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

3
votes
1answer
65 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
23 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
74 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
23 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
24 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
29 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
40 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
44 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
31 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
87 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
101 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
33 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
89 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
61 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
74 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
19 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
326 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
47 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 ...
1
vote
0answers
48 views

Fundamental group of quotient disk

Consider the disk $D^{2}$ in $\mathbb{R}^{2}$. By taking out two disjoint, smaller disks within $D^{2}$, we obtain a disk of genus 2. Now consider identifying the boundaries of the two deleted circles ...
2
votes
1answer
51 views

Computing fundamental groups of products

Let $X$ be a connected graph and $S^{1}$ the usual circle and consider the product $X \times S^{1}$. How would one compute the fundamental group $\pi_{1}(X \times S^{1})$ in this case? I know that one ...
0
votes
0answers
63 views

A short exact sequence of chain complexes with null-homotopic chain maps

Problem Suppose $0\to K'\xrightarrow iK\xrightarrow pK''\to 0$ is an exact sequence of chain complexes of modules over $R$, say. If chain maps $i,p$ are null-homotopic, then $K$ is contractible. ...
4
votes
0answers
43 views

Homotopic properties of the Spin group from geometric algebra

There are two possible ways to define the $\mathrm{Spin}(n)$ group of Euclidean $n$-space from $\mathrm{Pin}(n)$. First is that $\mathrm{Spin}(n)$ is the identity component of $\mathrm{Pin}(n)$. ...
3
votes
1answer
82 views

what is the two sheeted covering space of a sphere with a diameter?

I have calculated the fundamental group of sphere with a diameter using Van-Kampen theorem, which is $Z$. So corresponding to subgroup $2Z$ there exist a two sheeted connected covering space. So ...
1
vote
0answers
23 views

Proof of compression criterion (iff condition for representing zero in relative homotopy group)

I'm trying to prove (and understand) the compression criterion which states that a function $f\colon(I^n,\partial I^n,J^n)\to (X,A,x_0)$ represent zero in the relative homotopy group $\pi_n(X,A,x_0)$ ...
2
votes
0answers
49 views

Request for Details of Proposition 0.16 in Hatcher's Algebraic Topology

In Proposition 0.16 of Hatcher's Algebraic Topology, Hatcher claims that $ X^n \times I $ is obtained from $ (X^n \times \{0\}) \cup ((X^{n-1} \cup A^n) \times I) $ via attaching copies of $D^n \times ...
2
votes
2answers
60 views

A criterion to prove that a topological space is simple connected.

Let $\{U_i\}$ be an open covering of the space $X$ having the following properties: (a) There exist a point $x_0$ such that $x_0\in U_i$ for all $i$. (b) Each $U_i$ is simply-connected. (c) If ...
2
votes
1answer
38 views

CW approximation as an adjoint equivalence?

I have some intuitions that I want to make precise and accurate. I am very sure there are many mistakes in my understanding, but allow me to state it in the raw form as I sense it for now. Let ...
2
votes
1answer
49 views

Exercise 1.1.4 in Hatcher's Algebraic Topology, star-shaped

The following is Exercise 1.1.4 in Hatcher's Algebraic Topology: A subspace $X\subset \Bbb R^n$ is said to be star-shaped if there is a point $x_0 \in X$ such that, for each$x \in X$ , the line ...
3
votes
1answer
38 views

Comparison of direct limits and homotopy direct limits

Suppose we have a finite diagram in the category of topological spaces. What is the condition for homotopy equivalence of the natural map from homotopy direct limit of this diagram to it's direct ...
0
votes
1answer
29 views

punctured Mobius band in high dimension

Let $M^{n+1}$ be the non-trivial line bundle over sphere $S^n$. When $n=1$, $M^2$ is Mobius band and the punctured space $M^2\setminus *\simeq S^1$. How about $M^{n+1}\setminus *$ in general? Does ...
8
votes
1answer
193 views

Difference between homotopy equivalence and homeomorphism - dimensionality

(The most voted answer to) This question shows spaces of the same dimension can be homotopy equivalent but no homeomorphic. On the other hand "difference in dimension" is still a nice way to tell ...
1
vote
2answers
54 views

Topological group closed path

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$, $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let $f,h$ be closed paths ...
1
vote
1answer
35 views

homotopic closed paths in topological group

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$ $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let f,h be closed paths in ...
2
votes
0answers
20 views

Constant perimeter homotopy between a square and a circle

Is there a relatively simple homotopy between a square and a circle so that the perimeter of the curve is constant during the transformation?
1
vote
2answers
65 views

Show that the plane $\mathbb{R}^2 - \{(-1,0), (1,0)\}$ is homotopy equivalent to $C(1,0) \cup C(-1,0)$

I'm trying to find a deformation retract of the union of the two circles to $\mathbb{R}^2 - \{(-1,0), (1,0)\}$, any help is appreciated.
4
votes
0answers
61 views

Homology of $\mathbb{R}\setminus A_+$. [duplicate]

Let $A$ be the unit circle in the $xy$ plane in $3$-dimensional real space and let $A_+$ be a semicircle. I have to compute the homology of $\mathbb{R}^3\setminus A_+$. I was thinking that ...
0
votes
1answer
43 views

'Elementary' proof of $\tilde{X}$ is contractible iff $\pi_n(X) =0 \forall n \ge 2$.

I have studied algebraic topology, but have not studied $\pi_n(X)$ any further than $n=1$. Is there a proof of $\tilde{X}\cong 1 \Leftrightarrow \pi_n(X) =0 \forall n \ge 2$ that does not require ...
2
votes
0answers
53 views

In what sense is cohomotopy dual to homotopy?

I understand the duality in the case of homology and de Rham cohomology. Through integration a chain can be understood as a linear functional on differential forms and vice versa. Is there a way to ...