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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Fundamental group of quotient disk

Consider the disk $D^{2}$ in $\mathbb{R}^{2}$. By taking out two disjoint, smaller disks within $D^{2}$, we obtain a disk of genus 2. Now consider identifying the boundaries of the two deleted circles ...
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47 views

Computing fundamental groups of products

Let $X$ be a connected graph and $S^{1}$ the usual circle and consider the product $X \times S^{1}$. How would one compute the fundamental group $\pi_{1}(X \times S^{1})$ in this case? I know that one ...
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45 views

A short exact sequence of chain complexes with null-homotopic chain maps

Problem Suppose $0\to K'\xrightarrow iK\xrightarrow pK''\to 0$ is an exact sequence of chain complexes of modules over $R$, say. If chain maps $i,p$ are null-homotopic, then $K$ is contractible. ...
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Homotopic properties of the Spin group from geometric algebra

There are two possible ways to define the $\mathrm{Spin}(n)$ group of Euclidean $n$-space from $\mathrm{Pin}(n)$. First is that $\mathrm{Spin}(n)$ is the identity component of $\mathrm{Pin}(n)$. ...
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72 views

what is the two sheeted covering space of a sphere with a diameter?

I have calculated the fundamental group of sphere with a diameter using Van-Kampen theorem, which is $Z$. So corresponding to subgroup $2Z$ there exist a two sheeted connected covering space. So ...
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Proof of compression criterion (iff condition for representing zero in relative homotopy group)

I'm trying to prove (and understand) the compression criterion which states that a function $f\colon(I^n,\partial I^n,J^n)\to (X,A,x_0)$ represent zero in the relative homotopy group $\pi_n(X,A,x_0)$ ...
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47 views

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 ...
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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 ...
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1answer
34 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 ...
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1answer
43 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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33 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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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 ...
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186 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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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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29 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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18 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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60 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.
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60 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 ...
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1answer
42 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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45 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 ...
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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 ...
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84 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 ...
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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 ...
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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 ...
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1answer
62 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 ...
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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 ...
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2answers
75 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 ...
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1answer
21 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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20 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 ...
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1answer
57 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 ...
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52 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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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 ...
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1answer
31 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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219 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 ...
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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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22 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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74 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
37 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 ...
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1answer
47 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 ...
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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 $S^{1}$. ...
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86 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 ...
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1answer
45 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 ...
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1answer
47 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
126 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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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 : ...
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1answer
40 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 ...
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42 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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1answer
23 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: ...
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1answer
44 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 ...
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3answers
126 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 $f|_{X_n}$ ...