Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

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2answers
84 views

horn of a simplex

I'm reading one book of simplicial homotopy. It's just amazing. But I am stuck at the very beginning of the book. He let a simplicial set be a contravariant functor between the category $ \Delta $ of ...
11
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4answers
696 views

What are two continuous maps from $S^1$ to $S^1$ which are not homotopic?

This is an exam question I encountered while studying for my exam for our topology course: Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as ...
2
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0answers
51 views

Space of curves and differentiability

Let $A\subset\mathbb{R^2}$ be an open connected set, and $\Omega=\{c:[0,1]\rightarrow A| c\in C^1\textrm{ and } c(0)=c(1)\}$. Consider the $1$-form $\omega = F_1dx_1+F_2dx_2$ in $A$ such that ...
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1answer
103 views

$[M,\mathbb CP^\infty]=[M,\mathbb CP^2]$ where $M$ is a smooth closed orientable 3-manifold!

Prove the above result, where $[X,Y]$ means the set of all homotopy classes of maps from X to Y, two topological spaces. I have answered it below.
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1answer
147 views

Find a regular homotopy

Firstly, we define a regular homotopy between regular closed curves as a continuous map $F:$I x I$\rightarrow \mathbb{R}^n$ satisfying the following conditions: (i) for each fixed u $\in$ I, the map ...
6
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1answer
122 views

For a transitive action of a path-connected group, does every path lift?

Let $G$ be a Hausdorff, path-connected group acting transitively on a Hausdorff space $X$. Assume the action is continous (i.e. $(g,x) \mapsto g \cdot x$ is continuous) and transitive. It follows ...
8
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3answers
352 views

Spheres in different dimension are not homotopy equivalent

Is there a way to prove that $\textbf{S}^n$ and $\textbf{S}^m$ are not homotopy equivalent if $n\neq m$ without using the machinery of homology or higher homotopy groups?
2
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2answers
338 views

(weak) homotopy equivalence

I have a question arising from chapter 3, page 41, in Switzer. He says "Note that every homotopy equivalence (in $\mathscr{T}$ [this is the category of topological spaces]) is a weak homotopy ...
4
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2answers
194 views

Question about null-homotopic function

How to prove that every continuous function $f:M\to X$ from the Möbius strip $M$ into a simply connected space $X$ is null-homotopic? Thanks in advance.
3
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0answers
74 views

Is a simplicial subcomplex of a contractible simplicial CW-complex also contractible?

Let $Y,X$ be simplical CW-complexes (By that I mean functors $\Delta^{op}\to CW$. In other words, for each natural number $n$ there exist CW-complexes $X_n$ and $Y_n$ and there are face and degeneracy ...
3
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0answers
67 views

Removing the star without changing homology

I know that if the link of a simplex $\sigma$ in a finite simplicial complex $K$ is contractible then the two complexes $K$ and $K\setminus \text{Star}(\sigma)$ share the same homotopy type. ...
4
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0answers
142 views

algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
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2answers
142 views

Is the winding number of a map $\Omega: \; S^n \mapsto S^n$ dependent on the radius of both spheres?

I have 2 questions regarding homotopy groups. My first questions comes from the fact that I've read different definitions of homotopy groups. The book by Manton and Sutcliffe defines them as: ...
13
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1answer
353 views

What is $\pi_{31}(S^2)$?

What is $\pi_{31}(S^2)$ - high homotopy group of the 2-sphere ? This question has a physics motivation: 1) There are relations between (2nd and 3rd) Hopf fibrations and (2- and 3-) qbits (quantum ...
2
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0answers
232 views

Lemma of Whitehead

this is the lemma of Whitehead And i really don't understand the proof How to see that $k$ is well defined (i.e how to write an element from $X\cup_{\varphi_i}e^{\lambda} , i=0,1$ ) and how to ...
9
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3answers
943 views

An intuitive idea about fundamental group of $\mathbb{RP}^2$

Someone can explain me with an example, what is the meaning why $\pi(\mathbb{RP}^2,x_0)$ is $\mathbb{Z}_2$? We consider the real projective plane as a quotient of the disk. I don't receive and ...
6
votes
1answer
169 views

graphs and homotopy extension property

If $T$ is a spanning tree of a graph $X$. How to prove that the pair $(X,T)$ has the homotopy extension property, without using the definition of CW complexes? I mean I don't need the general case ...
6
votes
1answer
351 views

The action of the group of deck transformation on the higher homotopy groups

This is for homework. I'm supposed to do exercise 4.1.4 in Hatchers "Algebraic Topology", which is to show that given a universal covering $p: \tilde{X} \to X$ of a path-connected space $X$, the ...
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2answers
144 views

Is there a fibration whose base, total and fibre is a Moore space?

I want to find a example which base, total of fiber spaces is a Moore space. I don't want trivial fibration. For example the projection $p : M(G,n) \times M(H,n) \to M(G,n)$ and path fibration ...
8
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2answers
105 views

Can the rank of the homology group of an abstract simplicial complex be computed in polynomial time?

I want to write a function that does the following: Input: An integer $n$ A function $f$ that maps nonempty subsets of $\{1, \dots, n\}$ to "yes" or "no" such that (a) every singleton set gets ...
14
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1answer
194 views

Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
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3answers
322 views

What's the point of spectra?

I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and ...
3
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1answer
59 views

Naive question: Why $B:Mon\rightarrow Top^{*}, \Omega: Top^{*}\rightarrow Mon$ is an adjoint functor?

It is not clear to me why we have a bijection of the form $$Mor_{Top^{*}}(BY,X)\rightarrow Mor_{Mon}(Y,\Omega X)$$where $Mon$ is the category of topological monoids and $Top^{*}$ the based topological ...
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2answers
420 views

What is the homotopy colimit of the Cech nerve as a bi-simplical set?

Let $f:Y_\bullet\to X_\bullet$ be an epimorphism of simplicial sets and define the bi-simplicial set $$ F_{\bullet\bullet}=\ldots Y\times_X Y\times_X Y\underset{\to}{\underset{\to}{\to}}Y\times_X Y ...
5
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2answers
487 views

$SU(n)$ is simply connected (proof without fibrations, $n>2$)

How to show that $SU(n)$ is simply connected for $n>2$ if I don't know about fibrations yet? For $SU(2) \cong S^3$ the fact is said to be known. For any matrix $A \in SU(n)$ there is a matrix $S ...
6
votes
3answers
141 views

introductory reference for Hopf Fibrations

I am looking for a good introductory treatment of Hopf Fibrations and I am wondering whether there is a popular, well regarded, accessible book. ( I should probably say that I am just starting to ...
5
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1answer
367 views

Every principal $G$-bundle over a surface is trivial if $G$ is compact and simply connected: reference?

I'm looking for a reference for the following result: If $G$ is a compact and simply connected Lie group and $\Sigma$ is a compact orientable surface, then every principal $G$-bundle over $\Sigma$ ...
3
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0answers
51 views

How to show $\mathbb{S}^{2n-1}$ is a $\mathbb{S}^{n-1}$ bundle over $\mathbb{S}^{n}$ then $n=1,2$ or divisible by $4$?

I was asked to show if $$\mathbb{S}^{n-1}\rightarrow \mathbb{S}^{2n-1}\rightarrow \mathbb{S}^{n}$$ holds. Then $n=1,2$ or divisible by $4$. The $n=1,2$ cases in the converse are verfied. I am ...
2
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1answer
73 views

Are these 2 paths homotopic?

Let $X$ be a topological space. Let $a,b$ be points in $X$. Let $f,g$ be paths of length $1$ from $a$ to $b$ such that $Im \,f=Im \,g$. Does it follow that $f,g$ are homotopic ?
2
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1answer
187 views

Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.

First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes: The motivation for ...
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1answer
87 views

Question regarding the reduced suspension functor

There is a step in J.F.Adams book, Infinite Loop Spaces, which I don't quite understand. Here is the whole extract: Let $W$ be a further space (not sure what 'further' means, seems unnecessary), ...
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1answer
93 views

Is it true that any two tame knots are homotopic?

My understanding is that if the embeddings $f_0,f_1$ are tame knots then $H(t,\theta) = (1-t)f_0(\theta) + t f_1(\theta)$ is a homotopy between them, thus all tame knots are homotopic. Is this the ...
2
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1answer
110 views

Homotopy equivalence of torus and $\mathbb{C^2}$ without coordinate cross.

I have difficulties with one intermediate result. I know it is right, but it is not obviously for me. I'm trying to prove homotopy equivalence of $\mathbb{C^2}\setminus\{(a,b)\in\mathbb{C^2}\mid ...
1
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3answers
434 views

A question concerning fundamental groups and whether a map is null-homotopic.

Is it true that if $X$ and $Y$ are topological spaces, and $f:X \rightarrow Y$ is a continuous map and the induced group homomorphism $\pi_1(f):\pi_1(X) \rightarrow \pi_1(Y)$ is the trivial ...
2
votes
1answer
71 views

Proof for nonhomotopy

Let $Z =S^1 \times I$, and let $X = S^1 \times \{ 0 \}$ and $Y = S^1 \times \{1\}$ be two subspaces of $Z$. Let $f$ be a map from $Z$ to itself, sending $(z,t)$ to $(z \cdot e^{2 \pi it}, t)$, ...
3
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0answers
43 views

some help on the group of unknotted

Show that the group of the unknotted $K=\{(z_z,z_2)\in \mathbb{S^3} : |z_1|=1 \}$ is infinite cyclic. where $\mathbb{S^3}$ is to be considering as the unit vectors in $\mathbb{C^2}\cong \mathbb{R^4}$. ...
2
votes
1answer
107 views

Different possibilities defining $\eta^2$ in the ring of stable homotopy groups?

The Hopf fibration $\eta:S^3\to S^2$ represents the generator of the first stable homotopy group $\pi_1^s$. The direct sum of the stable homotopy groups $$ \pi_*^s=\bigoplus_{k\in\mathbb{Z}} \pi_k^s ...
1
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1answer
233 views

fundamental group of a torus

In finding the fundamental group of the torus by using the Van Kampen theorem, I was reading this: "Let us choose the covering consisting of the punctured torus U, an open disk V that covers the ...
3
votes
0answers
88 views

Relation between complex and real sphere

I want to understand relation between complex and real spheres. How to show? $S^1(\mathbb{C}) \approx \mathbb{R} \times S^1$ $S^3(\mathbb{C}) \approx \mathbb{R} \times S^3$ $\approx$ means homotopy ...
2
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1answer
181 views

reference request: Postnikov towers for non-simply-connected spaces

I've read that for a space $X$ which is connected but not necessarily simply-connected, we can no longer obtain the $n^{\rm th}$ layer $P_nX$ of the Postnikov tower for $X$ as the pullback of a ...
3
votes
1answer
173 views

Inducing a homotopy

I am trying to give proof of some fact regarding the model structure on the category of topological spaces. I think I am just a few steps away from a solution, but i am stumped on the following ...
3
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1answer
355 views

local homology group

The question is: $X\in \mathbb{R}^{n}$ is the subspace ${(x_{1},...,x_{n}\mid x_{n}\geq 0)}$, and let $Y$ is the subspace with $x_{n}=0$. let $x\in Y$, calculate the local homology of $X$ at $x$. ...
4
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0answers
79 views

Is there any relation between homotopy pushout and mapping cone?

Given two maps $f:A \to B$ and $g:A \to C$, we can have the homotopy push out square \begin{array}{rcl}A& \stackrel{f}{\rightarrow} &B\\ {\tiny g}\downarrow&& {\tiny b}\downarrow\\ ...
5
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1answer
126 views

spaces homotopic equivalent to $\mathbb R \backslash \mathbb Q$

I am looking for spaces who are homotopic equivalent to the irrational numbers! Can you help me, if you have seen some examples or references?
2
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0answers
44 views

Homotopy group of a path space.

I am studying now a homotopy theory and I have a question. Suppose that we have a connected space $X$, and a $\pi_{1}(X)$ action on $\pi_{2}(X)$ is trivial. In this case, is it true that ...
3
votes
2answers
253 views

When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
3
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1answer
57 views

Stable homotopy groups of $Sp(2n)$

What are the stable homotopy groups of the symplectic groups $Sp(2n)$? Is there a reference which contains a detailed treatment?
2
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75 views

Equivariant homotopy equivalence of based loop group

Consider a compact, connected, simply connected Lie group $G$ and consider $S^1$ as an additive group. Let $\Omega G = \{ \gamma: S^1 \to G: \gamma(0) = e_G\}$ be the corresponding based loop group of ...
2
votes
1answer
120 views

Showing $f$ extends to a continuous map on a disk if $f$ has degree $0$

How would you show that, if $f: S^1 \to S^1$ has degree $0$, (i.e. $\deg (f \circ \exp) = 0$ ) then $f$ extends to a continuous map $D \to S^1$?
3
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0answers
155 views

Proving that $f:S^1 \to S^1$ with closed degree $n$ is homotopic to the map $z \mapsto z^n$?

Suppose that $f:S^1 \to S^1$ is continuous and has closed degree $n$, how would you show that $f$ is homotopic to the map $z \mapsto z^n$? I know that by definition of closed degree, we have $\deg(f ...