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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24 views

$0$-th homotopy set of $G\times Z_2/H$

For a connected Lie group $G$ and its subgroup $H$, if $\pi_0(G/H) = 1$, is it true $\pi_0(\frac{G\times Z_2}{H}) = \{1,-1\}$ and $\pi_0(\frac{G\times Z_2}{H\times Z_2}) = 1$? I have to understand ...
2
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0answers
85 views

A(nother ignorant) question on phantom maps

My last question (Is such a map always null-homotopic?) is quite similar. If you do not care about my motivation for these questions, you can skip to the last line. I asked if some assumptions were ...
2
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1answer
19 views

map $GL^+(n)\rightarrow GL^+(n+1)$ homotopy equivalence

Let $GL^+(n)$ be the $n\times n$ real matrices with positiv determinant, and let $i\colon GL^+(n)\mapsto GL^+(n+1)$, $i(A)=\begin{pmatrix} 1 & 0 \\ 0 & A \end{pmatrix}$. Is $i$ a homotopy ...
1
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1answer
28 views

Homotopically equivalence for a 2 dimensional manifold

I have the following problem in my homework for algebraic topology: Does there exist a compact 2-dimensional manifold $M$ without boundary such that $M\times M$ is homotopically equivalent to $M$? I ...
1
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1answer
67 views

Give a direct simple proof that comb space cannot be strong deformation retracted to $(0,1)$?

Give a direct simple proof that comb space cannot be strong deformation retracted to $(0,1)$ where comb space is $\bigl(\bigcup_{n\in \mathbb{N}} \{\frac{1}{n}\}\times[0,1]\bigr)\cup ...
4
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0answers
55 views

representatives of $\pi_n(U(n))$

I know that $\pi_n(U(n)) \cong \mathbb{Z}$ for $n$ odd. I am looking for a description of an explicit map $S^n \rightarrow U(n)$ that represents the generator in this group. By precomposing with maps ...
7
votes
2answers
87 views

What's the “easiest” closed 3-manifold with a nonabelian fundamental group?

I'm looking for some easy compact, oriented 3-manifolds without boundary that have a nonabelian fundamental group. It needn't be perfect. "Easy" means that it has an easy Heegard diagram, say, one ...
2
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1answer
22 views

Is a map inducing surjections on all stable homotopy groups $\pi_k$ an epimorphism in the stable homotopy category?

Let $f\colon X\to Y$ be a morphism of spectra. The associated morphism in the stable homotopy category is an epimorphism, if and only if it fits into a distinguished triangle $$ X\xrightarrow{f} ...
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0answers
22 views

homotopy of simplicial maps between infinite complexes

I have such a problem. Let $K,L$ be simplicial complexes, where $K$ is connected, countable and locally finite. In particular we can express $$K=\bigcup_{i=0}^\infty K_i,$$ where $K_0$ is some full ...
2
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1answer
59 views

spaces with isomorphic homotopy groups, though not homotopy equivalent

Is there a way to prove that $S^2$ isn't homotopy equivalent to $S^3 \times \mathbb{CP^\infty}$ without homology theory? The tricky part is that all homotopy groups are isomorphic and I don't know ...
2
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55 views

What are the projective and the injective objects in the category of spectra?

What are the projective and the injective objects in the category of spectra (of simplicial sets)? Does the category of spectra have enough projectives and injectives? An object $P$ of a ...
2
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0answers
35 views

How can I show this is a homotopy.

Let $X$ be a topological space. I want to show that the following map is a homotopy. $h(t,s): [0,1] \times [0,1] \to X$, with: $$h(t,s) = \left\{ \begin{array}{c c} \alpha(\frac{s}{v(t)})& 0 \le s ...
1
vote
1answer
34 views

Help identifying two connecting homomorphisms

I'm running into a little trouble with the proof of Proposition 11.1 on page 64 of the first chapter of Goerss and Jardine's book Simplicial Homotopy Theory. Proposition 11.1. Suppose $X$ is a Kan ...
5
votes
2answers
241 views

A map which is trivial on homology but not on cohomology?

Is there a map $f:X\to Y$ of connected CW-complexes which induces the trivial map $f_*=0:H_i(X,\Bbb Z) \to H_i(Y,\Bbb Z)$ for all $i\ge 1$, but with the property that the induced map on cohomology ...
3
votes
0answers
68 views

The set of homotopy classes of maps is in bijection with homomorphisms between homotopy groups

My question comes from the fact that $\langle K(G,n),K(H,n)\rangle = \text{Hom}(G,H)$. The only proof of this fact that I know uses $\langle X,K(H,n)\rangle = H^n(X,H)$. However, this way of ...
2
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0answers
25 views

Two different notions of covering homotopy?

In Steenrod's The Topology of Fibre Bundles, there are two different notions of covering homotopy. One is furnished in Theorem 11.3: Let $\mathcal B,\mathcal B'$ be two bundles having the same ...
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2answers
80 views

Abstract homotopy invariance of homology

When topology is involved, we know (singular) homology is homotopy invariant. However, homology and homotopy can be discussed in much more general contexts. Living in ...
1
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1answer
38 views

Homotopy in $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ induced by homotopy in $\mathsf{Top}$?

I'm trying to put together the relationships between homotopy in $\mathsf{Top}$, chain homotopy, and homotopy in $\mathsf{Ch}_\bullet(R\mathsf{Mod})$. I more-or-less understand the connection between ...
2
votes
1answer
54 views

Is every point of a contractible space a deformation retract of that space?

Given a topological space $X$, I can show that the following are equivalent: X is contractible (that is, has the homotopy type of a point) There is some $x \in X$ such that $\{x\}$ is a deformation ...
2
votes
2answers
39 views

Induced homomorphisms on fundamental group

Define the map $f : S^{1} \times S^{1} \to S^{1}$ with $f(x,y) = xy$ and $g : S^{1} \times S^{1} \to S^{1} \times S^{1}$ with $g(x,y) = (xy,x)$ where $x, y \in \mathbb{C}$ on the unit circle $S^{1}$. ...
3
votes
2answers
99 views

$\Omega$ of a homotopy cofiber sequence

What is an example of a homotopy cofiber sequence $$ X\to Y\to Z $$ of well-pointed connected CW-complexes such that the associated sequence of loop spaces $$ \Omega X\to \Omega Y\to\Omega Z $$ is not ...
2
votes
1answer
48 views

calculating homology group of Real Projective Plane

I am reading an alternative way of calculating $H_1(\mathbb{R}P^2)$ not through the use of delta complexes and they have used the following fact: $H_1(M, \delta M) \cong H_1(\mathbb{R}P^2,D)$ where ...
3
votes
1answer
54 views

Homotopy and chain homotopy determine each other

In continuation of this previous question, I'm having problems with the following proposition from Kamps and Porter's Abstract Homotopy and Simple Homotopy Theory: Proposition (3.7). [page 210] ...
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1answer
133 views

Constructing model category from given category

Given a model category $\mathcal{M}$, Goerss and Hopkins constructed a subcategory (see Structured Ring Spectra, p. 160) $\mathbf{E}$ of $\mathcal{M}$ such that: If $X\in\mathbf{E}$ and $Y$ is ...
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0answers
28 views

Restricting a continuous function between simplicial complexes to $2$-skeleton

I have a continuous function $f : X \rightarrow X$, where $X$ is an $8$-dimensional simplicial complex. I want to homotope $f|_{\text{skel}^2(X)} : \text{skel}^2(X) \rightarrow X$ to a function $g : ...
0
votes
1answer
53 views

Zeroth homotopy group of the space $O(3)/H$

Short version of the question: Is $\pi_0(O(3)/C2) = Z_2$ as $\pi_0(O(3)) = Z_2$? Here $C_2$ is a cyclic group of order 2. Long version or the question: The zeroth homotopy group describes the ...
2
votes
0answers
51 views

Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
0
votes
1answer
30 views

Freely homotopic but not homotopic

I want to find a example of closed paths freely homotopic but not homotopic (I do not have many tools, like fundamental group, then has to be the simplest way possible). I thought at the following: ...
1
vote
1answer
45 views

A pushout of a homotopy equivalence along

Can anybody show me an example which prove that: A pushout of a homotopy equivalence along a arbitrary map (in Top) doesn't have to be a homotopy equivalence. I know that if we change "arbitrary ...
4
votes
3answers
143 views

Is such a map always null-homotopic?

Let $X,Y$ be CW-complexes with $X$ finite dimensional and $X = \bigcup_{n \in \Bbb N} X_n$ where the $X_n\subset X_{n+1}$ are finite sub-complexes of $X$. If $f: X \rightarrow Y$, with $f|_{X_n}$ ...
0
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0answers
34 views

Why use class multiplication in Homotopy groups?

This is a related to a physics question Why use class multiplication to describe topological entangling and merging?. In physics, the homotopy theory is used to describing topological defects in order ...
0
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0answers
29 views

Reference Request: James reduced product

I would like to quickly learn the basics of James reduced product (also called James construction). Anyone know some suitable material for beginners?
2
votes
1answer
30 views

Homotopically equivalent to Čech nerve?

I see a theorem without proof on Gelfand & Manin: Suppose $\mathfrak U=\{U_\alpha\}_\alpha$ is a locally finite open covering of the topological space $X$ such that each finite intersection ...
0
votes
1answer
54 views

Is the stable homotopy group of sphere a commutative ring? If not, are there easy examples?

Is the stable homotopy group of spheres a commutative ring? If not, are there easy examples? In the Adams spectral sequence converging to the stable homotopy group of spheres, it seems that any page ...
0
votes
1answer
58 views

Homotopy equivalence?

Can someone explaine what this means mathematicaly : "Let us denote by $h: X\rightarrow Y$ a homotopic equivalence map for which $h|_{Y}$ is the identity " Remark: $Y$ is include in $X$ Please ...
1
vote
0answers
19 views

use homology groups to obtain minimal cell structure

On Hatcher's book Algebraic Topology, Section 4.C Prop. 4C.1, for a simply-connected CW-complex $X$, if $H_*(X;\mathbb{Z})$ is known as a graded module over $\mathbb{Z}$, then the minimal cell ...
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1answer
61 views

what does Homotopy Tell?

What is the Homotopy geometrically?And what is path-homotopy? If two real valued functions are homotopic to each other, then what can we say about the geometry behind this? It need not be equal ...
1
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0answers
17 views

$S^1$ a p-local complex?

Let $p$ be a prime. Is $S^1$ a p-local CW-complex? Meaning, for any reduced homology theory $\overline{E}_*$, do we have $\overline{E}_*(S^1)=\overline{E}_*(S^1) \otimes_{\mathbb{Z}} ...
2
votes
0answers
50 views

Reduced homology and colimits

I would like to prove that colimits commute with reduced homology, i.e that if $ X = \operatorname{colim}\limits_{n \in \Bbb{N}} X_n$, then $$ \displaystyle\tilde H_k(X) = ...
0
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0answers
29 views

Serre fibration and CW-complexes

Suppose that $p:X\rightarrow E$ is a Serre fibration. I know the definition: then $p$ has the right lifting property with respect to all inclusions $I^n\rightarrow I^{n}\times I$. Now it seems that ...
0
votes
1answer
66 views

Which of these are homotopy equivalent? $S^1, \mathbb{R}, \{*\}$

Which of these spaces are homotopy equivalent: $S^1, \mathbb{R}, \{*\}$? I found a homotopy equivalence between $\mathbb{R}$ and the one point space $\{*\}$, so they are homotopy equivalent. The ...
1
vote
2answers
56 views

Proving that a continuous map is homotopic to the constant map

How can I prove that a continuous map $f : \mathbb{R}P^2 \to S^1$ is homotopic to the constant map? I know that in the projective space every point is a line but I do not get why the above has to be ...
0
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0answers
26 views

Is $EG \times EG \times Y$ homotopy equivalent to $EG \times Y$ through $G$-maps?

This is loosely related to my previous question Homotopy type vs. weak homotopy type, and repercussions for EG, and more closely related to the answer to the question Contractibility vs. ...
1
vote
1answer
31 views

Lifting properties of Serre fibrations

Suppose that $p:X\rightarrow B$ is a Serre fibration. I want to prove that $p$ has the right lifting property with respect to all maps of the form: $$S^{n-1}\times ...
2
votes
0answers
37 views

conjugation of Lie groups and homotopy group

Let $G$ be a Lie group. Let $\phi, \psi \in \pi_n(G)$. Consider $\theta \in \pi_n(G)$ defined by $\theta(x):= \phi(x)\psi(x)\phi(x)^{-1} \in G$, where we use multiplication and inversion in $G$ in ...
0
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1answer
39 views

Extending a Homotopy from $X \times [0,1]$ to $X \times \mathbb{R}$

In Jeffery M. Lee's Manifolds and Differential Geometry Exercise 1.77: For smooth manifolds $X$ and $Y$, show that if $f_0: X \rightarrow Y$ and $f_1: X \rightarrow Y$ are $C^r$ homotopic then ...
4
votes
0answers
113 views

The Seifert-Van Kampen theorem as a push-out

My question concerns the proof of the Seifert-Van Kampen theorem. The version of such a statement that interests me is the following. Let $X$ be a topological space, and $U, V\subseteq X$ two ...
1
vote
1answer
73 views

Adjoining an $(n+1)$-cell is an $n$-equivalence

Suppose $X$ is a topological space and $x_0 \in X$. Let $$ X' = X \cup e^{n+1} $$ be obtained from adding a $(n+1)$-cell (so $X'$ is the pushout of the map $\partial e^{n+1} \to e^{n+1}$ and the ...
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0answers
234 views

Homotopy classes of $*$-morphisms and unital $*$-morphisms

Let $A$ and $B$ be C*-algebras (non necessarily unital). A homotopy between two $*$-morphisms $\phi,\psi:A \to B$ is a $*$-morphism $A \to C([0,1],B)$ such that you can recover $\phi$ and $\psi$ from ...
0
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0answers
26 views

Fibration with CW-complex as basespace admits retraction

Suppose $f:E\rightarrow B$ has the right lifting property with respect to all CW-pairs $(X,A)$. Then $f$ is a Serre fibration and also a weak-homotopy equivalence. But want i want to prove is the ...