# Tagged Questions

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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### Are there nontrivial continuous maps between complex projective spaces?

Are there maps $f: \Bbb{CP}^n \rightarrow \Bbb{CP}^m$, with $n>m$, that are not null-homotopic? In particular, is there some non-null-homotopic map $\Bbb{CP}^n \rightarrow S^2$ for $n>1$? Can we ...
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### Homotopy groups of $S^2$

I'd like to understand higher homotopy groups better and I guess there's no simpler way than understanding them for as simple spaces as possible; therefore $S^2$. My question essentially has two ...
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### Brave New Number Theory

I suppose this is an extremely general question, so I apologize, and perhaps it should be deleted. On the other hand it's an awesome question. Is it clear exactly how much (assumedly algebraic) ...
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### 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 ...
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### Introductory book for homotopical algebra

I'm interested in learning homotopical algebra (by which I mean: model categories, simplicial methods, etc.) However, I've been unable to make heads or tails of any of the "standards" (Jardine&...
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### Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
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### $X$ and $Y$ are homotopy equivalent $\Leftrightarrow$ $\exists Z:$ $X,Y$ are strong deformation retracts of $Z$

This question is very similar to this one, but the difference is that I'm asking for a strong deformation retraction. Notation/Definitions: (all maps are by definition continuous) A homotopy between ...
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### An intuitive idea about fundamental group of $\mathbb{RP}^2$

Someone can explain me with an example, what is the meaning of $\pi(\mathbb{RP}^2,x_0) \cong \mathbb{Z}_2$? We consider the real projective plane as a quotient of the disk. I don't receive and ...
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### What is the best path to learn Category theory and Type theory?

I have little background in Programming in functional languages and wanted to learn type theory. I started with taking Homotopy type theory class Online videos of Robert Harper. I thought rather than ...
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### Homotopy equivalence of universal cover

As part of am exam question (Q21F here), I'm trying to prove that if $X$ and $Y$ are path-connected, locally path-connected spaces with universal covers $\widetilde{X}$ and $\widetilde{Y}$, ...
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### Is a bijective homotopy equivalence with bijective homotopy inverse a homeomorphism?

I've been thinking about this for a while, but didn't get very far. Maybe someone here can say something about it. I know of an example of two spaces $X, Y$ with continuous bijections in both ...
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### Is wedge sum for finite CW complexes cancellative in the homotopy category?

Let $X,Y,Z$ be finite pointed CW complexes. Is it possible that $X\vee Z$ and $Y\vee Z$ are homotopy equivalent, but $X$ and $Y$ are not? Remark 1: Without the finiteness assumption on $Z$, there are ...
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### Can Path Connectedness be Defined without Using the Unit Interval?

Can path connectedness be defined without using the unit interval or more generally the real numbers? I.e., do we need Dedekind cuts or Cauchy convergence equivalence classes of the rational ...
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### Homotopy equivalence of a space with the sphere

I have some trouble with the following problem. A space $X$ is obtained by gluing two $2$-cells to a circle $S^1$ using maps winding $2$-times and $3$-times around $S^1$. Show that $X$ is homotopy ...
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### Why is Top a model category?

Recall that a model category is a complete and cocomplete category with classes of morphisms called cofibrations, fibrations, and weak equivalences. These are closed under composition and satisfy ...
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### Is $\mathbb{R}^n$ properly homotopy equivalent to $\mathbb{R}^m$ if $n \neq m$?

$\DeclareMathOperator{\id}{id} \newcommand{\R}{\mathbb{R}}$ If $f,g : X \to Y$ are two maps (all maps considered are continuous here), a homotopy between $f$ and $g$ is a map $H : [0,1] \times X \to Y$...
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### What are the best known results for the stable homotopy groups of spheres?

There are a number of proposed ways to compute the stable homotopy groups of spheres. One can rather peculiarly consider stable (co)homotopy of an Eilenberg Maclane spectrum as a generalised (co)...
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### Based space, commuting in diagram up to homotopy, dual Barratt-Puppe sequence.

For a based map $f : X \to Y$, define the "homotopy fiber" $Ff$ to be$$Fd = X \times_f PY = \{(x, \chi) : f(x) = \chi(1)\} \subset X \times PY.$$Equivalently, $Ff$ is the pullback displayed in the ...
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### What are two continuous maps from $S^1$ to $S^1$ which are not homotopic?

This is an exam question I encountered while studying for my exam for our topology course: Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as well....