Questions tagged [higher-homotopy-groups]

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46 views

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 ...
2
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1answer
30 views

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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1answer
55 views

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}(...
5
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1answer
69 views

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 ...
2
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1answer
57 views

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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1answer
59 views

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 ...
3
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0answers
31 views

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 ...
8
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1answer
187 views

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 ...
3
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2answers
97 views

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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1answer
33 views

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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0answers
44 views

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 ...
3
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1answer
64 views

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 ...
1
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1answer
79 views

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 ...
2
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2answers
85 views

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....
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1answer
51 views

Finite injectivity radius implies compactness

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 ...
4
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1answer
71 views

$\mathbf{B}A$ as a 2-group in a long fiber sequence

I am trying to digest the following statement about 2-group: From nlab Observation 4.2: "Let $A \to \hat G$ be the inclusion of a subgroup, exhibiting a central extension $A \to \hat G \to G$ ...
2
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1answer
113 views

Question involving homotopy groups of quotients

Let $X,Y$ be topological spaces and suppose $$\pi_2(Y)=0$$ $$\pi_2(X/Y)=0$$ Can we deduce that $\pi_2(X)=0$?
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2answers
112 views

Higher homotopy groups of surface of genus 2, without using universal cover

In a course of algebraic topology I am self-studying I encountered the following problem: calculate the homotopy groups $\pi_n(S_2)$, where $S_2=T\#T$. Now I found a solution online involving taking ...
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1answer
102 views

Concerning homotopy groups of a finite wedge of spheres

Let $X$ be a finite wedge of $m$-spheres containing some circles. Is $\pi_n (X)$ a free $\mathbb{Z}\pi_1 (X)$-module, for all $n\geq 2$?
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0answers
55 views

How many cells do we need in $\mathbb{S}^n$ to induce $\pi_n(\mathbb{S}^m)$?

Take on the spheres $\mathbb{S}^n$ and $\mathbb{S}^m$ some "easy" simplicial structures $\Sigma^n$ and $\Sigma^m$ (e.g. as surfaces of the corresponding simplices). Then by simplicial approximation ...
4
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1answer
125 views

What are the homotopy groups of the space of matrices with rank bigger than $k$?

Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. What are the homotopy groups $\pi_n(H_{>k})$? In particular, I would like to know whether ...
2
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1answer
53 views

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 ...
2
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3answers
118 views

Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$?

Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$? I can't find any reference about who did it first.
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1answer
128 views

On the homotopy group of a mapping cylinder

Suppose that a space $A$ homotopy dominated by a space $X$. i.e., there exist continuous maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq 1_A$. Also, let $\phi :K\...
2
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0answers
90 views

Why is the $J$-homomorphism an isomorphism for $n=1$?

I am trying to proove that $\pi_{n+1}(S^n) \cong \mathbb{Z}_2$ using the Pontryagin-Thom construction and the special case $n=1$ of the $J$-homomorphism $$ J_1:\pi_1(SO(n))\rightarrow \pi_{n+1}(S^n). $...
3
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1answer
97 views

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 ...
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0answers
27 views

From what manifold do we need to start to build an eversion for $any$-tridimensional object?

If a sphere eversion is possible using a half-way model how model is used for $cylinder$ eversion ? I need to make some premises to be able to frame the true nature of the problem In sphere eversion ...
3
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3answers
150 views

Higher homotopy groups in terms of the fundamental groupoid

Let $X$ be a topological space. Then we can construct the following structure. Let an $n$-morphism be a map $I^n\to X$. We can view $n+1$ morphisms exactly as homotopies between $n$-morphisms. Let $f,...
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1answer
45 views

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 ...
2
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1answer
80 views

$\pi_n(S^n \vee S^n)$ 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 ...
4
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1answer
261 views

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 ...
2
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0answers
79 views

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 ...
2
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1answer
256 views

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 ...
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0answers
32 views

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'...
4
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1answer
1k views

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 ...
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1answer
2k views

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 ...
1
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0answers
67 views

Is there any precise sense in saying that before Serre the homotopy groups of sphere could in principle have contained an infinite amount of data?

I have heard it stated vaguely more than once in discussions that before the work of Serre on homotopy groups of spheres, the answer to, for given natural numbers $n$ and $k$, the question "What is $\...
3
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1answer
78 views

An orientable 4-manifold with completely nontrivial 2-type

I'm looking for a closed, orientable 4-manifold $M$ with completely nontrivial 2-type. By "completely nontrivial", I mean the following: The fundamental group $\pi_1(M)$ has to be nontrivial. The ...
0
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1answer
87 views

On the homology of $K(G,n)$

I am wondering (but I may be wrong) if one can say that the homology of a K(G,n) is finite dimensional for G finitely generated abelian group. I looked up on the internet for some counterexample and I ...
4
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1answer
448 views

Do Homotopy Groups commute with generalized filtered colimits?

I know that if X is a topological space such that $X= \underset{i}{\bigcup} X_i$ where $X_0 \subset X_1 \subset ... \subset X_n \subset ...$, where $X_i$ are all hausdorff, then the functor $\pi_n(\_)$...
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1answer
68 views

Categorical meaning of the two definitions of $\pi_n(X)$

I know that the higher homotopy groups $\pi_n(X)$ can be defined in two way. Homotopy classes of the continuos maps $(I^n,\partial I^n)\longrightarrow (X,x_0)$. Homotopy classes of the continuos maps ...