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 ...
3
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1answer
142 views

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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37 views

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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37 views

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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32 views

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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31 views

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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1answer
28 views

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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20 views

Question about isomorphism of induced homomorphism in the proof of CW approximation.

I am currently working through the proof of one exercise which is Let $(X,x_0)$ be a path-connected space. Show that there is a $CW$-Complex $(Z,z_0)$ together with a map $f\colon (Z,z_0)\to (X,...
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1answer
38 views

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 ...
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40 views

$f:X\to Y$ 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 ...
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40 views

Is composing homotopies of spheres bilinear?

$\newcommand{\SS}{\mathbb{S}}$ For nonnegative integers $n, m$ and $k$ consider the map $$ c: \pi_n(\SS^m) \times \pi_m(\SS^k) \to \pi_n(\SS^k) $$ given by composition, i.e. $([f], [g]) \mapsto [g \...
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2answers
54 views

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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29 views

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. $$\pi_2(S^1)\rightarrow\pi_2(K)\rightarrow\pi_2(S^1)\rightarrow\pi_1(...
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40 views

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 ...
3
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1answer
42 views

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. ...
3
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1answer
65 views

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$ $...
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1answer
66 views

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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1answer
80 views

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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27 views

Homotopy groups of weighted projective space via homotopy sequence for Serre-fibrations

I am trying to compute the homotopy groups of the weighted projective space $\mathbb{P}(w) = S^{2n+1}/S^1$ with weights $w=(w_0,...,w_n)$ which is the orbit space of the $S^1$-action $$ s\cdot (x_0,......
3
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1answer
58 views

Proof of a property of Whitehead product

Let $\alpha\in \pi_n(X)$ and $\beta\in\pi_k(X)$. Let $[\alpha,\beta]\in \pi_{n+k-1}(X)$ be the Whitehead product of $\alpha$ and $\beta$. I am having trouble understanding the following property of ...
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54 views

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 ...
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1answer
85 views

Allen Hatcher algebraic topology proof of theorem 4.5 Whitehead

It was stated as the last answer in this post: Idea behind the proof of Whitehead's Theorem and Compression Lemma I copy here what is asked there: I am reading as well the proof of Whitehead's ...
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1answer
37 views

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 ...
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1answer
49 views

$\pi_n((S^1 \vee S^n)\cup e^{n+1})$, Example 4.35, Hatcher, Algebraic Topology

In Example 4.35, I can't see how the underlined statement holds. I see that $\pi_n(S^1\vee S^n)$ is isomorphic to $\Bbb Z[t,t^{-1}]$ as algebras, but why $\pi_n(X)$ is the ring $\Bbb Z[t,t^{-1}]/(2t-1)...
2
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1answer
56 views

Bijection between homotopy classes

Let $X$ be a CW-complex and $w: Z \to Y$ a weak homotopy equivalence. Show that $$w_*: [X, Z] \to [X, Y]: [f] \mapsto [w \circ f]$$ is a bijection. Hint: use mapping cylinders. I am having trouble ...
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2answers
112 views

$\pi_1(A,x_0)$ acts on the long exact sequence of homotopy groups for $(X,A,x_0)$

In the last paragraph in page 345 of Hatcher's Algebraic Topology(link:http://pi.math.cornell.edu/~hatcher/AT/ATch4.pdf), Hatcher says that $\pi_1(A,x_0)$ acts on the long exact sequence of homotopy ...
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81 views

Compute $\pi_{2}(S^2 \vee S^2).$

Compute $\pi_{2}(S^2 \vee S^2).$ Hint: Use universal covering thm. and use Van Kampen to show it is simply connected. Still I am unable to solve it, could anyone give me more detailed hint and the ...
3
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1answer
89 views

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 ...
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1answer
36 views

Is a formula for the *number* of Lyndon words of a fixed length over a given number of generators known?

In this paper, P.J. Hilton describes the homotopy type of a wedge of spheres (in terms of the homotopy types of the spheres in that wedge). Therein, he describes what he defines as basic products. He ...
4
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1answer
94 views

Space of Non-Surjective maps between Spheres

Let $\text{Map}_{ns}(S^n,S^n)$ denote the space of all continuous, non-surjective maps from $S^n$ to $S^n$ carrying the compact-open topology. There is the following fibration:$$ \text{Map}_{ns}((S^n,...
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0answers
26 views

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 ...
3
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2answers
162 views

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 ...
4
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80 views

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 ...
3
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0answers
38 views

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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28 views

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 ...
1
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2answers
80 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
44 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
70 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}(...
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1answer
89 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
61 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 ...
2
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1answer
100 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
268 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 ...
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2answers
140 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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50 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
84 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
86 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
108 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
86 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 ...