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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6
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4answers
690 views

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 ...
6
votes
3answers
658 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 ...
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 ...
2
votes
3answers
742 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 ...
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 $\widetilde{Y}$, ...
13
votes
1answer
323 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
382 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
1answer
494 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 ...
4
votes
3answers
427 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 ...
7
votes
3answers
396 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 $\mathbb{Z}_2$? consider this quotient on the disk representing the situation: $\mathbb{RP}^2$ (sorry ...
2
votes
2answers
193 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 ...
2
votes
3answers
83 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 ...
19
votes
4answers
890 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 ...
5
votes
2answers
195 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)) ...
6
votes
1answer
351 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 ...
19
votes
1answer
700 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) ...
8
votes
3answers
419 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 ...
4
votes
3answers
464 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 ...
14
votes
1answer
188 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 ...
10
votes
0answers
212 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
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 $\operatorname{con}(X)$ ...
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?
6
votes
1answer
166 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 ...
4
votes
2answers
112 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, ...
1
vote
1answer
85 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 ...
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 ...
6
votes
1answer
95 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 ...
6
votes
1answer
413 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
2answers
169 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 ...
5
votes
3answers
209 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 ...
4
votes
2answers
182 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
4
votes
1answer
1k 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. ...
3
votes
1answer
150 views

Homotopy problem for infinite dimensional topological space III

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : $X_{n}$ is a $n$-dimensional regular CW complex. ...
3
votes
2answers
481 views

Hatcher chapter 0 exercise.

Show that $f:X \rightarrow Y$ is a homotopy equivalence if there exist maps $g,h:Y \rightarrow X$ such that $fg \simeq \mathbb{1}$ and $hf \simeq \mathbb{1}$. Why isn't this trivial. Surely if f is a ...
2
votes
1answer
113 views

Solving an exercise in Milnor-stasheff's “characteristic classes”

I am trying to solve the following exercise (which is an exercise in Milnor-Stasheff's book). It basically says the following: Let $ M =S^n $ be the $n$-sphere and let $TM$ be its tangent ...
2
votes
1answer
84 views

Understanding the inclusion of sets in the open category of X $Op_X$ and what \{pt\} denotes

What I am trying to understand is what is going on with the inclusion of sets, as if I understand correctly they are the morphisms of the category of open sets on X: $Op_X$ is the category of open ...
2
votes
1answer
74 views

Should the first be the last by composition of paths?

Given two paths $f,g:\mathbb{I}\rightarrow X$ with $f\left(1\right)=g\left(0\right)$ there is a composite $f.g$ defined by $t\mapsto f\left(2t\right)$ if $2t\leq1$ and $t\mapsto g\left(2t-1\right)$ ...
2
votes
3answers
232 views

$\pi_1(S^n)=0$ for $n\geq2$

Hello friends of math :D I want to prove the result named in the heading. I have some hints but i can't imagine how wo work with this to conclude the result: Consider $S^n\subset\Bbb{R^{n+1}}$ (as ...
1
vote
1answer
200 views

What is a homotopy equivalence?

I have found the following problem. Let $A\subseteq X$ be a contractible space. Let $a_0\in A$. Is the embedding $X\setminus A\to X\setminus\{a_0\}$ a homotopy equivalence? I don't understand the ...
5
votes
3answers
617 views

Showing that the loopspace $\Omega S^{\infty}$ is homotopic to $S^{\infty}$.

Showing that the infinite dimensional sphere $S^{\infty}$ is contractible is rather easy by constructing an explicit contraction (Hatcher gives a nice one). I thought it might be a nice exercise to ...
5
votes
3answers
205 views

Covering map on the unit disk

Let $f: D^2 \rightarrow X$ be a covering map. I am trying to show that $f$ must in fact be a homeomorphism. To do so, I believe it suffices to show that $f$ is injective. Moreover, if only one point ...
5
votes
4answers
3k views

Fundamental group of the double torus

In May's "A Concise Course in Algebraic Topology" I am supposed to calculate the fundamental group of the double torus. Can this be done using van Kampen's theorem and the fact that for (based) spaces ...
4
votes
2answers
246 views

Is there a initial “bordism-like” homology theory?

Let $X$ be a space and $x_0\in X$ a base point. The Hurewicz map $$\pi_k(X,x_0)\longrightarrow H_k(X)$$ factors through oriented bordism $$\pi_k(X,x_0)\longrightarrow MO_k(X)\longrightarrow H_k(X).$$ ...
3
votes
0answers
31 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
3
votes
2answers
201 views

When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
3
votes
0answers
146 views

Why is well-pointedness necessary for $X\hookrightarrow M_f$ to be a pointed cofibration?

In Jeffrey Strom's Modern Classical Homotopy Theory on page $125$, it is stated that "Now we come to one of the crucial differences between the pointed and the unpointed categories. The mapping ...
3
votes
2answers
240 views

Kan fibrations and surjectivity

I have a basic question on the usual model structure on simplicial sets. What is the relation between being a Kan (trivial maybe ?) fibration and surjectivity ? Surjectivity here means either ...
2
votes
1answer
132 views

Intuition behind a retraction from the cylinder onto the mapping cylinder.

Please excuse me for including pictures, but I thought it was easier than trying to redraw them here. I am right now reading Strøm's book Modern Classical Homotopy Theory. I have encountered a ...
2
votes
3answers
223 views

A question about the proof of the fact that contractible spaces are simply connected

In greeberg's algebraic topology, the following fact is used in the proof that contractible spaces are simply connected without justification: Let $p:\mathbb{I}\rightarrow X$ be a continuous function ...