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

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
53 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
47 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
69 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
17 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
61 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
110 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 ...
6
votes
1answer
57 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 ...
0
votes
5answers
96 views

Examples of Same fundamental group but not homeomorphic

Can you give me some example that their fundamental group (which is non-trivial) is same but their topological spaces are not homeomorphic? $i.e$, \begin{align} \pi_1(X) = \pi_1 (Y), \qquad X \ncong ...
1
vote
1answer
15 views

Let $\omega_n: I \to S^1$ by $\omega_n(t)=\exp (2\pi int)$. Show that $\omega_p * \omega_q$ is homotopic, rel endpoints, to $\omega_{p+q}$.

Define $\omega_n: I \to S^1$ by $\omega_n(t)=\exp (2\pi int)$. Show that the concatenation $\omega_p * \omega_q$ is homotopic, rel endpoints, to $\omega_{p+q}$. (Consider first the special case of ...
3
votes
1answer
42 views

CW complex and equivalence of $[X,K(\mathbb{Z},n)]$ and $\langle X,K(\mathbb{Z},n)\rangle$

So Hatcher remarks that this is true when $X$ is connected and $n>0$, I was wondering if the result holds even if $X$ is not connected. If this isn't true what are some weaker assumptions we can ...
0
votes
1answer
15 views

Prove |[X,Z]| = |[Y,Z]| where X, Y are homotopic and Z is a top space

Since $X$ and $Y$ are homotopically equivalent there are two maps $f:X \to Y$ and $g:Y \to X$ which composites are homotopic to the appropiated identity maps. Now if I pick a representative of an ...
0
votes
1answer
18 views

Proving homotopy of 2 paths

$A$ and $B$ are $2$ points of affixes $i$ and $1$ respectively, $O$ is the origin. $γ_1=[AO]∪[OB]$ and $γ_2=[AB]$ are $2$ paths. I know how to prove that $2$ paths are homotopy but in this case I ...
4
votes
0answers
53 views

Homotopy of boundary paths

Let $G$ be a bounded, simply connected, open set in $\mathbb{R}^2$, and let $\gamma_1$ and $\gamma_1$ denote two paths such that $\gamma_0(0)=\gamma_1(0)$, $\gamma_0(1)=\gamma_1(1)$, and the following ...
6
votes
1answer
91 views

Can we always find homotopy of two paths which lies “between” the paths?

Let $\gamma_0,\gamma_1:[0,1]\to\mathbb{R}^2$ be paths such that $\gamma_0(0)=\gamma_1(0)$ and $\gamma_0(1)=\gamma_1(1)$. I wish to show that there is a homotopy ...
1
vote
2answers
40 views

$X$ is contractible if and only if $X \simeq \{ * \} $ - A three-part question

First some background: The topological spaces, $X, Y$, are homotopically equivalent if and only if there are continuous functions, $f \colon X \longrightarrow Y$ and $ g \colon Y \longrightarrow X $ ...
1
vote
1answer
10 views

Showing $\bar{p} * \alpha * p \simeq_{x_0} \bar{p} * \beta * p$, path-homotopy

For given path-connected topological spaces $X$, $x_0 , x_1 \in X$, and given loops $\alpha$, $\beta$ $: I \rightarrow X$ with base point at $x_1$ and a path $p: I \rightarrow X$ such that $p(0) = ...
0
votes
1answer
39 views

Homotopy of induced homomorphism

What i want to do is prove homotopy \begin{align} f \circ (\alpha * \beta) \simeq_{\{ f(x_0)\}} (f \circ \alpha) * ( f \circ \beta) \end{align} where $\circ$ is a composition $f \circ \alpha = ...
2
votes
0answers
25 views

Compute directly that the mapping cone of a homotopy equivalence is contractible

Let's consider the category $Ch_R$ of cochain complexes of modules over a commutative ring $R$. I'm trying to prove that if the chain map $\phi:M\rightarrow N$ is a homotopy equivalence then its ...
4
votes
1answer
78 views

Is the special orthogonal group really rationally homotopy commutative?

It is a classical result that the rational cohomology of $SO(n)$ is given by: $$H^*(SO(2m); \mathbb{Q}) = \begin{cases} S(\beta_1, \dots, \beta_{m-1}, \alpha_{2m-1}) & n = 2m \\ S(\beta_1, \dots, ...
3
votes
0answers
63 views

Can some Lie groups ($S^3$ in particular) be converted to simplicial groups?

I'm trying to understand the concept of Kan simplicial sets and simplicial groups in particular. My question is, in a nutshell: What is the relation between the group operation in a simplicial ...
3
votes
2answers
45 views

Winding number of composition of maps

If $f,g:S^{1}\rightarrow S^{1}$ maps, show that $N(f\circ g)=N(f)N(g)$, where $N(f)$ is the winding number of $f$. We defined the winding number of $f$ to be $N(f)=\frac{1}{2\pi}(\tilde{f}(1)- ...
1
vote
1answer
27 views

Model structure induced by a combinatorial model category.

In Hirschhorn's Model categories and their localizations he gives a sufficient condition to induce a cofibrantly generated model structure on a category $\mathcal{N}$, given an adjoint pair of ...
1
vote
0answers
37 views

Proving exactness in homotopy exact sequence

I am trying to work through Hatcher's proof (page 344 of his book) that the homotopy sequence of a triple $$... \rightarrow \pi_n(A,B,x_0) \stackrel{i_*}{\rightarrow} \pi_n(X,B,x_0) ...
1
vote
1answer
35 views

Square is homotopy Cartesian if horizontal maps are weak equivalences

This is probably trivial but I'm not the best with category theory. Let $M$ be a right proper model category (that is pullbacks of weak equivalences along fibrations are weak equivalences). The ...
4
votes
0answers
29 views

How to find fundamental groups and covering spaces of $\mathbb{RP}^2\vee \mathbb{RP}^2$?

The following is an exercise I was assigned in homotopy theory. Defined $X = \mathbb{RP}^2\vee \mathbb{RP}^2$. a) Find $\pi_1(X)$. b) Find the universal cover of $X$. c) Find all of its connected ...
1
vote
2answers
42 views

Homology of $\mathbb R^n \times \mathbb R^n \setminus \Delta_{\mathbb R^n \times \mathbb R^n}$

Here $\Delta_{\mathbb R^n \times \mathbb R^n}= \{(x,y) \in \mathbb R^n \times \mathbb R^n \mid x=y\}$. My idea was that for $n=1$ we have $$\mathbb R \times \mathbb R \setminus \Delta_{\mathbb R ...
0
votes
1answer
83 views

Isomorphism between homotopy groups of Lie group, Grassmann manifold

It is asserted without proof in a book edited by Novikov and Rokhlin that $$\pi_{k - 1}(\text{GL}_n^+(\mathbb{R})) \cong \pi_k(\tilde{G}_n).$$ I know how to show that these two spaces are bijective. ...
0
votes
0answers
17 views

Homotopy equivalence of a shrinked full torus

My task is to find $n,k$ such that $ S^{n} \vee S^{k} $ is homotopy equivalent to $[S^{1} \times D^{2}] /S^{1} \times S^{1} $ .By lookig at $S^{1} \times D^{2}$ as an "full" torus I've figured ...
4
votes
1answer
209 views

A map of a compact connected manifold with non-empty boundary to a sphere of the same dimension is null-homotopic

In Milnor & Kervaire's Groups of Homotopy Spheres paper, this claim: A map of a compact connected manifold with non-empty boundary to a sphere of the same dimension is null-homotopic is made ...
0
votes
1answer
51 views

Unique way to show $S^n$, $n \geq 2$ is simply connected.

This questions is asked in Armstrong's Topology book, and I am totally stuck.... I could really use a major hint: Think of $S^n \subset \mathbb{E}^{n+1}$. Given a loop $\alpha \in \pi_1 (S^n , ...
2
votes
1answer
15 views

any continuous function is null homotopic for convex set.

Let $X$ be a topological space. and suppose $B$ is a convex subset in $\mathbb{R}^n$. Prove that any continuous map $f: X \rightarrow B$ is null-homotopic. My strategy is following the defintion of ...
1
vote
1answer
45 views

Example of building a classifying space

I'm reading some things about algebraic topology, and they mention the classifying space of a group $G$ as $BG$, but they doesn't build one, so I want to ask if someone knows where can I find the way ...
1
vote
1answer
20 views

How to show the constant path is the identity element in fundamental group?

Let $X$ be a topological space and $q$ is a point in $X$. Denote the fundamental group of $X$ based at $q$ by $\pi _1(X,q)$. Then how should I verify that the constant path $c_q(s)\equiv q$ is the ...
1
vote
2answers
45 views

For which values of $k$ is there an $X$ with $\Omega^kX \cong X$?

Bott periodicity can be formulated as $\Omega^2 U \cong U$ where $\Omega$ denotes the based loop space functor and $U$ is the direct limit of unitary groups. The real version can be formulated as ...
3
votes
1answer
38 views

Classification of line bundles by group homomorphisms from the fundamental group to $\mathbb{Z}_2$

Let $X$ be a path-connected manifold. Then by the classification of vector bundles, the collection of all (real) line bundles on $X$ is $$ \text{Vect}^1(X)\cong ...
2
votes
1answer
34 views

Is it true that $\pi_n(X^{n-1})=0$ for $n>1$ if $X$ is $K(G,1)$ space?

In this post, Joe Johnson 126 mentioned the above fact, which I'm skeptical of. It is well-known that $\pi_n(X^{n+1})=\pi_n(X)$, but being a $K(G,1)$ space doesn't seem to imply the identity in the ...
0
votes
0answers
18 views

Difficulties with the description of $p*$ in the serre spectral sequence(Bullet 7 of Mosher Tangora Page 76):

For the first difficulty, let $E^r$ be the rth page of a first quadrant spectral sequence with elements $E^r_{p,q}$ , where $p$ is the filtering degree. Difficulty 1: On bullet 7 of Mosher and ...
1
vote
1answer
78 views

Cobordism and h-cobordism

Is there a way to simply explain cobordism and h-cobordism? I am not looking for a math based explanation, but rather, just the main ideas behind the concepts.
5
votes
1answer
65 views

Why is the group $[\Sigma\Sigma X, Y]_{\ast}$ commutative?

Can anyone give a reference (or explain here), why the group $[\Sigma\Sigma X,Y]_*$ is commutative? How is it related to the fact that $\Sigma X$ is a co-H-space?
1
vote
0answers
14 views

What is the topology of uniform convergence in this case of $P=P(x_{0},M)$ of all paths in $M$ starting at $x_{0}$?

The following definition I found it in a text on Lie groups: Let $M$ be a connected smooth manifold and $x_{0}\in M$. A path in $M$ starting at $x_{0}$ is a continuous curve $\gamma ...
0
votes
1answer
31 views

free commutative graded algebra

Let $V$ be a free graded module. $\wedge V$ be the free commutative graded algebra. $\wedge V$ = symmetric algebra $(V^{even})\otimes$ exterior algebra $(V^{odd})$ I don't understand this equation ...
0
votes
0answers
16 views

Formulation of zeroth order deformation equation (HAM)

I am reading the paper by Liao 'An explicit, totally analytic approximate solution for Blasius’ viscous flow problems', although this question equally applies to the other formulations of the homotopy ...