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
votes
4answers
218 views

What is an “inner isomorphism” between different groups?

It is well known that if $X$ is a path-connected topological space containing points $x$ and $y$, then the fundamental groups $\pi_1(X,x)$ and $\pi_1(X,y)$ are isomorphic. Wikipedia makes the further ...
3
votes
0answers
47 views

The relation between homotopy equivalence and contractible mapping cone?

In this MO thread, the OP claimed that it is obvious that homotopy equivalence implies the mapping cone contractible, whereas the converse proposition is wrong. I hate to admit that it's not obvious ...
1
vote
0answers
53 views

How do I show that this function is continuous? [duplicate]

I'm interested in showing that $CX=\frac{I\times X}{\{1\}\times X}$ is contractible. I defined the d.r $F(s,[t,x])=[(1-s)t+s,x]$ and the only missing part for me is to show that it is continuous. How ...
2
votes
0answers
44 views

Homotopy groups of mapping spaces

If I have an $\infty$-category $\mathcal{C}$ (AKA quasi-category), can I say anything about the homotopy groups of the mapping spaces $\mathrm{Hom}_\mathcal{C}(X,Y)$ for two objects $X$ and $Y$? ...
1
vote
0answers
64 views

“Stable model categories are categories of modules” - Clarification about a few things

I was reading Schwede and Shipley's "Stable model categories are categories of modules", I needed clarification about a few things: 1 - When they say that stable model categories are categories of ...
1
vote
1answer
29 views

Let $X$ be the figure 8 space embedded in $S^2$. Find $\pi_2(S^2/X)$.

Let $X$ be the figure 8 space embedded in $S^2$. Find $\pi_2(S^2/X)$. The problem is in page 457 of "Topology and Geometry" written by Glen E. Bredon. I think I need to use a long exact sequence of ...
0
votes
1answer
26 views

Is the skeleton-coskeleton adjunction $sSet$-enriched?

Let $n\geq 0$ be an integer. Is the adjunction $$ \mathbf{sk}_k\colon sSet \leftrightarrows sSet\colon\mathbf{cosk}_k $$ of the skeleton and coskeleton an $sSet$-enriched adjunction?
1
vote
1answer
66 views

Pull-back of a fibration along a homotopy equivalence

Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\rightarrow B$ and $g:B\rightarrow B'$ be homotopy inverses. Denote by ...
3
votes
0answers
78 views

Criterion for homotopy equivalence in the category of pair of spaces

I am trying to prove the following statement : Let $\{ * \} \subset A \subset B \subset X$ be a chain of topological spaces (all subsets have the subspace topology). It is given that $A \to X$ ...
0
votes
0answers
26 views

Long exact sequence of homotopy groups $\pi_n$ for a pointed homotopy pullback square

Let \begin{align} A &\to B\\ \downarrow &~~~~~\downarrow\\ C &\to D \end{align} be a homotopy pullback square of pointed simplicial sets. One gets a long exact sequence $$ ...
0
votes
2answers
50 views

How to compute a homotopy to show the operation on the fundamental group is assoicative?

By definition $$[(\alpha *\beta) *\gamma ] (s) = \begin{cases}\alpha (4s) & 0 \leq s\leq \frac{1}{4} \\ \beta(4s-1) & \frac{1}{4}\leq s\leq \frac{1}{2}\\ \gamma(2s-1) & \frac{1}{2}\leq ...
3
votes
1answer
64 views

The Plate Trick and $SO(3)$

This is a followup of sorts to Plate trick demonstrating SO(3) not simply connected. . Suppose I do the ``plate trick'' to demonstrate the existence of an order-2 element in $\pi_1(SO(3))$. That is, ...
3
votes
1answer
39 views

Fundamental question about cohomology on the stable homotopy category

I've gotten myself tied in knots about this elementary derivation of a ludicrous conclusion. Appreciate a hand straightening myself out! (1) A fibration $F\to E \to B$ of CW complexes gives rise to a ...
2
votes
0answers
38 views

Question on the coskeleton functor $\mathbf{coskel}'$ for pointed simplicial sets

There is an abstract construction of the coskeleton functor for simplicial set as follows: Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq ...
5
votes
1answer
62 views

Bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_n/\pi_1$

I would like to prove that there is a bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_{n}(X,x_{0}) / \pi_{1}(X,x_{0})$ where the action of $\pi_{1}(X,x_{0})$ over ...
0
votes
1answer
52 views

relative homotopy groups

I study relative homotopy groups and I have a question: Let $A\subseteq X$ (not necessarily CW complex) and $\pi_{n}(X,A)$. Is it always possible to find a pointed space Y for which ...
4
votes
0answers
38 views

Transversality and homotopic maps

I'm trying to solve some problems in differential topology, and I came across the following: suppose $f:M\times [0,1]\rightarrow N$ is a homotopy, where $M$ is a compact manifold, such that $f_0$ and ...
4
votes
1answer
134 views

If $f:S^1\to S^1$ doesn't have any fixed point then it is homotopic to the identity

How to show that every continuous function $f:S^1\to S^1$ without fixed points is homotopic to the identity? (without using homology nor the concept of degree).
7
votes
1answer
114 views

Homotopic equivalence for $S^5$ without three $S^1$

Consider a standard embedding of $S^5$ in $\mathbb R^6$: $S^5: \; x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 1.$ And consider three circles, which are sections of $S^5$ by $x_1 x_2$, $x_3 x_4$, ...
0
votes
0answers
37 views

Question on cofiber sequence map in equivariant homotopy theory

Let $G=C_2$ denote the cyclic group with two elements. Up to isomorphism there are only two irreducible $C_2$-representations, the identity representation, $\mathbb{R}^{1,0}$, and the sign ...
1
vote
1answer
30 views

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph, and the question asks further "which graph" Let ...
0
votes
1answer
31 views

Simple homotopy construction

I'm sure this isn't too difficult but i can't seem to do it if you have two loops $p_0 = e*g $ and $p_1 = g*e$ where $e$ is the trivial loop How would i construct an explicit homotopy between the ...
4
votes
1answer
64 views

Relationship between cohomology and higher-homotopy

Let $M$ be a connected, compact, and orientable 3-manifold ($H^3(M)\cong\mathbb{Z}$), and let $G$ be a simple Lie group satisfying $\pi_1(G)=\pi_2(G)=0$. Let $\pi_M(G)$ denote the set of homotopy ...
1
vote
2answers
72 views

Show that the Möbius band has its central circle $C$ as a deformation retract

I have started this problem by using the planar representation of the Möbius band and noted that a line down the middle is probably what is meant by the central circle, since travelling from top to ...
1
vote
0answers
26 views

Tools for proving map is homotopy equivalence

General situation I'm preparing a geometry exam, and a lot of exercises from past years' exams are of the form «given $f$ the map so-and-so, prove (or determine whether) it is a homotopy ...
4
votes
3answers
103 views

The significance of filtered colimits in homotopy theory

I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask ...
1
vote
1answer
159 views

How to show $S^n$ is not contractible without using Homology..

I know the prove $S^n$ is not contractible using homology.But I don't know how to prove it from definition of contractibility.Can anyone help me in this direction? Thanks.
5
votes
1answer
137 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
1
vote
1answer
48 views

Question about homotopic functions and homotopy classes

If $X = [0,1] \times [0,1]$ and $Y = [0,1] \cup [2,3]$. I have to give an example of two continuous functions $f$ and $g$ that are not homotopic. I was thinking of $f(x,y) = x$ and $g(x,y) = y +2$ ...
1
vote
0answers
33 views

functors with a morphism lifting property

By analogy to the familiar situation in homotopy theory (i.e., (unique) path lifting in covering spaces), it is natural to consider the following. Let $P:C\to D$ be a functor. Say that $P$ has ...
1
vote
1answer
17 views

Relative homotopy and composition of maps

I am trying to prove something and am stuck on the following issue : Suppose $\Psi, \Phi : I^n \to Y$ are two maps and $q:Y \to Z$ is a homotopy equivalence such that $q \Phi \cong q \Psi $rel ...
2
votes
1answer
57 views

Find $f$ and $g$ homotopic s.t. induce different homomorphisms

Let $X$ and $Y$ be two topological spaces. Are there continuous functions $f,g:X\to Y$ satisfying the following conditions? $f(a)=g(a)=b$ for some $a\in X$, $f$ and $g$ are homotopic, and the ...
3
votes
2answers
98 views

How to construct $K(\mathbb{Z}/5\mathbb{Z},1)$?

From the Wikipedia article on Eilenberg-MacLane spaces: A $K(G, n)$ can be constructed stage-by-stage, as a CW complex, starting with a wedge of n-spheres, one for each generator of the group ...
1
vote
0answers
19 views

Calderón–Zygmund lemma and Lebesgue measure

Can someone please explain the relationship between the Calderón–Zygmund lemma and Lebesgue measure, thanks.
0
votes
1answer
11 views

proof of the decomposition of a fibration by a trivial fibration followed by a minimal fibration

In Hovey's book theorem 3.5.9 asserts that any fibration $p:X\rightarrow Y$ can be factored as a trivial fibration followed by a minimal fibration. The proofs begins by defining a set T of simplices ...
2
votes
1answer
50 views

A quotient map $X\to X/A$ that is not a Serre fibration

What is an example of a CW-pair $(X,A)$ such that the quotient map $X\to X/A$, i.e. the map obtained from the pushout \begin{eqnarray} A &\to& X\\ \downarrow &&\downarrow\\ * ...
4
votes
0answers
75 views

What is this relation on the set of paths called in graph theory?

Suppose I have a directed simple graph $\Gamma$ (no edge loops or multi-edges) and a directed path $v_0v_1\cdots v_k$ joining vertex $s=v_0$ to vertex $t=v_k$. By directed path I mean that each pair ...
3
votes
0answers
151 views

A construction with homotopy colimits and homotopy pullbacks for descent.

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...
0
votes
1answer
31 views

Based on the Andreotti-Frankel theorem, what is the CW complex homotopy equivalent to $x^2 + y^2 - 1$?

I am referring to this theorem: http://en.wikipedia.org/wiki/Andreotti%E2%80%93Frankel_theorem I have no idea how to begin thinking about this.
5
votes
1answer
95 views

Finding a good cover such that its lifting is still a good cover

Let $Y$ be a compact manifold, $X$ a topological space and $f: X \to Y$ a surjective map. Suppose further that every point in $Y$ has arbitrarily small open neighbourhoods such that their preimages in ...
0
votes
1answer
50 views

Homotopy proof of Cauchy's Theorem

The proof of Cauchy's theorem in these notes http://people.math.gatech.edu/~cain/winter99/ch5.pdf rely on the concept of homotopy. But it seems to me that the proof did not use any property special to ...
1
vote
1answer
52 views

Homotopy and Semidirect Product

I know there is a relation in homotopy theory which is $\pi(G\times H) = \pi(G)\times\pi(H)$. However, does this relation still hold for $\pi_0$ which may not be a group? Moreover, is there such a ...
2
votes
1answer
82 views

When a homotopy equivalence of the closed unit ball in the Euclidean n- dimensional space is injective?

I have a compact connected metric space $X$ of dimension $n$ which is homotopically equivalent to the closed unit ball $D^n$ in the n-dimensional Euclidean space. I am wondering if there is an ...
-2
votes
1answer
40 views

Homotopy split monomorphisms [closed]

Let $C$ be a model category. Recall that a morphisme $f : X \to Y$ is called a homotopy monomorphism if the diagonal $X \to X \times^h_Y X$ induces an isomorphism in the homotopy category. Suppose ...
1
vote
1answer
51 views

Question on showing a bijection between $\pi_1(X,x_0)$ and $[S^1, X]$ when X is path connected.

I am trying to do this question taken from Hatchers algebraic topology and I am struggling to understand the notation and the concepts. As far as I know $\pi_1(X,x_0)$ is the set of end point ...
3
votes
3answers
97 views

Reference: In every free homotopy class is a unique minimizing closed geodesic

Does anyone know a reference for the following result: Let $M$ be a compact hyperbolic manifold/manifold with strict negative curvature . Then in every non-trivial free homotopy class of $M$ there ...
4
votes
0answers
121 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
0
votes
1answer
36 views

Simplicial approximation

One of the definition of simplicial approximation says that: a simplicial map $h:|K|\rightarrow|L|$ is a simplicial approximation of a continuous map $f:|K|\rightarrow|L|$ if and only if $$\forall ...
1
vote
0answers
28 views

Criterion for projectively cofibrant diagrams?

Let $\mathbf{D}$ be a small category, $\mathbf{Set}$ the category of sets and $\mathbf{sSet}$ the category of simplicial sets with its usual model structure. The category of functors $\mathbf{D}\to ...
3
votes
2answers
85 views

Residue Theorem and Homologous to zero

This is a very basic question and I couldn't find it posted yet but here it goes; The Residue Theorem states that if $f:G\to \mathbb{C}$ is analytic on $G$- a region and $f$ has isolated singularities ...