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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Request for Details of Proposition 0.16 in Hatcher's Algebraic Topology

In Proposition 0.16 of Hatcher's Algebraic Topology, Hatcher claims that $ X^n \times I $ is obtained from $ (X^n \times \{0\}) \cup ((X^{n-1} \cup A^n) \times I) $ via attaching copies of $D^n \times ...
2
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2answers
55 views

A criterion to prove that a topological space is simple connected.

Let $\{U_i\}$ be an open covering of the space $X$ having the following properties: (a) There exist a point $x_0$ such that $x_0\in U_i$ for all $i$. (b) Each $U_i$ is simply-connected. (c) If ...
2
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1answer
28 views

CW approximation as an adjoint equivalence?

I have some intuitions that I want to make precise and accurate. I am very sure there are many mistakes in my understanding, but allow me to state it in the raw form as I sense it for now. Let ...
2
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1answer
36 views

Exercise 1.1.4 in Hatcher's Algebraic Topology, star-shaped

The following is Exercise 1.1.4 in Hatcher's Algebraic Topology: A subspace $X\subset \Bbb R^n$ is said to be star-shaped if there is a point $x_0 \in X$ such that, for each$x \in X$ , the line ...
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1answer
32 views

Comparison of direct limits and homotopy direct limits

Suppose we have a finite diagram in the category of topological spaces. What is the condition for homotopy equivalence of the natural map from homotopy direct limit of this diagram to it's direct ...
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1answer
28 views

punctured Mobius band in high dimension

Let $M^{n+1}$ be the non-trivial line bundle over sphere $S^n$. When $n=1$, $M^2$ is Mobius band and the punctured space $M^2\setminus *\simeq S^1$. How about $M^{n+1}\setminus *$ in general? Does ...
8
votes
1answer
158 views

Difference between homotopy equivalence and homeomorphism - dimensionality

(The most voted answer to) This question shows spaces of the same dimension can be homotopy equivalent but no homeomorphic. On the other hand "difference in dimension" is still a nice way to tell ...
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2answers
45 views

Topological group closed path

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$, $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let $f,h$ be closed paths ...
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1answer
20 views

homotopic closed paths in topological group

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$ $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let f,h be closed paths in ...
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0answers
17 views

Constant perimeter homotopy between a square and a circle

Is there a relatively simple homotopy between a square and a circle so that the perimeter of the curve is constant during the transformation?
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2answers
57 views

Show that the plane $\mathbb{R}^2 - \{(-1,0), (1,0)\}$ is homotopy equivalent to $C(1,0) \cup C(-1,0)$

I'm trying to find a deformation retract of the union of the two circles to $\mathbb{R}^2 - \{(-1,0), (1,0)\}$, any help is appreciated.
4
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0answers
59 views

Homology of $\mathbb{R}\setminus A_+$. [duplicate]

Let $A$ be the unit circle in the $xy$ plane in $3$-dimensional real space and let $A_+$ be a semicircle. I have to compute the homology of $\mathbb{R}^3\setminus A_+$. I was thinking that ...
0
votes
1answer
41 views

'Elementary' proof of $\tilde{X}$ is contractible iff $\pi_n(X) =0 \forall n \ge 2$.

I have studied algebraic topology, but have not studied $\pi_n(X)$ any further than $n=1$. Is there a proof of $\tilde{X}\cong 1 \Leftrightarrow \pi_n(X) =0 \forall n \ge 2$ that does not require ...
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0answers
37 views

In what sense is cohomotopy dual to homotopy?

I understand the duality in the case of homology and de Rham cohomology. Through integration a chain can be understood as a linear functional on differential forms and vice versa. Is there a way to ...
0
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0answers
23 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
80 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
16 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
vote
1answer
26 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
vote
1answer
48 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
68 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
votes
1answer
20 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
19 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
votes
1answer
54 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
votes
0answers
49 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 ...
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vote
0answers
31 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
27 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 ...
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votes
2answers
185 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
54 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 ...
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0answers
20 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
60 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 ...
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1answer
35 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
40 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
36 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
67 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
39 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
46 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] ...
1
vote
1answer
120 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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votes
0answers
27 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
32 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
38 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 ...
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votes
1answer
19 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
41 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
104 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
28 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 ...
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0answers
22 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?
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1answer
21 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 ...
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1answer
44 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 ...
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1answer
55 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 ...
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16 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 ...