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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171 views

Where to get help with Homotopy type theory?

I'm trying to understand the Homotopy Type Theory book. I find myself completely lost in Chapter 2, especially when it starts using higher groupoids. What is the recommended background for this book? ...
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53 views

Will the pullback of homotopic maps give rise to isomorphic fibre bundles?

I know it's certainly right for the case of vector bundles, but what about fibre bundles?
3
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0answers
45 views

Understanding J homomorphism

I was trying to understand J homomorphism $J:\pi_r(SO(q)\rightarrow \pi_{r+q}(S^q)$from the Wikipedia page http://en.wikipedia.org/wiki/J-homomorphism. It's clear that an element of $\pi_r(SO(q)$ ...
0
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1answer
43 views

Does the structure group of S^n homotopic to O(n+1)?

It is easy to show that Diff(S^1) is homotopic to O(2), but in the case of n bigger than 1 things become really complicated, I cannot see the conclusion directly.
2
votes
1answer
105 views

Intuition behind a retraction from the cylinder onto the mapping cylinder.

Please excuse me for including pictures, but I thought it was easier than trying to redraw them here. I am right now reading Strøm's book Modern Classical Homotopy Theory. I have encountered a ...
4
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3answers
385 views

A confusion about the fact that contractible spaces are simply connected

Question 1: Greenberg's Algebraic topology has a proof that contractible spaces are simply connected. In the middle of the proof, the book makes use of the following fact without justifying it ...
2
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3answers
215 views

A question about the proof of the fact that contractible spaces are simply connected

In greeberg's algebraic topology, the following fact is used in the proof that contractible spaces are simply connected without justification: Let $p:\mathbb{I}\rightarrow X$ be a continuous function ...
2
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2answers
96 views

Question on homotopy

What is the relation between the definition of homotopy of two functions " a homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined ...
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2answers
74 views

Intuition behind definition of homotopic equivalence and distinction with homeomorphism

I am a physics student and have come across the definition of homotopic equivalence of two spaces as existence of two functions $f:X \to Y,g: Y \to X$ such that $g \circ f$ and $f \circ g$ are ...
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1answer
89 views

Meaning of Fundamental group of a graph

I am a computer science student working in graph algorithms. I am unable to understand what the fundamental group of a graph means. I have some intuition regarding the fundamental group of a ...
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3answers
368 views

Showing continuity of partially defined map

There is a theorem in Note on Cofibrations by Arne Strøm. It says Let $A$ be a closed subspace of a topological space $X$. Then $(X,A)$ has the HEP if and only if there are (i) a neighborhood ...
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49 views

Is this a correct alternative definition of 'cofibration'?

We work in category $\mathbf{Top}$. Let $i:A\rightarrow X$. It is a cofibration if: For every space $Y$, arrow $f:X\rightarrow Y$ and homotopy $F:A\times\mathbb{I}\rightarrow Y$ with $F_{0}=fi$ ...
3
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1answer
92 views

Bijection between homotopy classes and basepoint-preserving homotopy classes

$[X,Y]$ is the homotopy classes of maps from $X$ to $Y$ and $[X,Y]_0$ is the based homotopy classes of based maps. If $Y$ is path-connected and $\pi_1(Y)$ is abelian, then is the inclusion $$[X,Y]_0 ...
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2answers
26 views

Seeking 'simple' space with specified homotopy

I am looking for a 'named' space $S$ such that $\pi_1(S) = \mathbb{Z}_2$ and $\pi_n(S) = \star$ (the one-point group) for all $n\geq 2$. Commentary: I know that the projective plane fits the first ...
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0answers
69 views

Why is this space aspherical?

Let $X = Y \cup Z$ be a connected, path-connected Hausdorff space. Suppose that $Y$, $Z$, and $Y\cap Z$ are all connected, path-connected, and aspherical, and that the homomorphism $\pi_1(Y\cap Z) ...
1
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1answer
88 views

Difference between free homotopy and isotopy. Numer of non-isotopic curves.

I realize that whenever I think of two simple closed curves in a surface being isotopic I actually think of them as being freely homotopic (intuitively). I am really confused now. So I have the ...
1
vote
1answer
43 views

Is the continuous map between CW-complexes a cofibration?

If $f:A \rightarrow X$ is a continuous map between CW-complexes, then is $f$ necessarily a cofibration? I know that when $A$ is a subcomplex of $X$ and $f$ is the inclusion, the conclusion is true. ...
2
votes
1answer
70 views

Should the first be the last by composition of paths?

Given two paths $f,g:\mathbb{I}\rightarrow X$ with $f\left(1\right)=g\left(0\right)$ there is a composite $f.g$ defined by $t\mapsto f\left(2t\right)$ if $2t\leq1$ and $t\mapsto g\left(2t-1\right)$ ...
4
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2answers
143 views

Can abstract nonsense be helpful here?

Here a question for those among you, who teach Homotopics/Algebraic Topology at university. I encountered some questions that were in my view quite easier to solve in category hTop instead of Top ...
3
votes
1answer
46 views

Path homotopy in the plane

Let $C$ be a closed and simply connected subspace of the Euclidean plane $\mathbb{R}^2$. Suppose we have two simple paths in $C$, continuous functions $\alpha, \beta : [0,1] \to \partial C$, and ...
4
votes
1answer
83 views

Smooth torus eversion

I asked a vague question about torus eversion earlier, with no hard math, so while I'm at it, how about this one, which may involve hard math: "Everybody knows" that Stephen Smale showed us how to ...
2
votes
1answer
74 views

Show that two different embeddings of the figure-eight in the torus are not homotopic

Note, we can express the torus $|X.| \cong T$ as a square with edges denoted by $e$ and $f$, the diagonal by $g$, and faces $T_1$ and $T_2$, and a single vertex $v$, with appropriate identifications. ...
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24 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic) [closed]

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
3
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0answers
81 views

Homotopy of a CW complex

I have a CW complex constructed as follows: (The circle and the rectangles are 2-cells, different 1-cells are denoted by different colors, and there is one 0-cell). We can see it as gluing two Klein ...
3
votes
1answer
50 views

Let $\left(X,A\right)$ be a cofibered pair. Has pair $\left(X\times\left\{ t\right\} ,A\times\mathbb{I}\right)$ the gluing property?

Let $\left(X,A\right)$ be a cofibered pair. Then: $X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is a retract of $X\times\mathbb{I}$. Pair $\left(X\times\left\{ 0\right\} ...
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3answers
74 views

$\pi_1(X)\cong \mathbb Z_{p^n}$

If $p$ is a prime. Can one construct a space $X$ such that $\pi_1(X)\cong \mathbb Z_{p^n}$, for any $n\in \mathbb N$?
2
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0answers
74 views

Are cofibrant/fibrant replacements homotopically unique?

Let $\mathcal{M}$ be a model category in the old sense, i.e. with factorisations that are not necessarily functorial, and let $X$ be an object in $\mathcal{M}$. The category of cofibrant replacements ...
2
votes
2answers
178 views

$A$ retract of $X$ and $X$ contractible implies $A$ contractible.

I have constructed the following proof of the statement and have some questions (a question) about the correctness of the proof: Statement: $A$ retract of $X$ and $X$ contractible implies $A$ ...
5
votes
2answers
193 views

Who proved that existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ was sufficient for HEP?

It is well known that the existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is necessary to make $\left(X,A\right)$ a pair having the homotopy ...
2
votes
1answer
71 views

Calculating the homotopy groups of a complex

I'm trying to compute the homotopy groups of the complex obtained by gluing two Klein bottles along the generator that preserves orientation. It's not dificult to compute the fundamental group, ...
5
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344 views

Statement about Homotopy in Brown's “Topology & Groupoids”

I am trying to understand a statement in Brown's Topology and Groupoids, 7.2.5 (Corollary 1), page 270. Let's first have some preliminary remarks Let $X,Y$ be topological spaces. The track groupoid ...
2
votes
2answers
156 views

Show that $X$ deformation retracts to any point in the segment $[0,1]\times \lbrace 0 \rbrace$, but not to any other point.

I'm trying to solve a problem from Hatchers "Algebraic Topology" - exercise 0.6 (a): Let $X$ be the subspace of $\mathbb{R}^{2}$ consisting of horizontal segment $[0,1]\times \lbrace 0 \rbrace$ ...
0
votes
2answers
145 views

Is a path connected subspace of a simply connected space simply connected?

This is sort of a lemma I'm trying to prove for a larger proof. It seems intuitively true: if a space has trivial fundamental group, any two loops based at a point are homotopic. A subspace of such a ...
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2answers
356 views

Deformation retract and homotopy equivalence

If $A\subset X$ is a deformation retract of $X$. Are $X$ and $A$ homotopy equivalent?
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1answer
56 views

A particular Homotopy group.

How can we find $\pi_{d}(U(N+M)/U(N) \times U(M))$ ? Is there any way to visualize the target space ? I am specifically interested in the $d=1,2$ & $(N,M)=(2,2),(2,4)$ cases. Thanks.
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1answer
61 views

Is there a natural example of a $K(\hat{\mathbf Z}, 1)$?

Does there exist a nice classifying space for $\hat{\mathbf Z}$, the profinite completion of $\hat{\mathbf Z}$?
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91 views

What is $K (S^1,1)$?

It is known that if $G$ is a discrete group, then $BG= K(G,1)$. Originally I was interested in comparing classifying spaces of topological groups with the classifying spaces of the same groups ...
1
vote
1answer
49 views

fundamental group of punctuated plane

Let $X$ be the plane punctuated at the origin. Let $C$ be the unit circle, with each point being identified by an angle between $0$ and $2\pi$. $f$ is a function $[0,1] \rightarrow C$ so that ...
3
votes
0answers
47 views

Does the $\mathcal{E}G$-construction arise form a comonad?

Let $G$ be a simplicial group and consider the adjunction $$ Fr:sSets\rightleftarrows G-sSets:U $$ where the left adjoint $Fr$ maps a simplicial set $X$ to the free $G$-simplicial set $G\times X$ and ...
3
votes
1answer
139 views

Homotopy problem for infinite dimensional topological space III

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : $X_{n}$ is a $n$-dimensional regular CW complex. ...
0
votes
1answer
35 views

Deformation retract needs to be smooth?

So I am not quite sure that why none of these three is a deformation retract - is that because of the corners? But I don't remember deformation retract rely on smooth criteria, instead, on continuous ...
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0answers
34 views

Equivalence and complexes homotopically-minimal

Let $A$ and $B$ be two finite-dimensional algebras over a field $k$ and $G\colon \mathcal{K}^{-}(\mathcal{P}_A) \to \mathcal{K}^{-}(\mathcal{P}_B)$ be an (triangulated) equivalence. By [Krause-05], a ...
0
votes
1answer
93 views

Equivalence of path-connected CW-complexes and CW-complexes with one 0-cell

Proposition Any path-connected CW-complex is homotopy equivalent to a CW-complex with precisely one 0-cell. Proof (Sketch) Let $X$ be a path-connected CW-complex, so $sk_1(X)$ is a connected graph. ...
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vote
1answer
47 views

Why is an $E_\infty$-operad a kind of ''strictification'' for a non-commutative operation?

A topological space $X$ which is an algebra over an $E_\infty$-operad $E$ consists of a sequence of maps $$ \mu_n':E(n)\times X^n\to X $$ with compatibility conditions. The spaces $E(n)$ are ...
1
vote
1answer
48 views

Is a weakly contractible connected metric space, uniquely geodesic?

A topological space is weakly contractible if all the homotopy groups are trivial. It's connected if it's not the union of two disjoint nonempty open sets. A metric space $(X,d)$ is uniquely geodesic ...
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52 views

Does an equivariant weak equivalence induce weak equivalences on all orbits?

This question arose from another, which was not well formulated and completely answered by this MO thread as pointed out by the user roman. Let $G$ be a discrete group, $X$ and $Y$ be $G$-spaces (and ...
5
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0answers
84 views

$\pi_2(G)$ for $G$ a Lie group. [duplicate]

It is well known that $\pi_2(G)$ is trivial for any Lie-group $G$. Is there an elementary proof of this, say, that can be understood with minimal homotopy theory? Also, who gave the first proof of ...
1
vote
1answer
77 views

Are these two definitions of $EG$ equivalent?

Let $G$ be a topological group with multiplication $\sigma:G\times G\to G$. The simplicial topological space $\mathcal{E}G$ defined by $$ \ldots \begin{array}{c}\to\\\to\\\to\\\to\end{array}G\times ...
4
votes
2answers
102 views

Examples of failure of excision for homotopy groups ($\pi_k(X, A)$ is not $\pi_k(X/A, *)$)

Let $A$ be a subcomplex of CW-complex $X$. The excision axiom for homology implies that $H_i(X, A)\cong H_i(X/A, *)$, and it is widely known that homotopy groups don't have this property. However, ...
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2answers
151 views

Computing $\pi_3(\mathrm{Gr}_2(\mathbb{R}^4))$

How can one go about computing the 3rd homotopy group of the Grassmannian manifold of 2-planes through the origin in $\mathbb{R}^4$? I don't want to be more general in the question, because: 1) I ...