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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19
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4answers
897 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 ...
19
votes
1answer
709 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) ...
18
votes
2answers
288 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 ...
17
votes
4answers
353 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 ...
15
votes
1answer
281 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
181 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 ...
14
votes
1answer
189 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
505 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
4answers
460 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 ...
13
votes
1answer
327 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 ...
12
votes
7answers
1k 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?
12
votes
3answers
357 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 ...
12
votes
2answers
551 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" ...
12
votes
2answers
526 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 ...
11
votes
1answer
421 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 ...
11
votes
2answers
384 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, ...
11
votes
3answers
383 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 ...
10
votes
4answers
543 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 ...
10
votes
1answer
495 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 ...
10
votes
3answers
315 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
2answers
455 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
1k 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 $\widetilde{Y}$, ...
10
votes
1answer
97 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 ...
10
votes
1answer
317 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. ...
10
votes
0answers
214 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 ...
9
votes
2answers
379 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. ...
9
votes
1answer
124 views

Are close maps homotopic?

Consider a smooth manifold $M = M^m$ and a smooth submanifold $N = N^n \subset M$. Suppose that two maps $f, g: M \to N$ are close to each other, in the sense that there exists $\epsilon > 0$ such ...
9
votes
3answers
375 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 ...
9
votes
2answers
684 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 ...
9
votes
1answer
582 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 ...
9
votes
1answer
194 views

Does smashing always increase the connectivity of a space?

Does smashing of a pointed CW complex $X$ with an arbitrary pointed CW complex $Y$ increase the connectivity? The connectivity of a pointed space $X$ is the maximal number $\operatorname{con}(X)$ ...
9
votes
1answer
210 views

Is the quotient map a homotopy equivalence?

It is well known that, if $A \subset X$ is a reasonable contractible subspace, then the quotient map $X \to X/A$ is a homotopy equivalence ("reasonable" means that the pair $(X,A)$ has the homotopy ...
9
votes
0answers
281 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 ...
8
votes
3answers
299 views

Spheres in different dimension are not homotopy equivalent

Is there a way to prove that $\textbf{S}^n$ and $\textbf{S}^m$ are not homotopy equivalent if $n\neq m$ without using the machinery of homology or higher homotopy groups?
8
votes
2answers
156 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 ...
8
votes
1answer
803 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 ...
8
votes
3answers
420 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 ...
8
votes
3answers
243 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 ...
8
votes
2answers
340 views

What is the point of a lift in topology?

I've just covered 'lifts' in topology and also homotopy lifting to a covering map but I'm struggling to understand the intuition behind lifts and essential the 'point' of them. Could someone please ...
8
votes
2answers
203 views

Why does the loopspace $\Omega$ induces a weak equivalence on mapping telescopes?

I am trying to answer an exercise of Hatcher's "Algebraic Topology", Section $4$.F, exercise $3$. Suppose we are given a sequence of pointed topological spaces : $Z_0\rightarrow Z_1\rightarrow Z_2 ...
8
votes
2answers
311 views

What is the homotopy colimit of the Cech nerve as a bi-simplical set?

Let $f:Y_\bullet\to X_\bullet$ be an epimorphism of simplicial sets and define the bi-simplicial set $$ F_{\bullet\bullet}=\ldots Y\times_X Y\times_X Y\underset{\to}{\underset{\to}{\to}}Y\times_X Y ...
8
votes
2answers
548 views

Path homotopy and separately continuous functions

Two paths $f$ and $f'$ mapping the interval $I=[0,1]$ into $X$, are said to be path homotopic if they have the same initial point $x_0$ and the same final point $x_1$, and if there is a continuous ...
8
votes
1answer
280 views

How does hocolim relate to Hom?

In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like ...
8
votes
1answer
485 views

Fundamental group of the Klein bottle, using action of a group?

Let $G$ be the group of homeomorphisms of $\mathbb{R^{2}}$ generated by the two elements: $g:(x,y)\rightarrow (x+1,y),h:(x,y)\rightarrow(-x,y+1)$. Then it is clear that this group is isomorphic to ...
8
votes
1answer
153 views

How does this map fail to be a homotopy?

Consider the map $H: [0,2\pi) \times I \rightarrow \mathbb R^2 \setminus \{0\}, h(x,t)=e^{itx}$ which maps the interval $[0,2\pi)$ to a circular arc whose length grows with $t$, from just one point to ...
8
votes
1answer
90 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 ...
8
votes
2answers
100 views

Can the rank of the homology group of an abstract simplicial complex be computed in polynomial time?

I want to write a function that does the following: Input: An integer $n$ A function $f$ that maps nonempty subsets of $\{1, \dots, n\}$ to "yes" or "no" such that (a) every singleton set gets ...
8
votes
1answer
169 views

Hopf fibration and homotopy of spheres

Let $$ S^3 \to S^7 \to S^4 $$ an the Hopf fibration. We con consider the induced sequence in homotopy $$ \pi_i(S^3) \to \pi_i(S^7) \to \pi_i(S^4) \to \pi_{i-1}(S^3) \to \pi_{i-1}(S^7) \to \cdots $$ ...
8
votes
1answer
80 views

Is every good enough space homotopy equivalent to a compact one?

Well, the question is more or less completely contained in the title. I found a partially related question on MO, namely this, and googling around reveals an amazing theorem of Browder, Levine, and ...
8
votes
0answers
243 views

The status of $\mathbb{R}$ in homotopy theory.

The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...