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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5
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1answer
187 views

What is the $p$-primary component?

I got stuck when reading Differential topology 46 years later in the last section of the article ("Further details"). It is a summary of what is known about stable homotopy groups of spheres $\Pi_n$. ...
2
votes
1answer
156 views

Exotic spheres and homotopy groups: sanity check

There is a homomorphism $\Theta_n \to \Pi_n/J_n$ where $\Theta_n$ is used to denote the group of diffeomorphism classes of $n$-spheres (with connected sum), $\Pi_n$ denote the $n$-th stable homotopy ...
3
votes
0answers
504 views

Action of the Fundamental Group on Higher Homotopy Groups.

First: here are a couple links of which I am looking at. I try to add the relevant information (at least to my understanding) from them. ...
3
votes
1answer
57 views

$J(X)$ and exotic spheres.

I read that we can realize exotic sphere as the coker of $J$-homomorphism. SO we can consider the exotic sphere $S^7$ realized using an identificantion of $B^4 \times S^3$ (where I donote the four ...
1
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0answers
16 views

Homotopy equivalence of C-modules

A topological leftmodule $_CX$ is a topological functor $C \to Top$ for a Category $C$. A morphism $_CX \to _CY$ of leftmodules is a natural transformation of functors. Now such a morphism is called ...
1
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0answers
54 views

Cobordant to sphere or homotopy sphere

In Ranicki's notes (here) remark 6.19 distinguishes between cobordant to a sphere and cobordant to a homotopy sphere. Earlier in these notes right after example 1.6 he writes that homotopy equivalent ...
1
vote
1answer
76 views

Question about Lemma 7.7.1 from Hirschhorn's Model Categories and Their Localizations

The Lemma states the following. Let M be a model category. If $g:X\rightarrow Y$ is a weak equivalence between cofibrant objects in M, then there is a functorial factorization of $g$ as $g=ji$ where ...
5
votes
2answers
131 views

Path Connectedness in Van Kampen Theorem

On page 17 of this pdf, http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, the Van Kampen Theorem is proven. That is it is shown that for any covering of a space $X$ by a family of open ...
3
votes
1answer
74 views

$\pi_0$ in the long exact sequence of a fibration and quaternionic projective space

I am doing a past paper for an introductory course in algebraic topology. The question is Calculate the homology of the quaternionic projective space. What can you say about its homotopy groups? ...
1
vote
1answer
135 views

Homotopy Theory and extensions/liftings.

I found the statement: suppose that in the extension problem we have a map f': A -> E homotopic to f, and f' extends. Then it does not follow that f extends. Similarly, if the map g in the lifting ...
0
votes
0answers
105 views

Showing that $S^2\times S^3$ is not homotopic to $S^2 \vee S^4 \vee S^6$

In Switzer's algebraic topology book on page 285 he shows that $X = S^2 \times S^3$ is not homotopic to $Y = S^2 \vee S^4 \vee S^6$ by showing their cohomology rings are different. In doing so, he ...
4
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0answers
61 views

$J$-homomorphism and homotopy

We have Bott periodicity theorem for unitary group $U(n)$: $$ \pi_{i-1}^{s}(U) = \pi_{i-1}(U(m)) \simeq \pi_{i}(Gr_m(\mathbb{C}^{2m})) \simeq \pi_{i+1}(SU(2m)) \simeq \pi_{i+1}^{s}(U) .$$ So we can ...
4
votes
1answer
63 views

Is this theorem of Dold easy to prove with the modern language at hand?

There is a theorem of Dold (in Partitions of unity in the theory of fibrations) saying that if $X$ is a CW-complex and $Y\to X$, $Y'\to X$ are two fibrations connected by a map $f:Y\to Y'$ over $X$ ...
2
votes
1answer
91 views

Homotopy of spheres $\pi_{n+1}(S^n) \simeq \mathbb{Z}_2$

I have a problem: I have to prove that $$ \pi_{n+1}(S^n) \simeq \mathbb{Z}/2\mathbb{Z} $$ when $n \ge 3$. I know the Freudenthal suspension theorem and the Hopf fibration. Is there an easy method to ...
9
votes
1answer
234 views

Hopf fibration and homotopy of spheres

Let $$ S^3 \to S^7 \to S^4 $$ an the Hopf fibration. We con consider the induced sequence in homotopy $$ \pi_i(S^3) \to \pi_i(S^7) \to \pi_i(S^4) \to \pi_{i-1}(S^3) \to \pi_{i-1}(S^7) \to \cdots $$ ...
1
vote
1answer
84 views

What is metastable range?

Wikipedia states that In 1953 George W. Whitehead showed that there is a metastable range for the homotopy groups of spheres. What is the metastable range? I was unable to find the ...
2
votes
1answer
246 views

On framed cobordism

If two manifolds are h-cobordant then their homotopy groups agree. The notion of framed cobordism is supposedly a weaker notion. How much weaker than h-cobordism is it? What can be said about two ...
2
votes
1answer
84 views

Bott periodicity and homotopy groups of spheres

I studied Bott periodicity theorem for unitary group $U(n)$ and ortogonl group O$(n)$ using Milnor's book "Morse Theory". Is there a method, using this theorem, to calculate $\pi_{k}(S^{n})$? (For ...
3
votes
1answer
65 views

How to find stable homotopy group given the quotient group?

If $\Theta_n$ is the group of exotic spheres in dimension $n$ and $\mathrm{bP}_{n+1}$ is the group of spheres that bounds parallelizable $(n+1)$-manifolds, $\pi_n^S$ is the $n$th stable homotopy group ...
5
votes
1answer
300 views

If two maps induce the same homomorphism then they are homotopic

This is exercise 15.11(d) in C. Kosniowsky book A first course in algebraic topology: Prove that two continuous mappings $\varphi,\ \psi:X\to Y$, with $\varphi(x_0)=\psi(x_0)$ for some point ...
4
votes
1answer
101 views

Deformation retract of $\mathbb{R}^3$ minus a tetrahedron

Question from a past exam in an introductory course in algebraic topology. Call $X$ the space $\mathbb{R}^3$ minus the edges (one dimensional) of a tetrahedron. Does $X$ deformation retract onto a ...
5
votes
2answers
173 views

Framed Cobordism Classes of links in $\mathbb R^3$

We know that every link in $S^3$ is framed cobordant to the unknot with some framing. The idea is to study smooth homotopy classes of maps from $S^3$ to $S^2$. Actually in the title I have given ...
3
votes
0answers
76 views

How should I imagine the cup product on a topological space/manifold/variety

Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the ...
1
vote
2answers
108 views

Stronger two-space formulation of Hurewicz theorem about homotopy and homology groups

The following theorem of Hurewicz holds (let $\cdot$ be the one-point space and $n\!\geq\!2$): If $\pi_i(X)\cong\pi_i(\cdot)$ for $i\!<\!n$, then $H_i(X)\cong H_i(\cdot)$ for $i\!<\!n$ and ...
0
votes
1answer
84 views

On wedge of H-spaces

I've read that the wedge of two cyclic maps, $f\vee g$, does not need to be cyclic. Well, I understood the counter-example (see below) except by the fact that $S^1\vee S^1$ is not an H-space. Where ...
6
votes
2answers
313 views

double comb space is not contractible

I'm trying to show that the double comb space is not contractible. Intuitively I can see why this is true, but I can't seem to formalize a prof. I try to do the following: Let $D$ be the double ...
6
votes
1answer
169 views

Why are the integers appearing in lens spaces coprime?

I have a past paper question for a first course in algebraic topology, which asks one to calculate the first three homology and homotopy groups for the space $L_n$, defined as follows: Let ...
4
votes
2answers
199 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
3
votes
1answer
94 views

Visualizing the group operation in higher homotopy groups

I'm having trouble picturing the homotopy group operation of concatenation between two pointed spaces. For $n$-spheres, we have for $f,g: S^n \to X$ $$(f * g)(s_1,\ldots, s_n) = \begin{cases} ...
0
votes
1answer
56 views

Isomorphism of Covers

On page 26 of Peter May's A Concise Course on Algebraic Topology, it is claimed that given any two covers of a space $X$, $(E, p)$ and $(E', p')$ are isomorphic iff for any points $e \in E, e' \in E'$ ...
7
votes
2answers
578 views

Two spaces homotopy equivalent to eachother, attaching maps, Algebraic Topology.

I have a question regarding algebraic topology with which I was hoping someone could help me with. I've managed to show the following: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ ...
0
votes
0answers
80 views

Question about an isomorphism result in homotopy theory

I have another question regarding homotopy theory and winding numbers (or degrees). In Manton and Sutcliffe they state the following theorem: $\pi_2(G/H)=\pi_1(H)$ provided $G$ is a compact, ...
1
vote
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
votes
4answers
729 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
votes
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 ...
1
vote
1answer
105 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.
0
votes
1answer
150 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
votes
1answer
125 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
votes
3answers
359 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
votes
2answers
367 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
votes
2answers
213 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
votes
0answers
76 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
votes
0answers
69 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
votes
0answers
143 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 ...
1
vote
2answers
145 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
votes
1answer
361 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
votes
0answers
233 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
votes
3answers
1k 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
172 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
359 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 ...