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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35 views

Space deformation retracts to a point.

If space $X$ deformation retracts to a point $x\in X$, then for each open $U\in X$ containing $x$ there exists an open $V\in U$ again containing $x$ s.t. inclusion of $V$ into $U$ is nullhomotopic. ...
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1answer
44 views

Principal bundles with compact simply connected structure group over 2-manifolds

I'm reading Thomas Friedrich's "Dirac Operators in Riemannian Geometry," where the following is stated (in the Remark on page 42 before section 2.2 begins, if anyone is following along with the ...
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25 views

When is a fibration (canonically) a principal fibration over its group of automorphisms?

The question is inspired by the following observation: Let $p: X'\to X$ be a connected covering space where both spaces are suitably nice (say they are CW complexes), then $p: X' \to X$ is a principal ...
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45 views

commutivity of suspension and quotient space

Let $(A,A_0)$ be a $CW$-pair such that $A_0$ is a $CW$-subcomplex of $A$. Let $\Sigma$ be the suspension. Is $\Sigma(A/A_0)$ homotopy equivalent to $\Sigma A/\Sigma A_0$ or not?
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1answer
32 views

Direct Limit of Grassmannians

Let $X$ be a topological space and $G_n(\mathbb{C}^m)$ be the space of vector subspaces of $\mathbb{C}^m$ of codimension $n$. Let $G_n(\mathbb{C}^\infty):=\bigcup_{m=n}^{\infty}G_n(\mathbb{C}^m)$ ...
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1answer
43 views

Finding an explicit homotopy to prove inverses exist in fundamental group

The problem statement is: Suppose $X$ is a topological space with base point $x$. Let $\gamma_0:I\to X$ be the constant map $\gamma_0(s)=x, \forall s\in I$. Suppose $\gamma:I\to X$ is a continuous map ...
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1answer
50 views

Is there a rational homotopy equivalence between $\Omega S^3$ and infinite complex projective space?

The singular cohomology of the loop space $\Omega S^3$ of the 3-sphere is a divided power algebra $\Gamma_{\mathbb Z}[s]$ on one generator $s$ of degree 2, so the rational cohomology is a the ...
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1answer
76 views

Moore Spaces: explicit CW-complex for $M(\mathbb{Z}_m, n)$

Given an abelian group $G$ and an integer $n \ge 1$ we can construct a $CW$ complex such that $H_n(X) \cong G$ and $\tilde{H}_i(X)=0$ for all $i \neq n$. We call this $CW$ complex a Moore space and ...
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97 views

Does a group homomorphism up to homotopy induce a map between classifying spaces?

Let $H$ and $G$ be topological groups and denote by $BH$ and $BG$ their classifying spaces. If $$f\colon H\rightarrow G$$ is a continuous group homomorphism, we get an induced map of spaces ...
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1answer
74 views

Homotopy commutative diagrams and homotopy equivalent spaces

The question is fairly general. Suppose I have a homotopy commutative diagram of the form \begin{equation} \require{AMScd} \begin{CD} A @>{f}>> B\\ @V{h}VV @V{i}VV \\ C @>{g}>> D ...
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55 views

Curves Knotted in the Torus

I've been struggling with the following problem. I suspect it's actually not so hard, but I'm missing something relatively obvious about the proof. The attached image will help. Suppose $K$ and $L$ ...
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2answers
89 views

One point union, second homotopy group is not finitely generated?

Let $X$ be the one-point (wedge sum) union of the circle $S^1$ and the sphere $S^2$. What is the easiest way to see that the abelian group $\pi_2(X)$ is not finitely generated?
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1answer
64 views

Free Product of Groups with Presentations

There is a highly believable theorem: Let $A, B$ be disjoint sets of generators and let $F(A), F(B)$ be the corresponding free groups. Let $R_1 \subset F(A)$, $R_2 \subset F(B)$ be sets of relations ...
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27 views

Homotopy fixed points of connective K-theory

Let $ku$ be the $p$-completion of the connective complex K-theory spectrum. The group $\Bbb{Z}_p^\times\cong \Delta \times \Bbb{Z}_p$ acts on $ku$, where $\Delta$ is the cyclic group of order $p-1$. ...
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18 views

Conjugate paths have free homotopic circle representations?

Is this statement true? In a path connected space $X$, conjugate elements of $\pi_1(X,p)$ have free homotopic circle representations. This is related to my other question here. Basically, I am ...
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2answers
92 views

Where to learn about model categories?

A model category (introduced by Quillen in the sixties) is a category equipped with three distinguished classes of morphisms (weak equivalences, fibrations and cofibrations) satisfying some axioms ...
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140 views

When should one learn about $(\infty,1)$-categories?

I've been doing a lot of reading on homotopy theory. I'm very drawn to this subject as it seems to unify a lot of topology under simple principles. The problem seems to be that the deeper I go the ...
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1answer
48 views

Bijection induced by mapping loops to their circle representations

Let $X$ be a path connected space, and $p \in X$. I wish to show that the map $f \mapsto \tilde{f}$ induces a bijection between the conjugacy classes of $\pi(X,p)$ and $[\mathbb{S}^1: X]$, the free ...
5
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0answers
52 views

$M$ is homotopy equivalent to $S^n$. [duplicate]

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. How do I see that $M$ is ...
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1answer
118 views

Does it follow that $Y$ is homotopy equivalent to $S^{n-1}$?

Let $M$ be a compact connected $n$-manifold (without boundary), where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. Does it follow that $Y$ is ...
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24 views

Homotopy fixed points for product of groups

I want to show the following: Let $G$ and $H$ be groups, and let $X$ be a $G\times H$-space/spectrum. Then, $(X^{hH})^{hG}\simeq X^{h(G\times H)}$ with the obvious actions of $G$ and $H$ on $X$. I ...
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2answers
53 views

What is the fundamental group of the torus with two segments attached?

I'm trying to calculate the fundamental group of the following space: I've been thinking that I should apply Seifert - Van Kampen theorem but I haven't been able to choose some nice open sets $U$ ...
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1answer
211 views

Compact $n$-manifold has same integral cohomology as $S^n$?

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. Does $M$ have the same integral ...
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38 views

Group structure on $[X, \Omega^{2}(Y)]$ is abelian

I'm very begginer when it comes to algebraic topology. I have no idea how to prove (firstly see why it could be true or even start) this statement: Composition of loops gives the structure of group ...
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1answer
137 views

Existence of Closed Curves around Bounded Components

I am stuck on part of a complex analysis proof that I think needs more justification than given. It's pretty purely a topological statement, but it may be that complex-analytic techniques would be ...
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12 views

$\int_{\gamma(.,s)}f(z)dz$ is independet from s (homotopy lemma)

Let $U\subset\mathbb{C}$ open, $f:U\to\mathbb{C}$ continuous and complex differentiable. Let $\gamma:[a,b]\times [c,d]\to U$ be in $C^2([a,b]\times [c,d])$. And assume that one of the following ...
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1answer
23 views

How to prove that $\phi:G\rightarrow \pi_1(X/G,p(x_0))$ is a homomorphism of groups?

Let $G$ be a discontinuous group (this means that it acts discontinuously with finite stabilizers) of homeomorphisms of a simply connected, locally compact metric space $X$. Let $p:X\rightarrow X/G$ ...
2
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1answer
31 views

Why is a continuous injective map of a closed ball in $\mathbb{R}^2$ nulhomotopic?

In Munkres' Topology, the proof of theorem 62.3 goes as follows: Let $U$ be an open subset of $\mathbb{R}^2$ and let $f:U\rightarrow S^2$ be continuous and injective. Then let $B$ be any closed ball ...
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25 views

Geometric realization of a “simplicial space up to homotopy,” part two

This question is a follow-up, and my initial motivation for asking Is there a sensible way to form the geometric realization of a "simplicial space up to homotopy"? Given that the questions ...
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1answer
39 views

Is there a sensible way to form the geometric realization of a “simplicial space up to homotopy”?

I am new to simplicial methods, and have some naive questions. As such, the questions may be malformed and the terminology might not even be right. I assume these are answered in some reference and ...
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1answer
71 views

Example of a cogroup in $\mathsf{hTop}_{\bullet}$ which is not a suspension

Let $\mathsf{hTop}_{\bullet}$ denote the homotopy category of pointed topological spaces. More precisely, the objects are pointed topological spaces and for two objects $X$ and $Y$, the morphisms from ...
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1answer
56 views

How can I prove that the horn torus and $\mathbb{T} \cup D_1$ have the same homotopy type?

I'm trying to prove that the horn torus ($W$) defined by rotating the circumference $(x-1)^2+z^2=1, y=0$ around the z axis and $A=A_1 \cup A_2$ where $A_1$ is the torus obtained rotating ...
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1answer
23 views

Is the projection of two homotopic maps path homotopic?

Let $\alpha$ and $\beta$ be two homotopic paths in a path connected topological space $X$. Let $\alpha(0)=x_0$ $\alpha(1)=x_1$ $\beta(0)=x_2$ and $\beta(1)=x_3$. Let $p:X\rightarrow Y$ be a continous ...
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1answer
25 views

Is $\gamma$ homotopic to $g\circ\gamma$?

Let $X$ be a simply connected topological space. Let $x_0,x_1\in X$ and let $\gamma$ be a path in $X$ from $x_0$ to $x_1$. Let $g$ be a homeomorphism of $X$ with itself. Then $g\circ\gamma$ is a path ...
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1answer
86 views

Does having the same fundamental group imply that two spaces have the same homotopy type?

I have to prove that two different topological spaces $X,Y$ have the same homotopy type. I've been able to prove so far that $\pi_1(X)=\pi_1(Y)$ but I don't know if this is enough to say that $X$ and ...
1
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1answer
24 views

Equivalent statements to simply connectedness

Show that the following are equivalent for a path connected space $X$. (a) $X$ is simply connected. (with the definition that the fundamental group is trivial) (b) If two paths $\alpha$ and $\beta: ...
2
votes
0answers
57 views

Not every path connected space is contractible.

I wrote a proof that any path connected space is contractible which is completely wrong but i was not able to see what goes wrong in my proof: Let $X$ be a path connected space. Let $P$ be point in ...
2
votes
3answers
99 views

Torus cannot be embedded in $\mathbb R^2$

I've shown that $T^2$ can be embedded in $\mathbb R^3$. I just can't see why it can not be embedded in $\mathbb R^2$. Ideas: suppose $F: \mathbb S^1\times \mathbb S^1 \to \mathbb R^2$ is continuous ...
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1answer
42 views

Do 2 homotopic paths always have the same lenght?

We learned that two paths are homotopic if they can be continuosly transformed into each other by keeping their start and endpoints fixed. Does that always mean that two homotopic paths have the same ...
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0answers
13 views

Deformation retract, $\Pi_1 (RP^2/ \{p\}) = \Pi_1(S^1)$

Here What i try to do is by using deformation retract, compute its homotopy. First \begin{align} \Pi_1(\ddot{M}) = \Pi_1(S^1) \end{align} Where $\ddot{M}$ is Mobius. Second \begin{align} \Pi_1 ...
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1answer
44 views

Is there a long exact cofiber sequence for a homotopy pushout?

Let $f:Z \to X$, $g: Z \to Y$ and let $M(f,g)$ be the corresponding mapping cylinder. Does the homotopy pushout diagram induce a long cofiber sequence? If so what does it look like?
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38 views

Picard group and $\mathbb{A}^1$ homotopy

In Morel-Voevodsky $\mathbb{A}^1$-homotopy there is a famous theorem that states $\mathrm{Pic}(X)=[X,\mathbb{P}^{\infty}].$ Can you give me an example of computation of the Picard group of a ...
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0answers
44 views

Is the unique map $\emptyset\to \Delta^0$ really a horn inclusion? [duplicate]

Is it true that $\emptyset \to \Delta^0$ should be considered a horn inclusion? This seems to imply some terrible things, unless I'm missing something. In particular, it implies that a Kan fibration ...
6
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1answer
81 views

The ring of stable homotopy groups of spheres is not noetherian

On page 22 of this thesis, it is written that $\pi_*(\Bbb{S})$ is not noetherian. After a bit of thinking and looking online, I haven't found why this is true. A graded ring is noetherian if its ...
3
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1answer
51 views

Homotopy between cellular maps: an additional property

Let $f,g \colon X \to Y$ two cellular maps between (say) finite CW complexes such that $f\sim g$ via the homotopy $H \colon X \times I \to Y$. Are there any results that permits to modify the ...
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0answers
38 views

Are there any resolutions for $\mathbb{Z}$ over group algebra without topological <<model>>?

Let $G$ be group. Each cell partition of the universal cover of $K(G, 1)$ delivers a (projective?) resolution of $\mathbb{Z}$ over group algebra $\mathbb{Z}G$. Can one construct a pair of ...
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57 views

Construct a connected CW complex $K(\pi, 1)$ such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$? [closed]

Let $\pi$ be any group. How do I construct a connected CW complex $K(\pi, 1)$ such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$?
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2answers
75 views

Suppose $X$ is a space and $\pi_1(X, x)=\{e\}$, the trivial group. Show that there is a homotopy

The condition is that $\gamma_0,\gamma_1$ are paths in $X$ such that $\gamma_0(0)=\gamma_1(0)=x$ and $\gamma_0(1)=\gamma_1(1)=y$, then there is a homotopy $\{f_t\}_{t\in I}$ with ...
4
votes
1answer
84 views

Degree of maps and coverings

Following a recent question I had concerning degree $1$ maps from spheres, I came up with an assumption, which might either be very easily proven false, or, if not, still hasn't been answered. It goes ...
5
votes
0answers
77 views

Random sphere-valued fields

I would like to generate random functions from an $m$-sphere $S^m$ to an $n$-sphere $S^n$ that are not too wild, some kind of generalization of random Gaussian fields. More precisely, I want $f(x)$, ...