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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2
votes
1answer
56 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
97 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
47 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
72 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
146 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
93 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
45 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
51 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
79 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
39 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
48 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
92 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
116 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
74 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 ...
2
votes
1answer
41 views

Showing that identity and g are not homotopic (without Homology)

Question: Are the identity mapping on $S^1$ and the reflection about the $x$-axe homotopic? This is a question which I already know the answer. The objective is to find better answers and suggestions ...
3
votes
1answer
61 views

Aspherical but not contractible

Let $X$ be the topologist's sine curve (i.e. $\left\lbrace (x,y): y=\sin\left(\frac{1}{x}\right),x\in ]0,1]\right\rbrace\cup \lbrace (0,y): y\in [-1,1]\rbrace$) with an arc joining $(0,0)$ and ...
0
votes
0answers
22 views

Show $X$ is a H-space?

Let $Y$ be a H-space and suppose $X$ is a pointed retract of $Y$ with continuous pointed maps $s,r$. My thinking so far; If Y is a H-space then there is a map $m:Y$x$Y \rightarrow Y$ such that $m$ ...
1
vote
1answer
36 views

what is the inclusion map for $Y$ to $Y$ x $Y$?

I am studying homotopy and homology and one map we have been using is the left and right inclusion maps $i_L$, $i_R$, for example from the space $Y$ to the cartesian product $Y$ x $Y$. Whilst I ...
0
votes
2answers
71 views

Explicit construction of Eilenberg-Maclane spaces with n=1

Is there any examples of explicit construction of Eilenberg-Maclane spaces $K(G,1)$ for concrete groups except for G=$\mathbb Z$ and $\mathbb Z_n$? I know about general simplicial bar construction, ...
1
vote
0answers
21 views

Homotopy equivalence between O-O and $\theta$

Show that the dumbbell O-O (where there's no space between the "O" and "-") and the letter $\theta$ are homotopy equivalent, using the definition. So, let $X$ be the set of points in the dumbbell, ...
3
votes
1answer
50 views

Weak equivalence testable on invariant open covers?

Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't want to work in any convenient category of topological spaces. Let ...
0
votes
0answers
23 views

Criterion for a map to be homotopy equivalence in terms of its mapping cylinder

I am trying to prove that a map $f: X \to Y $ is a homotopy equivalence iff $j : X \to Z(f)$ is a deformation retract where $Z(f)$ is the mapping cylinder and $X \to Z(f) \to Y$ is the decomposition ...
1
vote
0answers
25 views

Lifting property of a covering map, product topology version

Suppose I have the following theorem (1): If $C,X$ are spaces, $p:C\to X$ is a covering map, $Y$ is a "nice" topological space (I think simply connected and locally path-connected is sufficient), ...
-1
votes
1answer
41 views

Is the following true about maps on unit closed disc? [closed]

(a) If $f : D_2 \to D_2$ is a map such that $f(x) = x$, for $x \in S_1$, then there exists an interior point $z$ in the disk (a point $z ∈ D_2 − S_1$) such that $f(z) = z$. (b) If $f : D_2 \to D_2$ ...
2
votes
0answers
39 views

Kan's loop group construction

I'm looking for a good place to read about the loop group construction $G : \mathbf{sSet_0} \to \mathbf{sGrp}$ taking a reduced simplicial set $X$ and producing a simplicial group $GX$. I would also ...
0
votes
0answers
53 views

Show that if $f:S^8 \to S^8$ and $\|f(x) + x\| < 1$ for all $x$ then $f$ is not homotopic to the identity map

I need to show that if a map $f:S^8 \to S^8$ satisfies $\|f(x) + x\| < 1$ for all $x \in S^8$ then $f$ is not homotopic to the identity map. Progress I saw some two related propositions: ...
0
votes
1answer
43 views

Homotopy Extension Property as a pushout?

The usual diagram for the homotopy extension property is: where $i_t^X:X\rightarrow X\times I,x\mapsto(x,t)$. Isn't this the same as saying the following square is a pushout? $$\require{AMScd} ...
4
votes
1answer
103 views

Utility of the 2-Categorical Structure of $\mathsf{Top}$?

It's well known that $\mathsf{Top}$ is a 2-category with homotopy classes of homotopies as 2-arrows. I'm a bit afraid to ask this question, but what is the utility of this 2-categorical structure? ...
1
vote
0answers
30 views

The “Wiggle Room” intuition for cofibrations

Often enough - for instance in the answer to this question - I have encountered the idea that an inclusion $i:A\subset X$ is a cofibration if $A$ has enough "wiggle room" in $X$. Although I have a ...
2
votes
1answer
85 views

How was the functorial factorization axiom “frequently misstated”?

In modern times, a model category is frequently required to have functorial factorizations of morphisms into (fibration/acyclic cofibration) and (acyclic fibration/cofibration). But the precise ...
6
votes
2answers
121 views

Categorification of geometry

I don't know if this idea is known, relevant or dumb, but I noticed that one could define abstract connectedness with groupoids. Let us forget about topology for a while, and let us think ...
3
votes
1answer
43 views

Is the induced map $\varphi$ on the homotopy cofibers null-homotopic in this situation?

Let \begin{eqnarray} X & \xrightarrow{f} & * \\ \downarrow & & \downarrow\\ Y & \xrightarrow{g} & Z \end{eqnarray} be a (strictly) commutative diagram of pointed CW-complexes ...
4
votes
2answers
97 views

can we derive integral cohomology from rational cohomology and mod p cohomology?

Let $X$ be a topological space. If we know that for $\mathbb{F}=\mathbb{Q}$ and $\mathbb{Z}/p$, for any prime $p$, $$ H^*(X;\mathbb{F})=0$$ for any $*\geq n+1$, can we conclude that $$ ...
3
votes
1answer
32 views

Universal spaces are homotopy equivalent

Consider a group $G$. We want to determine the universal space $EG$. Is it true that all universal spaces are homotopy equivalent. That is, to find $EG$ we only need to find a weakly contractible ...
3
votes
1answer
76 views

$\pi_n(X^n)$ free Abelian?

I have encountered a problem which states that denote $X$ as an Eilenberg-MacLane space $K(G,1)$ and is a CW complex, show that $\pi_n(X^n)$ is free Abelian for $n \geqslant 2$. However, I think I ...
1
vote
0answers
32 views

what are Classifying spaces actually classifying

Let $G$ be a group. When we say the classifying space of $G$ we are actually meaning the classifying space of the principal $G-$bundles because the notion of classifying spaces is about classifying ...
0
votes
0answers
36 views

Mapping cones for homotopic maps are homotopic

Suppose you have two homotopic maps $f,g:X\rightarrow Y$, with homotopoy $F:X \times I \rightarrow Y$. I want to show that the mapping cones for $f$ and $g$ are homotopic, so I want to show that $C_f ...
2
votes
2answers
85 views

Show $S^2$ is not homeomorphic to the closed unit disk.

How to show unit closed disk is not homeomorphic to sphere $S^2$?
3
votes
2answers
60 views

Double mapping cylinder of an open cover

Let $X$ be a topological space and $X=U\cup V$ an open cover of $X$. Let $Z$ be the double mapping cylinder of the inclusions $U\leftarrow U\cap V\rightarrow V$. One has an obvious map $Z\rightarrow ...
1
vote
1answer
75 views

Finding Fixed Point

If there is a continuous function from the closed unit disk to itself such that it is identity map on boundary, must it admit a fixed point in the interior of the disk?
2
votes
1answer
73 views

Mapping between disk

If there is a continuous function from the closed unit disk to itself such that it is identity map on boundary, must be onto?
0
votes
0answers
18 views

Are levelwise endofunctors of simplicial sets homotopical?

Let $Set$ be the category of sets and $sSet$ the category of simplicial sets (the category of functors from $\mathbb{\Delta}^{op}$ to $Set$). Every functor $F:Set \to Set$ induces a functor ...
7
votes
2answers
319 views

The fundamental group of a compact, locally simply connected space is finitely generated

Let $X$ be a compact space that is also locally simply connected (any point has a local base of simply connected open sets). Prove that the fundamental group at any point is finitely generated.
0
votes
2answers
62 views

Reversed composition, proving homotopy

I am working on a problem on path homotopy out of Mukres' topology (2ed., chapter 9 on Fundamental Group, section 51) that goes like this: Given that $h, h': X \to Y$ and $k, k': Y \to Z$ are ...
1
vote
2answers
46 views

Homotopy class of a loop in the $2$-skeleton of a simplicial complex

Suppose I have a loop $\sigma : [0, 1] \rightarrow X$ in a path-connected finite simplicial complex $X$. I know that $\sigma$ can be homotoped so that it lives in $\text{sk}^2(X)$, but is this ...