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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5
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0answers
30 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 $f:...
0
votes
2answers
45 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
34 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 $X^n$ ...
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
31 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
18 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 $V^\infty_n=...
0
votes
0answers
24 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
55 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 $\text{id} ...
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
59 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
57 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
75 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) = f(...
0
votes
0answers
31 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
25 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
64 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
118 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
26 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
27 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 ...
12
votes
3answers
132 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
101 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
16 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 $p=...
3
votes
1answer
45 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
95 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 $\Gamma:[0,1]\times[0,1]\to\mathbb{R}^...
1
vote
2answers
41 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 = f(\...
2
votes
0answers
31 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
83 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
64 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
49 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)- \tilde{...
1
vote
1answer
31 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
39 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) \stackrel{j_*}{\...
1
vote
1answer
39 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
37 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
44 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 \...