# Questions tagged [higher-homotopy-groups]

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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 ...
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### 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 ...
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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 $f$ ...
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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 ...
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### Infinitely generated $\pi_2(M)$ where $M$ is closed smooth manifold

I am looking for a closed smooth manifold with infinitely generated $\pi_2$. I know there is an easy example with some universal cover i think, but all i have in my head is $S^2\vee S^1$, which is not ...
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### Show that the space $Y = S^3 \vee S^6$ has precisely two distinct homotopy classes of comultiplications.

Here is the question: A comultiplication for a pointed space $X$ is a map $\phi : X \rightarrow X \vee X$ so that the composite $$X \xrightarrow{\phi} X \vee X \xrightarrow{i_{X}} X \times X$$ is ...
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### How to show that : $\mathrm{Hot} \simeq W_{ \infty }^{-1} \mathcal{T} \mathrm{op}$?

How to show that : $$\mathrm{Hot} \simeq W_{ \infty }^{-1} \mathcal{T} \mathrm{op}$$ For information : $\mathrm{Hot}$ is the homotopy category defined as being the category where the objects are ...
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### How does the Hopf map generate $\pi_3(S^2)$?

I have been studying the Hopf fibration which is an example of a map from $S^3$ to $S^2$. It is a member of $\pi_3(S^2)$ and shows that this group is non-trivial. It can be shown using a long exact ...
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### Explicit isomorphism $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_1(\mathbb{RP}^{n-1})$

From covering space theory we know that $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_{n+1}(\mathbb{S}^n)$. From wikipedia I can notice that $\pi_{n+1}(\mathbb{S}^n) \cong \pi_1(\mathbb{RP}^{n-1})$.* My ...
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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. ...
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### Proof that higher homotopy groups of kan complexes are abelian using Eckmann-Hilton

I try to prove that higher homotopy groups of kan complexes are abelian using an Eckmann-Hilton argument. For the definitions I followed the book "Simplicial objects in algebraic Topology" by Peter ...
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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 ...
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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 ...
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### Behaviour of direct limits of topological spaces with respect to preimages

Given a continuous map $p:E\rightarrow B$ where $B$ is given by a colimit of $B_{0}\subseteq B_{1}\subseteq B_{2}\subseteq\dots$. We get the canonical induced map  colim_{n\in\mathbb{N}} p^{-1}(...
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### How do synthetic homotopy groups relate to the usual homotopy groups?

In Homotopy Type Theory (HoTT in what follows) one may compute homotopy groups of objects that bear names that are the same as some usual spaces: for instance one may consider $S^1$ which is defined ...
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### Evaluating a symplectic form on $\pi_2$ or its image through the Hurewicz map

Let $(M,\omega)$ be a symplectic manifold. There are a priori two ways of evaluating $\omega$ on an element $A \in \pi_2(M)$: we can integrate $\omega$ on any representative $u : S^2 \to M$ of the ...
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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 ...
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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 ...
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### Are the higher homotopy groups of a compact manifold finitely generated as $\mathbb{Z}[\pi_1]$-modules?

Let $M$ be a compact manifold. The homology and cohomology groups of $M$ are necessarily finitely generated, as is the fundamental group. Serre proved that a simply connected finite CW complex has ...
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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 ...
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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?
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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 ...
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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 ...
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### Must one space be the loop space of other

If X and Y be two topological spaces such that n-th homotopy group of X and (n+1)-th homotopy group of Y are isomorphic for all natural number n. Does it imply that X is homotopy equivalent to the ...
### Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim} \ Z_n$ and $\underset{\rightarrow}{\lim} \ \Omega Z_n$
Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....
Let $(M,g)$ be a complete Riemannian manifold. Let $\tau:SM\to\mathbb{R}$ denote the cut distance function, where $SM$ is the unit-tangent bundle of $M$. Let $i_0(p)$ denote the injectivity radius ...