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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7
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Two CW complexes with isomorphic homotopy groups and homology, yet not homotopy equivalent

A standard example of two CW complexes which have isomorphic homotopy groups but are not homotopy equivalent is $ RP^2 \times S^3$ and $RP^3 \times S^2$. The easiest way to see that they are not ...
2
votes
3answers
1k views

the cone is contractible

Let $X$ be a topological space. I want to show that the cone $CX$ is contractible. Here we construct a deformation retraction from $CX$ to the tip point of the cone $$H_t: CX\to CX;\; (x,t')\mapsto ...
7
votes
3answers
815 views

How to compute homotopy classes of maps on the 2-torus?

Let $\mathbb T^2$ be the 2-Torus and let $X$ be a topological space. Is there any way of computing $[\mathbb T^2,X]$, the set of homotopy class of continuous maps $\mathbb T^2\to X$ if I know, for ...
10
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 $\widetilde{Y}$, ...
19
votes
2answers
354 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 ...
5
votes
3answers
583 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 ...
5
votes
1answer
2k views

Fundamental group of complement of $n$ lines through the origin in $\mathbb{R}^3$

Just a quick question to verify whether I'm right. Claim: The fundamental group of the complement of $n$ lines through the origin in $\mathbb{R}^3$ is $F_n$, the free group on $n$ generators. ...
1
vote
1answer
47 views

Question on showing a bijection between $\pi_1(X,x_0)$ and $[S^1, X]$ when X is path connected.

I am trying to do this question taken from Hatchers algebraic topology and I am struggling to understand the notation and the concepts. As far as I know $\pi_1(X,x_0)$ is the set of end point ...
12
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1answer
563 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 ...
13
votes
1answer
362 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 ...
11
votes
3answers
404 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 ...
9
votes
1answer
402 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 ...
7
votes
3answers
565 views

Homotopy equivalence of two different gluings of $B^n$ and an arbitrary space $X$

Let $f, g: S^{n-1} \to X$ be a pair of homotopic continuous maps. Let $X \cup_f B^n$ and $X \cup_g B^n$ be the respective adjunction spaces (pushouts of $B^n \hookleftarrow S^{n-1} \rightarrow X$). I ...
10
votes
3answers
1k views

An intuitive idea about fundamental group of $\mathbb{RP}^2$

Someone can explain me with an example, what is the meaning why $\pi(\mathbb{RP}^2,x_0)$ is $\mathbb{Z}_2$? We consider the real projective plane as a quotient of the disk. I don't receive and ...
8
votes
1answer
622 views

Continuous maps between compact manifolds are homotopic to smooth ones

If $M_1$ and $M_2$ are compact connected manifolds of dimension $n$, and $f$ is a continuous map from $M_1$ to $M_2$, f is homotopic to a smooth map from $M_1$ to $M_2$. Seems to be fairly basic, ...
5
votes
3answers
337 views

Cylinder object in the model category of chain complexes

Let $\text{Ch}⁺(R)$ be the category of non-negative chain complexes of $R$-modules where $R$ is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex ...
6
votes
2answers
319 views

double comb space is not contractible

I'm trying to show that the double comb space is not contractible. Intuitively I can see why this is true, but I can't seem to formalize a prof. I try to do the following: Let $D$ be the double ...
4
votes
3answers
137 views

Is such a map always null-homotopic?

Let $X,Y$ be CW-complexes with $X$ finite dimensional and $X = \bigcup_{n \in \Bbb N} X_n$ where the $X_n\subset X_{n+1}$ are finite sub-complexes of $X$. If $f: X \rightarrow Y$, with $f|_{X_n}$ ...
3
votes
1answer
86 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
2
votes
2answers
223 views

$A$ retract of $X$ and $X$ contractible implies $A$ contractible.

I have constructed the following proof of the statement and have some questions (a question) about the correctness of the proof: Statement: $A$ retract of $X$ and $X$ contractible implies $A$ ...
5
votes
2answers
1k views

Proof that retract of contractible space is contractible

I'm reading Hatcher and I'm doing exercise 9 on page 19. Can you tell me if my answer is correct? Exercise: Show that a retract of a contractible space is contractible. Proof: Let $X$ be a ...
4
votes
2answers
126 views

Lifting cohomology-killing maps through the 3-sphere

In his first answer to this question, Jason deVito claimed that a map $f:X\to S^2$ kills $H^2$ if and only if it factors through the Hopf fibration $\pi:S^3\to S^2$. What's the justification for this ...
2
votes
3answers
91 views

Symmetry of “is homotopic to” detail in the proof

Let $f,g:X\rightarrow Y$. If $f$ is homotopic to $g$ then $g$ is homotopic to $f$. Let $F:X\times I\rightarrow Y$ be a homotopy from $f$ to $g$ so $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x \in ...
0
votes
1answer
133 views

canonical map of a monoid to its classifying space

Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.] Every small category $C$ has a classifying space $BC$, defined as the geometric realization ...
21
votes
4answers
555 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 ...
21
votes
4answers
1k 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 ...
12
votes
1answer
677 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 ...
9
votes
1answer
447 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 ...
5
votes
2answers
217 views

Homotopy groups of some magnetic monopoles

This is a list of homotopy groups which I (as a physics researcher) encounter when studying magnetic monopole under certain configuration of gauge field profiles. \begin{gather} \pi_2(SU(2)/U(1)) ...
20
votes
1answer
862 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) ...
11
votes
1answer
167 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
3answers
522 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 ...
13
votes
0answers
276 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
1answer
426 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 ...
6
votes
2answers
234 views

A map which is trivial on homology but not on cohomology?

Is there a map $f:X\to Y$ of connected CW-complexes which induces the trivial map $f_*=0:H_i(X,\Bbb Z) \to H_i(Y,\Bbb Z)$ for all $i\ge 1$, but with the property that the induced map on cohomology ...
5
votes
3answers
762 views

A confusion about the fact that contractible spaces are simply connected

Question 1: Greenberg's Algebraic topology has a proof that contractible spaces are simply connected. In the middle of the proof, the book makes use of the following fact without justifying it ...
4
votes
1answer
247 views

Homology of mapping telescope

It is stated here http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf that if $X$ is an increasing union of the type $X=\bigcup_{i \in I}X_i$ (where $X_i \subset X_{i+1}$), then we have an ...
14
votes
1answer
198 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 ...
12
votes
2answers
125 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 ...
9
votes
1answer
245 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 $\operatorname{con}(X)$ ...
8
votes
3answers
364 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?
6
votes
1answer
246 views

Mathematical background for TQFT

I am physicist. I`ve started studying Topological QFT. What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? What books/articles could help form ...
6
votes
1answer
391 views

A question about the contractibility of the Sierpinski space

The two-point Sierpinski space is usually defined as follows: Let $X =\{x,y\}$ be the two-point space where the only open sets are $X, \varnothing, \{x\}$. I think from this it can be inferred that ...
4
votes
2answers
154 views

Examples of failure of excision for homotopy groups ($\pi_k(X, A)$ is not $\pi_k(X/A, *)$)

Let $A$ be a subcomplex of CW-complex $X$. The excision axiom for homology implies that $H_i(X, A)\cong H_i(X/A, *)$, and it is widely known that homotopy groups don't have this property. However, ...
3
votes
1answer
61 views

The Plate Trick and $SO(3)$

This is a followup of sorts to Plate trick demonstrating SO(3) not simply connected. . Suppose I do the ``plate trick'' to demonstrate the existence of an order-2 element in $\pi_1(SO(3))$. That is, ...
2
votes
1answer
103 views

Are these two definitions of $EG$ equivalent?

Let $G$ be a topological group with multiplication $\sigma:G\times G\to G$. The simplicial topological space $\mathcal{E}G$ defined by $$ \ldots \begin{array}{c}\to\\\to\\\to\\\to\end{array}G\times ...
2
votes
1answer
788 views

cauchy theorem over cycles homologous to zero

Definitions: $i)$ A cycle $\gamma$ is a finite sequence of continuous oriented closed paths in the complex plane. We denote $\gamma = (\gamma_1,...\gamma_n)$ where $\gamma_k$ are the closed paths of ...
9
votes
1answer
155 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 ...
6
votes
1answer
117 views

Is this morphism of spectra zero in the stable homotopy category?

Let $f\colon A\to B$ a morphism of spectra and suppose that both spectra $A$ and $B$ have only one non-zero stable homotopy group $\pi_n$, more precisely $$ \pi_k(A)=0=\pi_k(B) $$ for $k\neq n$. ...
6
votes
1answer
140 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...