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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99 views

Map induced by $O(n)\hookrightarrow U(n)$ on homotopy groups

There is an inclusion $O(n)\hookrightarrow U(n)$ which views an $n\times n$ orthogonal matrix as a unitary matrix. It is also a theorem, sometimes called Bott periodicity, that we have the following ...
1
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1answer
97 views

Simply connected subset of $\mathbb R^3$

Let $C$ be the closed unit cube in $\mathbb R^3$, and let $A$ be one face of the cube $C$ (say the face above and parallel to $xy$-plane). Let $U\subset\mathbb R^2$ be open and path-connected such ...
1
vote
0answers
36 views

Finding a homotopy map

Let $K=\mathbb R^2\times (-\infty,0)\subset \mathbb R^3$, and let $Q$ be an open connected subset of $\mathbb R^2$. Is the fundamental group $\pi_1(Q\times [0,1)\cup K)$ trivial? And is it possible ...
2
votes
1answer
64 views

Collapse of a subspace - Cofibration

Let $i:A \rightarrow X$ be a (closed ) cofibration (i.e a cofibration in the Strøm Model structure). For a subspace $B \subset A \subset X$, when is it true that $A/B \rightarrow X / B$ is a ...
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2answers
81 views

Proof that two homotopy inverses are homotopic

Let $X$ and $Y$ be topological spaces. A continuous mapping $f : X \to Y$ is said to be a homotopy equivalence if there exists $g : Y \to X$ continuous such that $g\circ f$ is homotopic to $id_{X}$ ...
3
votes
0answers
242 views

Want to show two maps are homotopic

I am trying to solve the following problem but so far I cannot do it. Let $X$ be a connected CW-space such that its homotopy group is 0 except for the fundamental group. Let $M$ be a closed manifold ...
7
votes
3answers
465 views

Is there any example of space not having the homotopy type of a CW-complex?

What is an example of space not having the homotopy type of a CW-complex? Is there any general method that can prove that the given space does not have the homotopy type of a CW-complex? (added) It ...
6
votes
1answer
355 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? ...
4
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0answers
72 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 $\pi_r(SO(q)$ ...
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1answer
48 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
167 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 ...
5
votes
3answers
781 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
votes
3answers
268 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
votes
2answers
134 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 ...
4
votes
2answers
254 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 ...
2
votes
1answer
270 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 ...
11
votes
3answers
405 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 ...
4
votes
1answer
151 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
40 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 ...
7
votes
1answer
125 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) ...
2
votes
1answer
279 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
59 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
84 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 $t\mapsto g\left(2t-1\right)$ ...
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2answers
162 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
61 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 ...
5
votes
1answer
161 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
86 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. ...
0
votes
0answers
28 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
votes
0answers
185 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
55 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
86 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 $n\in \mathbb N$?
3
votes
1answer
123 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
226 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
270 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
87 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, ...
10
votes
3answers
444 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 ...
3
votes
2answers
262 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
301 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 ...
0
votes
2answers
1k 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
59 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.
3
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1answer
65 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 $\hat{\mathbf Z}$?
1
vote
1answer
79 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
71 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
174 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
41 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 ...
0
votes
1answer
186 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
95 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
86 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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68 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
votes
0answers
91 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 ...