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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2
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0answers
30 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 $p-1$. ...
1
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0answers
19 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 ...
5
votes
2answers
103 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 ...
6
votes
0answers
149 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 ...
0
votes
1answer
53 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
votes
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 ...
7
votes
1answer
123 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 ...
0
votes
0answers
26 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 ...
3
votes
2answers
58 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 $U$ ...
4
votes
1answer
219 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 ...
0
votes
0answers
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 ...
3
votes
1answer
148 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 ...
0
votes
0answers
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 ...
1
vote
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
votes
1answer
37 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 ...
1
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0answers
33 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 ...
2
votes
1answer
51 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 ...
5
votes
1answer
82 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 ...
2
votes
1answer
61 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 ...
1
vote
1answer
28 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 ...
1
vote
1answer
30 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 ...
1
vote
1answer
114 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
vote
1answer
26 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
64 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
115 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 ...
-1
votes
1answer
43 views

Do 2 homotopic paths always have the same lenght? [closed]

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 ...
0
votes
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 ...
2
votes
1answer
56 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?
1
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0answers
39 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 ...
0
votes
0answers
48 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
votes
1answer
98 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
votes
1answer
54 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 ...
2
votes
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 ...
2
votes
0answers
59 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 $q \neq 1$?
0
votes
2answers
76 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
97 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 $f(x)$, ...
0
votes
0answers
45 views

Weak equivalence iff isomorphism in homotopy category?

I know that a weak equivalence becomes an isomorphism in the homotopy category but is the opposite direction true? Suppose we have a map $f: C\rightarrow D$ in a model category. If $f$ becomes an ...
0
votes
1answer
33 views

Nice Categories of Based Topological Spaces Are Not Cartesian Closed

One of the issues with the category of all topological spaces is that it lacks exponentiation, and is therefore not Cartesian closed. Suppose we take a "nice" category of spaces, like compactly ...
5
votes
2answers
229 views

Degree 1 maps from $\mathbb S^n$

Suppose that $f:S^n \to M$ is a map from the $n$-sphere to a simply-connected $n$-dimensional manifold that induces an isomorphism on top homology. I wonder if it's true that $f$ is already a homotopy ...
1
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0answers
33 views

Compression Criterion for $\pi_n(X,A,x_0)$. Why do we need homotopies $\text{rel} \ S^{n-1}$?

Recall the Compression Criterion: A map $f\colon [D^n, S^{n-1},s_0] \rightarrow [X, A,x_0]$ represents zero in $\pi_n(X, A,x_0)$ if and only if $f$ is homotopic relative $S^{n-1}$ to a map ...
3
votes
2answers
63 views

Homotopy classes of maps from projective plane to projective plane

Maybe I should think a bit longer, but are there more than two homotopy classes of maps $\mathbb{RP}^2\rightarrow \mathbb{RP^2}$? I am interested in both based and unbased maps.
8
votes
2answers
128 views

For $X$ contractible, deformation retract of $CX$ onto $X$.

Suppose $X$ is a topological space which is contractible. I want to show that the cone on $X$ deformation retracts onto $X$. My retraction $r: CX \to X$ is just the homotopy which contracts $X$ to a ...
1
vote
0answers
43 views

Path space of suspension

Let $X$ be a pointed homotopy type (of a CW complex) and let $G = \Omega \Sigma(X)$ be the loop space of the suspension. Let $P$ denote the homotopy pushout of the diagram $$ G \gets G \times X \to ...
0
votes
1answer
56 views

punctured real projective space

Let $\mathbb{R}P^m$ be the real projective space and $X=\mathbb{R}P^m\setminus \{*\}$ be the punctured space by removing one point. How to get the cohomology ring of $X$ with integer coefficient? Is ...
0
votes
1answer
22 views

Given a map $h: S^n \to \mathbb{R}^{n+1}-{0}$ such that $x\cdot h(x)=0$ for all $x\in S^n$, $f:X\to S^n$ is homotopic to $-f$.

If there is a continuous map $h: S^n \to \mathbb{R}^{n+1}-{0}$ such that $x\cdot h(x)=0$ for all $x\in S^n$ then for any continuous map $f:X\to S^n$, $f$ is homotopic to $-f$. My work: Given any ...
1
vote
0answers
39 views

Model category that doesn't admit functorial factorizations?

I guess it's a modern convention that model categories are typically required to have functorial factorizations. In the cofibrantly generated case, the factorizations constructed by the small object ...
1
vote
1answer
29 views

If $f(x) \neq g(x)$ for every $x ∈ S ^n$, then $g$ is homotopic to $a ◦ f$

Is this generalization that any map $f: S^n → S^n$ with no fixed points is homotopic to the antipodal map true? Let $f , g : S ^n → S^n$. Show that if $f(x) \neq g(x)$ for every $x ∈ S ^n$, then $g$ ...
0
votes
0answers
65 views

Example of a curve that is homologous to zero but is not homotopic to 0.

What is needed is an example of a curve for which the index of every point not on the trace is zero, but the curve is not homotopic to zero. One example is Pochhammer Contour, but is there any other ...
5
votes
2answers
89 views

Formula relating Euler characteristics $\chi(A)$, $\chi(X)$, $\chi(Y)$, $\chi(Y \cup_f X)$ when $X$ and $Y$ are finite.

This is a followup to my question here. Let $A$ be the subcomplex of a CW complex $X$, let $Y$ be a CW complex, let $f: A \to Y$ be a cellular map, and let $Y \cup_f X$ be the pushout of $f$ and the ...