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

1
vote
0answers
12 views

Correct definition of model category

When answering this question, In a model category, is the full subcategory of fibrant objects a reflective subcategory? I realized that I wasn't even sure what the correct definition of a model ...
1
vote
1answer
23 views

Extend a map over a $n+1$-cell IFF $f_nϕ$ is nullhomotopic.

Prove: If the map $f_n$ is defined on the $n$-skeleton $X_n$ and you want to define it on an $n+1$-cell with attaching map $ϕ:S_n→X_n$, then you can do so if and only if $f_nϕ$ is nullhomotopic. I ...
1
vote
1answer
37 views

$T^2-D$ does not retract to the boundary $\partial D$

First of all: yes, there is already a post about it, but I missread retract as strong deformation retract and wanted to know if this solution is right if we really do assume the stronger assumption of ...
0
votes
0answers
23 views

Gluing along infinitely many trivial cofibrations

I am in a situation where I have a space Y obtained from a space X by gluing on infinitely many trivial cells. That is, I have a collection of maps $f_\alpha: A_\alpha \to X$ each of which is a ...
1
vote
0answers
8 views

Equicontinuous homotopies of families of uniformly equicontinuous functions

Let $f\colon X \to Y$ be a uniformly continuous function. Then I think it is "well-known" that it may be approximated by a Lipschitz function, and how well one can do this depends on the modulus of ...
0
votes
0answers
18 views

Show that $h$ is homotopic to the identity map relative to $C$.

This is problem 5.3 and 5.4 in Armstrong's Basic toplogy. They are very much connected and i have solved problem 3. 3: Let $D$ be the disc bounded by $C$, i.e. $S^1$, parametrize $D$ using polar ...
1
vote
0answers
45 views

Why is the punctured plane not homotopic to the circle?

I know that the fundamental group of $X = \mathbb R^2 \setminus \{(0,0)\}$ is the same as the fundamental group of the circle $Y = S^1$, namely $\mathbb Z$. However, $X$ and $Y$ are not homotopic, ...
-2
votes
0answers
44 views

Help needed with picking topic for Master's thesis [closed]

I am undergraduate student and I am currently trying to finnish masters degree. In order to do that I need to write Masters thesis, but currently I am stuck with picking actual topic. I did my ...
2
votes
2answers
59 views

Elementary geometric characterization of spheres?

I've read the following two theorems. Theorem. A compact connected metric space whose points are cuts points with the exception of at most two is homeomorphic to the unit interval. Theorem. A ...
0
votes
0answers
60 views

How much algebra one needs to study algebraic topology and homotopy theory?

I wonder how much algebra(group theory, abstract algebra, linear algebra, ring theory, field theory) is assumed as a prerequisite in most of the modern algebraic topology text. For example, these ...
3
votes
2answers
38 views

Trivial loop on the $1$-Skeleton

Following Hatcher's proof of Hurewicz Theorem (version of 1999) we arrive at the point that we must show that the loop in the picture, created following the path $0,1,2,3,1,0,1,3,0,3,2,0,2,1,0$, is ...
1
vote
0answers
13 views

Munkres positive linear map definition of path product (page 328)

I'm confused by Munkres' definition of the path product using the positive linear map. He defines the positive linear map $p: [a,b] \rightarrow [c,d]$ to be the unique map of the form $x \mapsto mx + ...
4
votes
1answer
46 views

Homology with local coefficients as a functor from pointed, path-connected spaces and $\pi_1$-modules.

A local system of coefficients on a space $X$ is a functor $F\colon \Pi(X)\rightarrow Ab$ from the fundamental groupoid to the category of abelian groups. From this, one can define the homology groups ...
1
vote
2answers
40 views

Homotopy equivalent but not deformation retraction [closed]

Can somebody come up with an example where $X \subset Y$, the inclusion gives a homotopy equivalence between $X$ and $Y$, but there is no deformation retraction from $Y$ onto $X$?
3
votes
1answer
85 views

Degeneracies of simplex $y$ which appears as any face of some simplex $x$

Let $K$ be simplicial set and $d_i:K_n\rightarrow K_{n-1}$, $s_i: K_n\rightarrow K_{n+1}$ ($i = 0,...,n$) face and degeneracy maps respectively. Suppose we have some $x\in K_n$ with $d_0x = ... = ...
1
vote
0answers
29 views

projective model structure on presheaves , hom-functors are always cofibrant

Why hom-functors are always cofibrant in the projective model structure in $[\cal T,\cal V]$? The claim is here on page 5.
4
votes
1answer
75 views

Fixed point property of spaces having same homotopy type

Suppose X and Y have same homotopy type.X as a topological space has fixed point property.Can we conclude anything about fixed point property of Y?
0
votes
1answer
54 views

Homotopy colimit,weighted colimit, homotopy theory

Let's take the definition of $\mathbb{hocolim}$ as the representation of the representable functor like this: $\underline{\cal M}(\mathbb {hocolim}_{ \cal D} F,m)\cong \mathrm {{sSet}^{\cal ...
0
votes
0answers
33 views

Topology of a specific shape

How to find topology of this shape? It's Fundamental group, homotopy type and some interesting information about it?
2
votes
2answers
33 views

On the surjectivity of the Hurewicz homomorphism

The Hurewicz homomorphism is a surjective homomorphism from $\pi_n(X) \to H_n(X)$ if $\pi_{n-2}(X)=0$ according to Wikipedia. But if it is surjective then how could the following (contradiction) I ...
1
vote
2answers
36 views

Suspension: if $X$ is $(n-1)$-connected CW, is $SX$ $n$-connected?

If $X$ is $(n-1)$-connected CW complex, is that true that $SX$ is $n$-connected? I'm trying to understand Freudenthal Suspension Theorem on Hatcher. We define the suspension map: $\pi_i(X)\simeq ...
0
votes
2answers
42 views

How to show a straight line homotopy is continuous?

Given $f$ and $g$ continuous maps from $X$ into $\mathbb{R}^{2}$, how to show that the straight line homotopy map $F(x,t)=(1-t)f(x)+tg(x)$ is continuous?
1
vote
2answers
53 views

Universal covering group and fundamental group of $SO(n)$

The universal cover of $SO(2)$ is $\mathbb{R}$, whilst the fundamental group is $\mathbb{Z}$. That is $$ SO(2) \cong \mathrm{universal\ cover}/\pi_1 $$ Likewise, I believe that the universal cover of ...
2
votes
1answer
16 views

Why does sheaf $\pi_0$ of a simplicial presheaf determine maps to sheaves?

This question refers to an argument from Section 6, p. 22 of Freed–Hopkins, "Chern–Weil forms and abstract homotopy theory." There's something presumably straightforward I'm just not ...
0
votes
0answers
13 views

Good reference for “solving equation $f(x)=0$ by homotopy and continuation methods”

I need a good reference for "solving equation $f(x)=0$ by homotopy and continuation methods". If $f:X\to Y$ is a continuous map between to linear space $X$ and $Y$, we want to find the roots of $f$. ...
2
votes
1answer
59 views

How to prove that the set of real $ n \times n $ PSD matrices of rank $\leq r < n $ and unit trace, is not contractible?

Extended Version of the Question: How to prove that the set of real $n \times n$, symmetric positive semidefinite (PSD) matrices of rank $\leq r$ ($ 1 \lt r \lt n-1 $) and unit trace, is not ...
2
votes
1answer
58 views

Eilenberg-MacLane Spaces $K(G,n)$: Construction!

I'm looking for an easy construction of $K(G,n)$, Eilenberg-MacLane spaces. I know I can use the Postnikov Towers for the upper part $\pi_i(X)=0$ for $i > n$. For the lower part $\pi_i(X)=0$ for ...
1
vote
0answers
36 views

Fundamental group of a circle with rational lines

Let $X$ be the subset of $\mathbb{R}^2$ given by the union of the unit circle the $y$-axis all lines through the origin with rational slopes equipped with the subspace topology. Is there a simple ...
0
votes
2answers
46 views

Is this 2-complex a $K(\pi,1)$?

Consider $CW$-complex $X$ obtained from $S^1\vee S^1$ by glueing two $2$-cells by the loops $a^5b^{-3}$ and $b^3(ab)^{-2}$. As we can see in Hatcher (p. 142), abelianisation of $\pi_1(X)$ is trivial, ...
1
vote
0answers
19 views

Why is symmetric group action needed for symmetric spectra?

I know that Boardman spectra aren't supposed to have an on-the-nose commutative smash product, and symmetric spectra -- which look like essentially the same thing, except the spaces $X_n$ have to come ...
3
votes
1answer
49 views

Long Exact sequence of Relative Homotopy Groups: examples and applications

I'm going to make a talk around higher homotopy groups, and the long exact sequence of relative homotopy groups. I would like to show some nice examples and applications of this theorem after the ...
0
votes
0answers
17 views

Parallely transported vector continuous with respect to homotopy of curve?

Let $M$ be a smooth manifold with connection. Choose two points $p,q \in M$ and connect them with curve $\gamma _0$. Suppose I take fixed vector at $x$ and parallel transport it to $y$. Denote this ...
3
votes
1answer
40 views

Is the smash product of two Moore spaces again a Moore space?

Write $M(G,n)$ for the Moore space with $\tilde{H}_\ast(M(G,n);\mathbb{Z})$ naturally isomorphic to $G$ concentrated in degree $n$. Now fix finitely generated (Abelian) groups $G$ and $H$ and ...
0
votes
1answer
56 views

Proof of $BGl(n)\simeq Gr(n,\infty)$

What is a direct proof for the existence of a weak homotopy equivalence between the Grassmanian $Gr(n):=Gr(n,\infty)$ and the classifying space $BGL(n)$ of $GL(n)$? They both represent the ...
2
votes
1answer
42 views

Van Kampen theorem application in a simple three-holed figure

The purpose of this question is to understand the computations to get the expression of the fundamental group in a simple case using the Van Kampen theorem. Let $X$ be the three holes object ...
2
votes
1answer
79 views

Maps from Sum of Projective Planes to Circle

this is a problem from Lee's Topological Manifolds, 11-21. It asks the following: What compact, connected surfaces $M$ admit a non-nullhomotopic map to the circle? So, we use the classification of ...
1
vote
0answers
48 views

how to evaluate homotopy group of this specific structure

I am a Ph.D. student of physics and now I have some problems regarding the evaluation of homotopy group of a specific structure. In a paper, a specific topological structure is defined. The structure ...
0
votes
0answers
13 views

Using sequence of least squares solutions to solve non-linear problems?

I do know about the iteratively reweighted least-squares and have played around with it to some success finding non-linear solutions (like minimizing non-2-norms to achieve solutions which seem to be ...
3
votes
1answer
105 views

The fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed

Determine whether the fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed is trivial, infinite cyclic, or isomorphic to the figure eight space. I found this answer: ...
1
vote
0answers
44 views

Theta-space is a deformation retraction of the doubly-punctured plane, how to find equations.

That theta space is given by $S^1\cup(0\times[-1,1]) \subset\mathbb{R}^2$ it is said that this space is a deformation retract of the doubly punctured plane, here is the explanation I found: The ...
2
votes
1answer
64 views

The fundamental group of $B^2\times S^1$

In one exercise we are supposed to find the fundamental group of $B^2\times S^1$. It is given that the fundamental group is $\mathbb{Z}$, because we can show that $S^1$ is a deformation retract of ...
5
votes
1answer
67 views

Homotopy cardinality of the category of categories

The category of finite sets has homotopy cardinality $e$, because $$ |{\bf FinSet}|=\sum_{n=0}^{\infty}\frac{1}{\left|\operatorname{Aut}\ [n]\right|}=\sum_{n=0}^{\infty}\frac{1}{n!}. $$ What is the ...
0
votes
2answers
26 views

$h,h':X\to Y$ are homotopic and $k,k':Y\to Z$ are homotopic, then $k\circ h$ and $k'\circ h'$ are homotopic.

I want to show that if $h,h':X\to Y$ are homotopic and $k,k':Y\to Z$ are homotopic, then $k\circ h$ and $k'\circ h'$ are homotopic. This means that there is a continuous map $F_1:X\times I \to Y$ ...
5
votes
1answer
27 views

Does a weak homotopy equivalence induce an equivalence of categories on the fundamental groupoids?

Let $f\colon X\rightarrow Y$ be a weak homotopy equivalence. ($\pi_0(f)$ is a bijection and $\pi_n(f,x)$ is an isomorphism for all basepoints $x\in X$ and all $n$.) It induces a functor ...
8
votes
1answer
102 views

Where is basic algebraic topology in basic algebraic geometry?

I'm a student meeting commutative algebra and algebraic geometry for the first time. The idea of studying every (commutative) ring geometrically via its spectrum (as a locally ringed space) is ...
2
votes
1answer
50 views

Let $X = \Sigma Y = Y \wedge S^1$, cup product $\tilde{H}^p(X) \otimes \tilde{H}^q(X) \to \tilde{H}^{p+q}(X)$ is the zero homomorphism? [duplicate]

We take cohomology with coefficients in a commutative ring $R$ and we write $\otimes$ for $\otimes_R$. Let $X = \Sigma Y = Y \wedge S^1$. How do I see that the cup product$$\tilde{H}^p(X) \otimes ...
2
votes
0answers
87 views

$E_{\infty}$ algebra in characteristic zero

Let $A^{\bullet}$ be a cosimplicial commutative algebra over a field $\Bbbk$. Denote with $N(A)^{\bullet}$ the conormalized Moore complex. Since $A^{\bullet}$ is equipped with a product, the ...
1
vote
2answers
56 views

A natural topology on space of continuous functions

Let $X$ and $Y$ be two topological spaces. Let $C(X,Y)$ be set of all maps from $X$ to $Y$. Does there exists a natural map topology on $C(X,Y)$? By main motivation is to define two maps $f,g$ as ...
2
votes
0answers
41 views

Relation between homotopy and homology groups of realization of simplicial abelian group

Let $X$ be a simplicial abelian group, $U(X)$ the corresponding simplicial set and $|U(X)|$ its geometric realization (assumed to be path-connected). Then, for $k \geq 1$, how are $\pi_k(|U(X)|)$ and ...
0
votes
0answers
17 views

Cellular map homotopic to geometric realisation of the simplicial map induced by cellular map

Following 31-34 of http://arxiv.org/pdf/1508.05446v1.pdf We have that for any regular cellular map $f:X \to Y$ between regular CW complexes induces a monotone map $\mathcal{F}(f):\mathcal{F}(X,X_{i}) ...