n^{th}$$homotopy$ $group$ of $X$ is :
The set of homotopy classes of…

Morettin

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### Find the homotopy fibre of the map $\pi$

Find the homotopy fibre of the map $\pi : X \vee Y \rightarrow X$ given that $\pi ( X, *) = X$ and $\pi ( *, Y) = *$, I was given a hint to use the second cube theorem but I do not know how to build the squares, could anyone help me please?

Emptymind

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### Allen Hatcher chapter 4 exercise 17

Show that if $X$ is a $m$-connected CW complex and $Y$ is a $n$-connected CW complex then $(X \times Y, X \vee Y)$ is $(m+n+1)$-connected.
I'm trying to use the equivalences in page 346 to state a proof but is being quite hard to set it up so far..…

astro

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### Showing that $\pi_n(X,v)$ satisfies inverse axiom.

Given a fibrant simplicial set $X$ (has lifting condition with all horns) and a vertex $v \in X_0$.
I want to show that the simplicial homotopy group $\pi_n(X,v)$ satisfies the inverse axiom.
The composition law for two map $f,g:\Delta^n…

Bryan Shih

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### Different definitions of the minimal Chern number and the monotonicity of symplectic manifolds

I am trying to understand the differences between several definitions used in many texts in symplectic topology. Let $(M,\omega)$ be a symplectic manifold, and $c_1 \in H_2(M,\mathbb{Z})$ be its first Chern class.
The minimal Chern number $N_M$ of…

BrianT

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### Weak Lefchetz for a quasiprojective variety and a non-generic hyperplane

In the remarks on page 153-154 of Stratified Morse Theory, Goresky and MacPherson make a claim that they say follows from the theorem on that page. It seems to be false and I'm wondering if I'm wondering how to fix it. Here is the claim (all…

ehd

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### How does the image of the Hurewicz map $\pi_n(X,x) \to H_n(X)$ depend upon the choice of the base point?

Let $X$ be a path connected topological space. I understand that the homotopy groups $\pi_n(X,x_0)$ and $\pi_n(X,x_1)$ are isomorphic to each other. However I do not understand whether the image of the Hurewicz map $\pi_n(X,x) \to H_n(X)$ is…

user90041

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### Attaching 1-cells to CW-complex affects homotopy groups?

Is it true that attaching 1-cells to a CW-complex doesn‘t change it’s higher homotopy groups $\pi_n$ for $n\ge 2$?
(I am aware that a corresponding result for cells of higher dimension is far from being true.)

user39082

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### Simplicial homotopy via simplicial spheres

A bit of confusion on one thing: whenever I see an explanation for the existence of a graded-commutative multiplication on $\pi_*(R) = \bigoplus_n \pi_n(R)$ for a simplicial ring $R$, it's usually a sketchy explanation along the lines of: if $S^n…

Ashwin Iyengar

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### Pontryagin-Thom construction references for homotopy groups of spheres

I'm trying to find the details of the Pontriagin-Thom construction proof about the isomorphism between framed cobordism groups and homotopy groups of spheres and I can't find any good reference.
I was reading Milnor's Topology From the Differential…

Luis

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