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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4
votes
2answers
70 views

Non-closed deformation retract

I am searching for a $T_1$ space $X$ which deformation retracts onto a non-closed subspace $A$. Such a space cannot be Hausdorff as any retract in a Hausdorff space is closed. I tried some spaces, ...
4
votes
2answers
61 views

$\pi_n(SU(2)/Z_N)\simeq?$, $\pi_n(SO(3)/Z_N)\simeq?$, $\pi_n(U(1)/Z_N)\simeq?$

So based on the tool, I have attempted to compute the following threes homotopy groups. $\pi_n(SU(2)/Z_N)\simeq?$ $\pi_n(SO(3)/Z_N)\simeq?$ $\pi_n(U(1)/Z_N)\simeq?$ With a fixed positive integer ...
5
votes
2answers
196 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)) ...
3
votes
2answers
181 views

Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell

Prove: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ and $X\sqcup_gD^n$ are homotopy equivalent. I think it can be proved by showing they are both deformation retracts of ...
2
votes
1answer
57 views

For which $n$, is any continuous map from $S^n$ to $S^1 \times S^1$ nulhomotopic.

For which $n$, where $n$ is a positive integer, is any continuous map from $S^n$ to $S^1 \times S^1$ nulhomotopic. If every continuous map from some $S^n$ to $S^1 \times S^1$ was nulhomotopic, would ...
10
votes
1answer
97 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 ...
3
votes
2answers
57 views

Why is the Cech nerve $C(U)$ of a surjective map $U\to X$ weakly equivalent to $X$?

Let $f:U\to X$ be a surjective map of sets and $$ ...U\times_XU\times_XU \substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] ...
3
votes
2answers
189 views

Hopf fibration and $\pi_3(\mathbb{S}^2)$

I am interested in Hopf's original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre ...
2
votes
1answer
55 views

need help with problem on homology group

Let $A_n=\{z\in \mathbb{C}\mid z^n$ is non-negative real number$\}$ then find $H_1(A_n,A_n-\{0\})$ $H_1(A_n,A_n-\{z\})$ when $0\not=z\in A_n$ show that $A_n$ is not homeomorphic to $A_m$ when ...
2
votes
1answer
113 views

Union of 2-sphere with line segment in $\mathbb{R}^3$ removing one point homotopy equivalence.

I am working on a problem from Lee's Introduction to Topological Manifolds where one is asked to compute fundamental groups using Van Kampen's theorem. I know how to use Van-Kampen's theorem but I ...
3
votes
2answers
77 views

Prove these 3 spaces are homotopy equivalent

The image is below. (a) $S^2$ with a diameter. (b) $T^2$ with a disk in the middle hole. (c) $S^2$ tangent with $S^1$ . I think they may the deformation retract of the same space. But I can't ...
2
votes
0answers
67 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
0
votes
1answer
60 views

How to show that: A path is homotopic to a given point, then it is homotopic to any other point.

Let $D \subset \mathbb C$ be a domain and $\gamma : [\alpha, \beta] \to D$ a closed path. Let $a \in D$ be a given point. Assume that $\gamma$ is homotopic to that point $a$. Prove that $\gamma$ is ...
1
vote
1answer
30 views

Prove that $S_R^+(b)$ and $S_r^+(a)$ are homotopic in a domain $D$.

currently I'm working on the following exercise: Let $D \subset \mathbb C$ be a domain. Let $a,b \in \mathbb C$ and $r,R > 0$ such that $B_r(a) \subset B_R(b)$ and \begin{align*} A := \{z ...
2
votes
1answer
34 views

Does the functor $\mathbf{cosk_n}:sSet\to sSet$ preserve Kan complexes?

On this nlab page a functor $\mathbf{cosk_n}:sSet\to sSet$ is constructed. Is $\mathbf{cosk_n}(K)$ of a Kan complex $K$ again a Kan complex?
4
votes
0answers
131 views

Constructing $\pi_1$ actions on higher homotopy groups.

I am working on exercise 4.2.7 of Hatcher, which is to construct a CW complex $X$ with arbitrary homotopy groups and a prescribed action of the fundamental group on these homotopy groups (so making ...
4
votes
1answer
85 views

What is $\pi_2 (S^2/ S^1 \vee S^1)$?

I've been working on the exercise "Find $\pi_2 (S^2 / X)$ where $X$ is the image of $S^1 \vee S^1$ under some embedding into $S^2$". First I tried to find some helpful fibrations/cofibrations as this ...
0
votes
2answers
45 views

Homomorphisms induced by $h$

Let $X$ and $Y$ be spaces. Let $x_0 \in X$ and let there be a continuous function $h$ such that $h(x_0)=y_0$. Then there is a homomorphism $h_{*}: \pi_1(X,x_0)\to \pi_1(Y,y_0)$ by the rule ...
1
vote
1answer
47 views

Retraction and Homotopy type

I need to describe an example of a subspace $A \subset X$ such that there is a retraction $r: X\rightarrow A$ but such that $A$ and $X$ do not have the same homotopy type? any hints? I was thinking ...
8
votes
2answers
202 views

Why does the loopspace $\Omega$ induces a weak equivalence on mapping telescopes?

I am trying to answer an exercise of Hatcher's "Algebraic Topology", Section $4$.F, exercise $3$. Suppose we are given a sequence of pointed topological spaces : $Z_0\rightarrow Z_1\rightarrow Z_2 ...
1
vote
1answer
92 views

Homotopy groups of a covering space

This is a question related to the exercise 2218 from the book "Problems and Solutions in Mathematics" by Ta-Tsien, $2^{nd}$ Ed. Let $Z$ denote the figure 8 space, $Z = X \vee Y$, $X$ and $Y$ circles. ...
0
votes
1answer
50 views

Need help on finding homotopy

Define a continuous map $\ell:(I,\partial I)\to (SO(3),1)$ by $\ell(t) = \left( \begin{array}{ccc} \cos 2\pi t & -\sin 2\pi t & 0 \\ \sin 2\pi t & \cos 2\pi t & 0 \\ 0 & 0 & 1 ...
1
vote
0answers
43 views

homeomorphism still isotopic to the identity after the deletion of points.

I have a surface $M$ (without boundary) and a homeomorphism $f:M \rightarrow M$, which is isotopic to the identity on $M$. If I delete two points $x$ and $f(x)$ from the surface, I get a ...
5
votes
3answers
263 views

Introductory books as preparation to read Voevodsky homotopy-theory (HoTT) book

I would like to read Voevodsky HoTT book. However, I lack a lot of the basics. I would need a few introductory books first that cover topics like groupoids, fibrations, W -types, Homotopy theory. ...
6
votes
2answers
199 views

What is a (the?) good starting point for learning the modern “higher” mathematics?

As many of you know, category theorists are currently doing, among other things, a great job in advertising their modern developments. And I must say, this works for me - in particular, I find myself ...
1
vote
1answer
39 views

Non self-intersecting representatives in fundamental class

If $X$ is a Riemann surface with boundary $\partial X$ and $\pi_1(X,p)$ is its fundamental group, $p \in X$, then we shall call class $[\gamma] \in \pi_1(X,p)$ primitive (or generator) if it can not ...
4
votes
1answer
95 views

Inverses in the homotopy classes of maps into $RP^{\infty}$

One can define bilinear maps $\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^{2n-1}$ by considering the elements in $\mathbb{R}^n$ as polynomials and doing multiplication. This defines an ...
2
votes
2answers
117 views

A standard proof of $\pi_1(\mathbb{S}^1) = \mathbb{Z}$ using universal covering spaces

I am looking for "a standard proof of $\pi_1(\mathbb{S}^1) = \mathbb{Z}$ using universal covering spaces", as suggested by the book Homotopy Type Theory (p. 255). What is this proof? Where can I find ...
2
votes
1answer
85 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 ...
6
votes
0answers
117 views

How strong is the analogy between spectra and abelian groups?

I am led to understand that spectra are some kind of $\infty$-analogue of (discrete) abelian groups, or perhaps more accurately, some kind of generalisation of chain complexes of abelian groups. How ...
1
vote
2answers
55 views

Need help with computing homology group.

Let $D=$$S^2\cup$ x-axis$\cup$ y-axis be surface in $R^3$ I want to compute the homology group $H_n(D,\mathbb{Z})$ forcannot all $n\geq 0$ using Mayer-Vietoris Exact sequence. There exists many open ...
0
votes
0answers
43 views

Covering Manifolds

Let M a 3-manifold not orientable with incompressible boundary toral (possibly empty) and N the 2-sheeted cover space orientable of M. Questions: a) Why N has boundary toral(possibly ...
9
votes
1answer
210 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 ...
2
votes
1answer
82 views

negative Euler characteristic $\Rightarrow$ homotopy unique up to homotopy

In a paper by John Franks I stumbled upon the following: Let $M$ be a surface and $f:M \rightarrow M$ be a homeomorphism, which is homotopic to the identity on $M$. That means, that there is ...
3
votes
2answers
89 views

Do pushouts of compactly generated Hausdorff spaces exist?

Let $A\to X$ and $A\to Y$ be maps of compactly generated Hausdorff spaces. Does the pushout $X\coprod_A Y$ in the category of compactly generated Hausdorff spaces exist? If necessary, one can assume ...
0
votes
1answer
58 views

Chain homotopy and compositions of morphisms.

Show that if $\alpha_1 \sim \beta_1$ and $\alpha_2 \sim \beta_2$ , then (whenever composition makes sense) $\alpha_1 \circ \alpha_2 \sim \beta_1 \circ \beta_2$. I have two questions. So are these ...
4
votes
0answers
50 views

local gauge invariance of field's homotopy class? Every map $S^2\rightarrow \mathrm{group } G$ is homotopic to a constant map?

In a discussion of a gauge field theory with gauge group $G$, someone says we can use a celebrated result of E. Cartan to show the gauge invariance of matter field's homotopy class. And Cartan's ...
2
votes
1answer
58 views

Spaces sharing all higher homotopy groups

Is it possible that two topological spaces share all higher homotopy groups, but are not homeomorphic? I should note that I have not studied much in the way of the theory of higher homotopy groups; I ...
0
votes
1answer
39 views

symmetry in the homotopy relation

Suppose $\alpha, \beta : I \to X$ are paths and suppose $\alpha $ is homotopic to $\beta$, $\alpha \cong \beta$. So, can find a continuous function $F(s,t) = f_t(s)$ such that $$ f_t(0) = \alpha(0) ...
7
votes
1answer
158 views

What exactly is duality?

In general, I am familiar with this notion of duality (i.e. in category theory, a statement is dualized simply by "reversing all arrows" and leaving objects unchanged). There are a couple of questions ...
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 ...
1
vote
3answers
73 views

Number of homotopy classes

For topological spaces $X$ and $Y$, let $[X,Y]$ denote the set of homotopy classes of continuous maps $X\to Y$. If $I=[0,1]$ is the unit interval, then $[X,I]$ has only one element. If $X$ is path ...
1
vote
1answer
46 views

How to show that homotopy is preserved after composition?

I have two homotopies: $f\simeq f'$ and $y\simeq y'$. How can I show that $fy\simeq f'y'$ is again a homotopy?
2
votes
0answers
33 views

2 out of 3 axiom and simplicial sets

Let $i\colon\mathcal W\to\mathcal C$ be the inclusion of a subcategory. Unless I'm mistaken, the 2 out of 3 axiom for $\mathcal W$ to be a category of weak equivalences can be expressed as the ...
0
votes
1answer
89 views

Isomorphism functorially

I was reading the lecture notes of Pierre Schapira http://www.math.jussieu.fr/~schapira/lectnotes/AlTo.pdf I am not able to understand one thing. Please help. In page 75, theorem 4.6.1, the author ...
2
votes
0answers
39 views

Clarification about a double delooped H-space.

I've just started reading J.P. May's book The Geometry of Iterated Loop Spaces and am misunderstanding something. Somewhere, it's asserted that if an H-space X can be delooped twice, the its ...
1
vote
0answers
63 views

finite covering space of non-orientable surfaces

Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers. A ...
1
vote
1answer
46 views

Maps between surfaces of genus n

Could anyone help me find maps $f:\Sigma_{n}\longrightarrow\Sigma_{m}$ with $0<n<m\ \ $ NOT homotopic to a constant map or prove its existence without using K-theory? Thanks
0
votes
1answer
71 views

Question on “Homotopy invariance”

i have this from Hatcher's book "Algebric topology" And i don't understand why $\displaystyle \partial P(\sigma)=\sum_{j\leq i}(-1)^i(-1)^j F\circ (\sigma\times ...
0
votes
2answers
52 views

Homotopy between $X\setminus\{x\}$ and $A$

Let $X$ be a space obtained from $A$ by attaching a n-cell $e$. If $x\in e$ how to prove that the inclusion map $i_{A\to X\setminus\{x\}}: A\to X\setminus\{x\}$ is a homotopy equivalence?