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 ...

learn more… | top users | synonyms

3
votes
0answers
36 views

Obstruction for extending a map along a CW-inclusion: sufficient conditions?

I'm interested in a clarification of the highlighted comment taken from "A User's Guide to Algebraic Topology" by Dodson & Parker. In order to make the setting clear, I uploaded definition ...
1
vote
1answer
72 views

Hatcher's proof of the van Kampen Theorem (injectivity of $\Phi$ – unique factorizations of $[f]$)

I am trying to understand the details of Allen Hatcher's proof of the Seifert–van Kampen theorem (page 44-6 of Algebraic Topology). My question is regarding the same part of the proof mentioned in ...
0
votes
0answers
58 views

Homotopy between $x\mapsto x$ and $x\mapsto x/|x|$

1 Let $a,b:\mathbb{C}^\times\rightarrow\mathbb{C}^\times$ with $$a(x)=x,$$ $$b(x)=x/|x|.$$ 2 Let $c,d:\mathbb{C}^\times\rightarrow\mathbb{C}^\times$ with $$c(x)=\bar{x},$$ $$d(x)=1/x.$$ 1 How do I ...
0
votes
0answers
33 views

Two self maps $f,g:S^n\to S^n$ are homotopic if there is no $x\in S^n$ with $f(x)=-g(x)$

What I want to prove: Let $f,g:S^n\to S^n$ be continuous. If there is no point $x\in S^n$ with $f(x)=-g(x)$ then $f\sim g$, then . I am using this as a lemma to prove a slightly bigger result: ...
4
votes
1answer
38 views

Which surface is homotopy equivalent to $\Bbb{R}^4$ minus the planes $x=y=0$, $z=w=0$?

In completing an exercise I have shown that $\Bbb{R}^3$ minus the axes $x=0$, $y=0$, and $z=0$ is homotopic equivalent to the cube graph $Q_3$. To visualize this, $\Bbb{R}^3-0$ is homotopy equivalent ...
0
votes
1answer
33 views

Continuous functions between topological spaces and their homotopy equivalence relations

Let $A,B,C$ be topological spaces and $\alpha,\alpha':A\rightarrow B$ continuous and $\beta,\beta':B\rightarrow C$ be continuous. Let $\sim$ be the homotopy relation (which I know/can use to be an ...
1
vote
1answer
46 views

Every “star-shaped” set is simply connected

This is based off of a question in Serge Lang's Complex Analysis book, though much harder than the version of the question in the text. Call a set $S\subseteq\Bbb{C}$ star-shaped if there exists a ...
1
vote
0answers
33 views

Which of these graphs are homotopic but not homeomorphic?

I am struggling with the 'homotopic' part of the question Which of these are homotopic but not homeomorphic? The number of vertices of degree $\neq 2$ is a topological invariant, thus is true ...
1
vote
0answers
25 views

Do $G$-spaces with equivalent orbit categories also have equivalent fundamental categories?

I have heard it mentioned before that $G$-spaces which have equivalent orbit categories must then have equivalent fundamental categories (sometimes called the equivariant fundamental groupoid). This ...
0
votes
1answer
32 views

Homology of contractible space

I understand that if $f,g: X \to Y$ are maps and $f$ is homotopic to $g$, then the induced maps on the homology groups $f_*$ and $g_*$ are equal. Why does this imply that if $X$ is contractible then ...
1
vote
0answers
11 views

Homeomorhism of pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})$

I'm having trouble following an argument in Bredon's Topology and Geometry. He says that the pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times ...
0
votes
0answers
14 views

Properties of free suspensions and free cones

In the category of based topological spaces, suspension $- \wedge S^1$ is adjoint to loop spaces $\text{Hom}(S^1,-)$ and the based cone $- \wedge I$ (where $I$ has zero as basepoint) is adjoint to the ...
2
votes
1answer
28 views

The significance of CW-complexes in homotopy theory

I try to understand the significance of CW-complexes in homotopy theory, in particular with respect to the classical models structure on $\mathbf{Top}$. Why do we chose Serre cofibrations for the ...
0
votes
0answers
59 views

Is this octogon topologically equivalent to the Klein Bottle?

Note: this is an extension of a previous problem (identify the topological type obtained by gluing sides of the hexagon ) where a hexagon was considered. Is the space below also a Klein bottle ...
4
votes
1answer
45 views

identify the topological type obtained by gluing sides of the hexagon

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the ...
0
votes
1answer
35 views

configuration spaces in mathematics and in physics

On the Wekipedia website Configuration space , there are two configuration spaces defined. One is Configuration spaces in physics, the other is Configuration spaces in mathematics. Question. Do ...
2
votes
1answer
111 views

Badly explained solution

My algebraic topology class is very bad at teaching, it just doesn't explain what's needed. Let me be specific, I am looking at this question, Q. Find the degree of $f_0 :S^1 \to S^1$ the constant ...
2
votes
1answer
35 views

Infinite wedge sum of circles and a space homotopy equivalent

Given a space $G = \bigcup_{n=1}^{\infty} A_n$ where $ A_n $ is a circle $ C[ (n,0),n] \in R^{2}$ I'd like to show that its fundamental group is an infinately generated free group. So let's say $a_i $ ...
2
votes
0answers
37 views

stable homotopy groups of the projective plane

The projective plane $\mathbb{R}P^2$ is obtained by attaching a $2$-cell to $S^1$ via a degree $2$ map $$ S^1\overset{[2]}{\longrightarrow}S^1\longrightarrow\mathbb{R}P^2. $$ Question 1: the ...
2
votes
2answers
48 views

Why are bordism groups of a point nontrivial

My definitions of Bordism are from Tom Dieck's Algebraic topology book. Briefly, a singular manifold, $M \xrightarrow{f} pt $, for a closed smooth oriented manifold $(M, \omega)$ without boundary is ...
5
votes
0answers
28 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces ...
0
votes
0answers
7 views

pointed simplicial set as coequalizer

Im studying simplcial sets and homotopy theory. I found this statement that seems quite immediate but for me it is not. Let $X$ be a pointed simplicial set, then $X$ can be realized as the ...
0
votes
0answers
46 views

Free homotopies and extensions

I am trying to prove the following. Lemma. Let $X^n$ be the $n$-skeleton of a CW complex $X$ with attaching functions $\phi_{\beta}:S^{n-1}_{\beta} \to X^{n-1}$, for all $\beta \in B$, and let ...
0
votes
2answers
34 views

Covering spaces of wedge sum of circles

Let's suppose I'd like to characterise all connected covering spaces of wedge sum of circles.When it comes to a torus, a sphere or $ S^{1} \vee S^{2} $ it is easy to point out exactly how do all ...
1
vote
1answer
33 views

Homotopy classes of manifolds of dimension $2n$ which are $(n-1)$ connected

In his essay "Classification of (n − 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres" Milnor states (in passing) "The manifold can be built up (up to homotopy type) by taking ...
0
votes
1answer
20 views

Characteristic function is an identification function

Every characteristic function $\Phi_{\beta}^b: E^n_{\beta} \to e^{-n}_{\beta}$ is an identification function. My book says the following: This follows from the fact the the CW complex $X$ has ...
1
vote
1answer
31 views

Annulus Homotopic to punctured plane

I know a circle is homtopic to a punctured plane, and by the same reasoning, the aannulus must also be, as it a "step" in the homotopy (IE the annulus is a "stretched" circle). The only trouble is it ...
1
vote
1answer
25 views

What is the map from $H_j( \Sigma MSO(k)) \to H_{j-k}(BSO(k))$ on Tom Dieck page 537

I am reading Tom Dieck's page 537 and I am not sure what the vertical map that I put in the title is in the diagram in the bottom of the page. This map is labeled Thom Isomorphism. Here $MSO(k)$ is ...
1
vote
1answer
28 views

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$?

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$? $\pi(X)$ denotes the fundamental group of $X$.
0
votes
1answer
17 views

Why is $BSO(n-1)$ the sphere bundle of the tautolocigal bundle on $BSO(n)$?

To try to show this I wrote down explicitly what the classifying spaces can be realized as. I am realizing the classifying spaces $BSO(n-1)$ as $V^\infty_n \times_{SO(n-1)} pt$, where ...
0
votes
0answers
23 views

Cone is contractible

For every space $X$, the cone $CX$ is contractible. My book's proof: Define $F: CX \times I \to CX$ by $F(\left<x,t \right>,s) = \left<x,s+(1-s)t\right>$ for $\left <x,t \right ...
0
votes
1answer
25 views

Path homotopy between $\alpha(t) = t-t^2$ and $\beta(t) = t^2-t$

I am trying to find a path homotopy between $\alpha(t) = t-t^2$ and $\beta(t) = t^2-t$ where $t\in[0,1]$ $\alpha$ and $\beta$ are path homo topic if they have the same endpoints, $p, q$ and $\exists ...
0
votes
1answer
13 views

Why is the fibration $BSO(n-1) \to BSO(n)$, $n-1$ connected?

In other words I want to show that the induced map $\pi_k BSO(n-1) \to \pi_k BSO(n)$ is 0 for $k\leq n-1$. The fiber of this fibration is $S^{n-1}$. I am having trouble with the case $k=n-1$. I am ...
4
votes
0answers
53 views

group actions of fundamental groups on homotopy groups

Let $\pi_n(\mathbb{R}P^n)$ be the $n$-th homotopy group of the $n$-dimensional projective space. Then by the long exact sequence of homotopy groups associated to the fibration $S^n\to \mathbb{R}P^n\to ...
4
votes
1answer
62 views

Prove homotopy equivalence of two spaces

How can I prove that $ [S^{1} \times D^{2}]/S^{1} \times S^{1}$ is homotopy equivalent to $S^{2} \vee S^{3} $? So far I have proved that $S^{2} \vee S^{3} \cong S^{3}/S^{1} $ Additionaly applying ...
0
votes
1answer
21 views

Contractible nullhomotopy

If $X$ is a contractible space, then for every neighborhood $W$ of $*$, there is a neighborhood $U$ of $*$ such that $U \subseteq W$ and $U$ is contractible in $W$. My book's proof: Let ...
0
votes
1answer
17 views

Deformation retraction confusion

I am looking at the retraction deformations and one I saw is $r: R^n \to E^n$ and $$r(x) = \begin{cases} \dfrac{x}{|x|} & |x| \geq 1 \\ x & 0\leq |x|\leq 1 \end{cases} .$$ ...
0
votes
1answer
57 views

Compute the winding number of $\alpha(z)=4z^4+2z^2+1$ and $\alpha(z)=6z^2+7z+2$

The winding number of $\gamma$ about $0$ is given as: $w(\gamma, 0)=\frac{arg(\gamma(1))-arg(\gamma(0))}{2\pi}$ $\gamma : [0,1] \rightarrow \mathbb{C}-\{0\}$ is a loop in $\mathbb{C}$ not passing ...
0
votes
0answers
55 views

Computing fundamental group of the complement to three infinite straight lines, and of complement to $S^1 \cup {Z} $

Question 1: Find the Fundamental group of the complement to three infinite straight lines that have no common points in $\mathbb{R^3}$ Question 2 Compute the fundamental group of the complement of ...
1
vote
1answer
19 views

Definitions of a homotopy

Why are the following two definitions for a homotopy equivalent? Let $f,g: X \to Y$ be maps. Then $f \simeq g$ if and only if there is a map $G:X \to Y^{I}$ such that $G(x)(0) = f(x)$ and $G(x)(1) ...
3
votes
1answer
73 views

When is $BG$ a topological group?

Let $G$ be a topological group, then it has a classifying space $BG$. When is $BG$ a topological group? My motivation for asking this question is that I was thinking about the $B$-analogue of ...
0
votes
1answer
18 views

Composition of homotopy classes

For a map $f: X \to Y$, we let $[f]$ denote the equivalence class containing $f$, called the homotopy class of $f$. Therefore, since homotopies are compatible with composition, it follows that if ...
0
votes
1answer
21 views

Homotopy is an equivalence relation

I am trying to prove that for all spaces $X$ and $Y$, homotopy is an equivalence relation on the set of maps from $X$ to $Y$. In my book they say if $f: X \to Y$, then $f \simeq_F f$ where $F(x,t) = ...
0
votes
0answers
29 views

Why “the line segment in $\mathbb{C}$ joining $f$ and $g$ does not pass through $0$”

Every single algebraic topology problem I've encountered is leaving me baffled. I just can't visualise what is going on and why. Q. $f,g:X \to S^1$ such that $f(x) \neq -g(x)$ for all $x \in X$. ...
0
votes
0answers
24 views

Help with Theorem 4.23 in Hatcher

In Hatcher's 'Algebraic Topology'(P.361), in the proof of Theorem 4.23 (the revised version, appearing in the online version of the book: https://www.math.cornell.edu/~hatcher/AT/AT.pdf), the sentence ...
-3
votes
1answer
63 views

Prove $X/A$ is Hausdorff if $A↪X$ is a cofibration [closed]

Let $X$ be a Hausdorff space and $A$ a closed subspace. Suppose the inclusion $A↪X$ is a cofibration. Prove $X/A$ is Hausdorff. I have no idea that how Hausdorff is related to the cofibration. I can ...
0
votes
2answers
29 views

Homotpoic and $\#$ of two maps

In general topology, and algebraic topology course i learn that If two maps are homotopic, their $\#$ are same. $i.e$, \begin{align} f \simeq g, \quad f_{\#} = g_{\#} \end{align} I want to know ...
8
votes
1answer
113 views

Is this comb-like space contractible?

For each $ \ n \in \mathbb{N}^* = \{ 1,2,3,4,... \}$, let $ \ S_n = \big\{ (t,1-nt) \in \mathbb{R}^2 : 0 \leqslant t \leqslant 1/n \big\}$, $Y_n = \big\{ (t,-nt-1) \in \mathbb{R}^2 : -1/n \leqslant t ...
1
vote
0answers
25 views

Is every infinite loop space a ring prespectrum?

Suppose you are given an infinite loop space $T_0$. The $i$th deloopings, $T_i$ then form a prespectrum with evaluation maps maps $\Sigma T_i \to T_{i+1}$. In the two examples that I know of, the ...
1
vote
0answers
26 views

Show that a set of homotopy classes has a single element

This is from Munkres section 51 problem 2b Given spaces $X$ and $Y$ let $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$. Show that if $Y$ is path connected, the set $[I,Y]$ has ...