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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25
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4answers
756 views

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 ...
23
votes
4answers
2k views

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 ...
21
votes
1answer
1k views

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) ...
20
votes
3answers
896 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 ...
20
votes
2answers
452 views

Failure of excision for $\pi_2$

Would anyone know an example of failure of excision for 2nd homotopy groups? Specifically, I am looking for $A,B$ open in $X$ such that $X=A\cup B$ and $A\cap B$ is connected and $\pi_2(X,A)\ne \pi_2(...
18
votes
4answers
400 views

What are the ramifications of the fact that the first homotopy group can be non-commutative, whilst the higher homotopy groups can't be?

Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to non-...
18
votes
4answers
874 views

Useful fibrations

What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to S^{...
18
votes
2answers
908 views

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&...
17
votes
0answers
361 views

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 ...
16
votes
2answers
817 views

$X$ 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 ...
15
votes
3answers
3k views

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 ...
15
votes
1answer
207 views

Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in $\pi_3(S^2)...
15
votes
1answer
321 views

Geometric interpretation of the map $SO(4) \to SO(3)$

Let me first explain the background of my question. As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
14
votes
1answer
388 views

What is $\pi_{31}(S^2)$?

What is $\pi_{31}(S^2)$ - high homotopy group of the 2-sphere ? This question has a physics motivation: 1) There are relations between (2nd and 3rd) Hopf fibrations and (2- and 3-) qbits (quantum ...
14
votes
6answers
2k views

Real world uses of homotopy theory

I covered homotopy theory in a recent maths course. However I was never presented with any reasons as to why (or even if) it is useful. Is there any good examples of its use outside academia?
14
votes
1answer
207 views

Are locally homotopic functions homotopic?

Suppose we have two (smooth) functions $f,g:X\to Y$, where $X,Y$ are smooth (second-countable, Hausdorff) manifolds which are locally homotopic (that is, any point in $X$ has a neighbourhood $U$ such ...
13
votes
3answers
585 views

How to prove a manifold is simply connected?… using geometry

I was Looking at another questions title, and given the tag of DG, I thought it would read a little more like this one. Or at least that answers to this question would be answers to that question. ...
13
votes
1answer
1k views

What are $E_\infty$-rings?

I've been working with DG-algebras for the last year, and was able to obtain using them some nice commutative homological algebra results. However, I keep hearing about a (more general???) concept of ...
13
votes
1answer
524 views

What is $\pi_2(\mathbb{R}^2 - \mathbb{Q}^2)$?

I'm working through Hatcher book and done $\pi_1(\mathbb{R}^2 - \mathbb{Q}^2)$ is uncountable. It's easy to see that it's true as you can imagine only non trivial maps contract in the space. But, ...
13
votes
1answer
632 views

If $f\!: X\simeq Y$, then $X\!\cup_\varphi\!\mathbb{B}^k \simeq Y\!\cup_{f\circ\varphi}\!\mathbb{B}^k$.

How can I prove that if two spaces $X$ and $Y$ are homotopy equivalent, then the corresponding spaces obtained by gluing a $k$-cell are also equivalent? In detail, if $\varphi\!:\mathbb{S}^{k-1}\!\...
13
votes
2answers
706 views

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 ...
13
votes
1answer
2k views

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 $\widetilde{Y}$, ...
13
votes
1answer
240 views

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 ...
13
votes
2answers
259 views

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 ...
12
votes
2answers
306 views

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 ...
12
votes
3answers
132 views

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 ...
12
votes
2answers
990 views

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 ...
12
votes
2answers
196 views

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 $H : [0,1] \times X \to Y$...
12
votes
3answers
455 views

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)...
12
votes
1answer
206 views

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 ...
11
votes
4answers
935 views

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....
11
votes
3answers
728 views

Introductory books as preparation to read Voevodsky homotopy-theory (HoTT) book

I would like to read Voevodsky HoTT book. However, I lack a lot of the basics. I would need a few introductory books first that cover topics like groupoids, fibrations, W -types, Homotopy theory. ...
11
votes
1answer
1k views

Diffeomorphism group of the unit circle

I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and $\...
11
votes
3answers
607 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people speaking ...
11
votes
2answers
696 views

Understanding the definition of a cofibration

I have difficulties assimilating the notion of cofibration. I seem to get lost in the diagrams and technicalities (e.g. Hatcher/Bredon/Spanier). I'd be grateful if someone helped me out with that. ...
11
votes
1answer
965 views

Courses on Homotopy Theory

This autumn I'm considering taking an "advanced" reading course in Algebraic Topology, more specifically homotopy theory. I could extend this reading course over a year and wouldn't mind studying hard ...
11
votes
1answer
215 views

How to construct a quasi-category from a category with weak equivalences?

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a ...
11
votes
3answers
417 views

Showing continuity of partially defined map

There is a theorem in Note on Cofibrations by Arne Strøm. It says Let $A$ be a closed subspace of a topological space $X$. Then $(X,A)$ has the HEP if and only if there are (i) a neighborhood $U$...
11
votes
1answer
385 views

Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?

This question has been cross posted on MathOverflow with some very interesting answers and discussion. I'm currently writing a project on the braid groups and their analogues on closed surfaces. It'...
11
votes
0answers
805 views

Mapping cylinder is a CW complex

If you read the question entirely, it is not a duplicate. The first time I asked the question, I already gave the link to the similar question and explained why the answer is not satisfying. If you ...
10
votes
3answers
667 views

What's the point of spectra?

I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and ...
10
votes
1answer
607 views

Can Spectra be described as abelian group objects in the category of Spaces? (in some appropriate $\infty$-sense)

I'm not a topologist and I'm trying to understand a little bit about spectra. I've been told that spectra are the homotopical version of abelian groups. Can anyone expand on this point? Apparently ...
10
votes
3answers
508 views

Two Homotopy Colimit Questions

I have two questions about homotopy colimits: What can we say about $\operatorname{hocolim}_j\operatorname{colim}_i F(i,j)$? Iterated homotopy colimits commute, but what can we say when the inner ...
10
votes
3answers
490 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 ...
10
votes
2answers
660 views

Homotopy pushouts and induced maps

Suppose we are in a proper closed model category and consider a commutative square $$ \begin{array}{rcl} A&\to& B\\ \downarrow&&\downarrow\\ C&\to&D \end{array} $$ in its ...
10
votes
1answer
541 views

How much is cohomotopy dual to homotopy?

To what degree can we dualize theorems regarding homotopy into theorems about cohomotopy (or is there a good source that tries to do this)? For instance, is there some kind of Hurewicz theorem ...
10
votes
1answer
171 views

Gap between “fibration” and “fiber bundle”.

There are fibrations $E \rightarrow B$ which are not fiber bundles. Example: $E = [0,1]^2 / \text{middle vertical line segment}$ and $B=[0,1]$. In this example, $E$ has the homotopy type of a total ...
9
votes
3answers
765 views

Eckmann-Hilton and higher homotopy groups

How does the Eckmann-Hilton argument show that higher homotopy groups are commutative? I can easily follow the proof on Wikipedia, but I have no good mental picture of the higher homotopy groups, ...
9
votes
3answers
790 views

Is there any example of space not having the homotopy type of a CW-complex?

What is an example of space not having the homotopy type of a CW-complex? Is there any general method that can prove that the given space does not have the homotopy type of a CW-complex? (added) It ...
9
votes
2answers
206 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 ...