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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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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45 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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83 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 ...
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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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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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58 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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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
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1answer
85 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 ...
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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 $f(x)$, ...
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44 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 ...
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30 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 ...
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212 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 ...
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31 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 ...
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62 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.
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115 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 ...
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38 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 ...
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1answer
51 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 ...
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1answer
21 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 ...
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35 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 ...
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1answer
27 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 $g$ ...
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35 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 ...
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2answers
81 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 ...
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53 views

Reference request: homotopy groups of $\mathbb{C}P^n$ in terms of homotopy groups of spheres?

Could anyone work out/supply a reference to the computation of homotopy groups of complex projective space $\mathbb{C}P^n$ in terms of the homotopy groups of spheres?
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1answer
57 views

If $G$ is a group of isometries of $X$ then prove that $X/G$ and $X/\bar{G}$ are homotopically equvalent

Let $X$ be a connected, locally path connected, locally compact metric space. Let $G$ be a group of isometries of $X$ (that is a group of homeomorphisms of $X$ with itself that preserves distance). ...
5
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73 views

Is $Y \cup_f X$ a CW complex?

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 inclusion $A \to X$. My question is, is ...
5
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1answer
45 views

What does “cst” stand for in algebraic topology?

This is a question on notation present in this post. Define $E_p = \{ (y, \gamma) \in E \times B^{[0,1]} \mid p(y) = \gamma(0) \}$. There's a map (in fact a fibration) $q : E_p \to B$, $(y,\gamma) ...
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53 views

Intuition on homotopy group

I just started learning some algebraic topology and trying to make sense of the following identity, $$ \pi_m(S^n) = \begin{cases} \mathbb{Z} , & m = n \\ 0 , & m < n \, . \end{cases} $$ I ...
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1answer
41 views

What, exactly, is a vertical homotopy?

As the question title suggests, what exactly is a vertical homotopy? Googling has failed to provide any results as so far as a clear definition goes...
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170 views

Proving Cartan's magic formula using homotopy

On page 198 of Arnold's Mathematical Methods of Classical Mechanics, he asks the reader to prove Cartan's formula $$\tag{1}L_X=\mathrm{d}i_X+i_X\mathrm{d}$$ where $L_X$ is the Lie derivative wrt. $X$, ...
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77 views

Why is the Quillen model structure so painful to find?

Proving that the category of simplicial sets carries the Quillen model structure is undoubtedly difficult; the book by May and Ponto "A more concise course in algebraic topology" makes a considerable ...
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119 views

Why care about the $(\infty, 1)$-category of topological spaces?

While learning about homotopy in my Algebraic Topology course I (as someone who is at least aware of higher category theory) noticed that it's possible to define a notion of "homotopy between ...
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45 views

Relations between the 2-disc operad and fractals?

As you can see, as of late I opened a thread on n-disc operads: Clarification regarding little n-discs operads The thing is, those drawings there could somehow be construed in the real world as ...
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67 views

Iterating the suspension-loop adjunction in two different ways

Let $X$ be a sufficiently nice topological space (i.e. an object of a category of spaces where the reduced suspension-loops, $(\Sigma, \Omega)$, holds.) There are two directed systems of spaces one ...
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52 views

Generalization for Leray Hirsch theorem for Principal $G$-bundle

This is a general question: Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...
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1answer
25 views

Some explanation regarding a diagram of homotopy

I found this visualization regarding homotpy in wikipedia: https://commons.wikimedia.org/wiki/File:Homotopy_curves.png I would be very grateful if you could explai me all the abbreviations being ...
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1answer
45 views

Homotopy equivalences between graphs realizing isomorphisms on $\pi_1$

my question is how to proof this statement: Any isomorphism $φ:π_1(G_1,u_1) \to π_1(G_2,u_2)$ can be realized by a homotopy equivalence $f:(G_1,u_1) \to (G_2,u_2)$ If $φ:π_1(G_1,u_1) \to ...
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54 views

Limits in a Model Category

I've become interested in how the axioms of a model category have changed, since originally posed by Quillen; in particular, that Quillen only originally required finite limits and colimits, however ...
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3answers
814 views

Have there been (successful) attempts to use something other than spheres for homotopy groups?

Homotopy groups are famous invariants in algebraic topology. They have a myriad of wonderful properties: For $n \ge 1$, $\pi_n(X,*)$ is a group; for $n \ge 2$, this group is abelian. $\pi_n$ defines ...
4
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1answer
31 views

Topologizing the $I$ in $\prod_{i\in I} X_i$

I know two constructions for producing topologies on a function space: $I$ is a set, $(X_i)_{i\in I}$ is a collection of topological spaces; in this case the initial topology w.r.t the projections ...
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1answer
118 views

Is a path homotopy equivalence class a path component?

Let $X$ be a topological space and let $\Omega=\Omega (X;a,b)$ be a path space of $X$ from $a\in X$ to $b\in X$ with a compact-open topology. For any $f\in \Omega$, we can consider a homotopy ...
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1answer
47 views

Étale morphism has all its Deck transformation homotopic to identity

Is there an example that étale morphism (of degree $d,d<\infty$) $\pi: X\rightarrow Y$, s.t. all its Deck transformations homotopic to $Id_X$,except the trivial one, where $Y$ is general Enriques ...
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1answer
36 views

Asperical manifolds and their homotopy type

I read somewhere the following sentence: the homotopy type of an aspehrical manifold is determined by its fundamental group. Recall that $M$ is called asperical if $\pi_n(M)=0$ for $n>1$. By ...
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32 views

Well-definedness of first obstruction (to extending a map or homotopy of maps over the skeleta of a CW complex) lying in a nonzero group

Let $X,Y$ be CW complexes and suppose for simplicity that $Y$ is simply connected. There is an obstruction $O(f_n)$ to extending a map of CW complexes $f_n:X^n\to Y$ defined only on the $n$-skeleton ...
3
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1answer
144 views

Clarification regarding little n-discs operads

I am reading the wiki page on operad theory and I am trying to figure out how exactly those "Little something" operads work which are mentioned there. Specifically, I am having a hard time, despite ...
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1answer
100 views

Closed, simply connected manifolds which are not spheres

In 2 or 3 dimensions, every closed simply connected manifold is a sphere. In the smooth category, I suppose you could take exotic smooth structures to give examples of closed simply connected ...
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1answer
84 views

Is the homotopy category cartesian closed?

The homotopy category $\mathsf{hTop}$ (topological spaces localized at the weak homotopy equivalences) doesn't have many limits / colimits, but it does have products, computed on the point-set level. ...
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1answer
57 views

Relationship between homotopy equivalence and $k$-connectedness [closed]

Prove that if $X$ is $k$-connected and $X$ and $Y$ are homotopy equivalent, then $Y$ is $k$-connected. If $X$ and $Y$ are both $k$-connected, then so is $X\times Y$. Suppose that $X$ is a space that ...
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1answer
38 views

Not all retractions come from deformation retractions

On page $3$ of Hatcher it says "Not all retractions come from deformation retractions" and gives an example of a retract to a point and a deformation retract to a point. If a deformation retract to a ...
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1answer
41 views

How to show the failure of the homotopy extension property for the pair $I=[0,1]$ and $A=\left\{ 0,1,\frac{1}{2},\frac{1}{3}…\right\}$?

I'm facing difficulties regarding a statement which Hatcher makes in page 14 of his 'Algebraic Topology' book-that there's no continuous retraction from $I\times I\rightarrow ...
2
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1answer
36 views

Does a cofiber sequence of CW complexes induce a cofiber sequence of skeleta?

Suppose $X\to Y \to Z$ is a cofiber sequence of CW complexes. We can replace the maps with homotopic cellular maps $X_n\to Y_n \to Z_n$ taking the $n$-skeleton of $X$ to the $n$-skeleton of $Y$ and ...