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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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, ...
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438 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
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2answers
261 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 $[0,1]\times \lbrace 0 \rbrace$ ...
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2answers
298 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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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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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.
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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
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1answer
78 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 ...
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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
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1answer
172 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
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1answer
40 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
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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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1answer
90 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 ...
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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 ...
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0answers
90 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 ...
2
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1answer
103 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 ...
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2answers
154 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
175 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 ...
3
votes
1answer
354 views

Continuous maps from $S^1 \to X$ equivalent conditions

The following are equivalent for a topological space X according to a problem in Hatcher. $1$)Every continuous map $S^1 \to X$ is homotopic to a constant map. $2$)Every continuous map $S^1 \to X$ ...
4
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1answer
138 views

Inclusion $O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $

How to show, that an inclusion of homogenious spaces $$O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $$ is homotopy equivalence? The big space is the space of complex structures on ...
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2answers
124 views

How are sets “trees with no symmetries”

In Vladimir Voevodsky's lecture Foundations of Mathematics and Homotopy Theory, in passing he mentions that the Zermelo-Fraenkel axioms describe objects which they call "sets", but which he would not ...
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0answers
108 views

simple proposition about homotopy group

Let $A$ be a topological subspace of $X$. Then we have exact sequences of homotopy groups for pair spaces: $$ \ldots \rightarrow \pi_n(A,x_0) \xrightarrow{i_{*}} \pi_n(X,x_0) \xrightarrow{j_{*}} ...
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1answer
109 views

On the commutativity of the relative homotopy groups

Can you explain to me why relative homotopy groups $\pi_{n}(X, A; x_0)$ are commutative for $n \geq 3$? It would be great if you will show me explicit homotopy.
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2answers
54 views

Which is the functor of this limit?

In the book "Simplicial Homotopy Theory" of Goerss and Jardine in the lemma 2.1 says that there is and isomophism \begin{align} X\cong&\varinjlim\Delta^n \\ &\Delta\rightarrow X \\ ...
2
votes
3answers
311 views

Prove that any two maps $f, g:X \to Y$ are homotopic

Let $X$ be a topological space and $Y $ be the upper open hemisphere. Prove that any two maps $f, g:X \to Y$ are homotopic. I have just started to learn algebraic topology and just learn the ...
1
vote
1answer
186 views

Loop spaces and filtered colimits

I have read many things now that lead me to believe that the loop space functor preserves filtered (and/or directed) colimits. Is this true? And can somebody give a (sketch of a) proof or point me in ...
1
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1answer
65 views

2-connected map between a connected sum and a gluing along one point.

I'm working with closed 3-dimensional manifolds $M_1$ and $M_2$. Consider their connected sum $M_1\#M_2$ and their gluing at only one point $M_1\vee M_2$. Intuitively I think that the map that ...
3
votes
1answer
540 views

Free crossed modules

A crossed module (over groups) $\mathcal{M} = (H,G,\partial)$ is a homomorphism $\partial\colon H \to G$ (called the boundary) together with an action $\alpha\colon (g,h) \mapsto {}^gh$ of $G$ on $H$ ...
4
votes
1answer
165 views

The two-sided simplicial bar construction is Reedy-cofibrant

Let $\mathcal{M}$ be a simplicial model category and let $\mathbb{C}$ be a small category. For a diagram $F : \mathbb{C} \to \mathcal{M}$ and a weight $G : \mathbb{C}^\mathrm{op} \to \mathbf{sSet}$, ...
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1answer
87 views

Homotopy extension property and contractibility

Definition. A pair $(X,A)$ of topological space $X$ and its subspace $A$ satisfies Borsuk property if for any topological space $Y$ and for any continuous map $f \colon X \to Y$ any homotopy $F_A ...
2
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0answers
45 views

Acyclic Hurewicz fibrations

If I understand correctly, the claim below follows from some well-known facts about the Quillen and Hurewicz model structures on the category of all topological spaces: If $p : X \to Y$ is a ...
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1answer
74 views

Showing homotopy of two paths if they are homotopic after a delay

Let $X$ be a topological space and let $\gamma, \delta : [0,1] \rightarrow X$ be two paths from $x$ to $y$. Now define $\widehat{\gamma}: [0,2] \rightarrow X$ by $$\widehat{\gamma}(t) = \begin{cases} ...
2
votes
1answer
310 views

Is there a definition of the transfer homomorphism (between cohomology of cover and base) without referring to chains?

Let $\pi: \tilde{X} \rightarrow X$ be an n-sheeted covering. Hatcher (section 3G), defines the transfer homomorphism, $\pi^*: H^k(\tilde{X}, Z) \rightarrow H^k(X, Z)$ on the chain level by sending ...
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1answer
89 views

Isomorphism relative homotopy groups

Suppose that we have $Y \subset X$ topological space such that $$ \pi_i(X,Y)=0$$ for all $0 \le i < k$. How can I prove that the homomorphism induced by inclusion $i: Y \hookrightarrow X$, say ...
8
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2answers
173 views

How to check if polylines can be untangled?

In a program I'm writing I need to be able to check whether a straight line between two points is homotopic to a polyline between them. For example in the below example the first one is equivalent to ...
2
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0answers
73 views

Hurewicz isomorphism in equivariant stable homotopy

Let $G$ be a finite group and let $X$ be a $G$-CW-complex. Denote by $\pi_{\ast}^G(X)$ the $G$-equivariant stable homotopy groups of $G$ and by $\mathrm{H}_{\ast}^G(X,A(-))$ the Bredon homology of $G$ ...
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1answer
50 views

Left and right homotopies have the same equivalence relations

Let $f, g \colon A \longrightarrow X$ be continuous functions of topological spaces. We say that $f,g$ are left-homotopic if there exists a continuous map $H \colon A \times I \longrightarrow X$ such ...
2
votes
1answer
55 views

Pointed space mapping clarification and isomorphisms between different ordered Homotopy groups.

If I am given a pointed pair of spaces $(X,A,x_{0})$ and define $P(X;x_{0},A) \subset X^{I}$ as the subspace given by the paths $\alpha$ in $X$ such that $\alpha(0) = x_{0} \text{ and } \alpha(1) \in ...
4
votes
2answers
110 views

Derivation and meaning of a long exact sequence of a Homotopy Groups for pairs of spaces

I read that there are long exact sequences of homotopy groups for each pair of pointed spaces $(X,A,x_{0})$. Now I know that for an exact sequence that, as the example below denotes $f \text{ and } ...
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0answers
42 views

Computing group of exotic spheres

In Levine on page 90 it is stated that the following sequence is exact $$ 0 \to bP^{n+1} \to \Theta^n \to Coker(J_n) $$ where $\Theta^n$ is the group of exotic spheres, $bP^{n+1}$ is the subgroup of ...
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0answers
87 views

What happens in dimension 125?

In Differential topology 46 years later (page 807, bottom of left column) Milnor states that for $n \neq 4, 125, 126$ if the order of the stable homotopy groups $|\Pi_n|$ is known then we can compute ...
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1answer
191 views

What is the $p$-primary component?

I got stuck when reading Differential topology 46 years later in the last section of the article ("Further details"). It is a summary of what is known about stable homotopy groups of spheres $\Pi_n$. ...
2
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1answer
158 views

Exotic spheres and homotopy groups: sanity check

There is a homomorphism $\Theta_n \to \Pi_n/J_n$ where $\Theta_n$ is used to denote the group of diffeomorphism classes of $n$-spheres (with connected sum), $\Pi_n$ denote the $n$-th stable homotopy ...
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519 views

Action of the Fundamental Group on Higher Homotopy Groups.

First: here are a couple links of which I am looking at. I try to add the relevant information (at least to my understanding) from them. ...
3
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1answer
57 views

$J(X)$ and exotic spheres.

I read that we can realize exotic sphere as the coker of $J$-homomorphism. SO we can consider the exotic sphere $S^7$ realized using an identificantion of $B^4 \times S^3$ (where I donote the four ...
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0answers
16 views

Homotopy equivalence of C-modules

A topological leftmodule $_CX$ is a topological functor $C \to Top$ for a Category $C$. A morphism $_CX \to _CY$ of leftmodules is a natural transformation of functors. Now such a morphism is called ...
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54 views

Cobordant to sphere or homotopy sphere

In Ranicki's notes (here) remark 6.19 distinguishes between cobordant to a sphere and cobordant to a homotopy sphere. Earlier in these notes right after example 1.6 he writes that homotopy equivalent ...
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1answer
77 views

Question about Lemma 7.7.1 from Hirschhorn's Model Categories and Their Localizations

The Lemma states the following. Let M be a model category. If $g:X\rightarrow Y$ is a weak equivalence between cofibrant objects in M, then there is a functorial factorization of $g$ as $g=ji$ where ...
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2answers
131 views

Path Connectedness in Van Kampen Theorem

On page 17 of this pdf, http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, the Van Kampen Theorem is proven. That is it is shown that for any covering of a space $X$ by a family of open ...