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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16 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
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1answer
55 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 ...
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2answers
45 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
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1answer
92 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 = ... = ...
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32 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.
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1answer
83 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?
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1answer
75 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 ...
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34 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?
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2answers
52 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 ...
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2answers
44 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 ...
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282 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?
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59 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 ...
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1answer
21 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 ...
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22 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
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1answer
70 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 ...
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1answer
70 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 ...
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41 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 ...
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2answers
50 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, ...
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0answers
21 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
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1answer
57 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 ...
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19 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 ...
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1answer
45 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 ...
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1answer
60 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 ...
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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 ...
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1answer
103 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 ...
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49 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 ...
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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
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1answer
122 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: ...
2
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75 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 ...
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1answer
74 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 ...
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1answer
69 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 ...
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2answers
30 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
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1answer
35 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 ...
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1answer
109 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 ...
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1answer
57 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
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0answers
92 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 ...
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2answers
64 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 ...
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0answers
19 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}) ...
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39 views

Space deformation retracts to a point.

If space $X$ deformation retracts to a point $x\in X$, then for each open $U\in X$ containing $x$ there exists an open $V\in U$ again containing $x$ s.t. inclusion of $V$ into $U$ is nullhomotopic. ...
2
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1answer
52 views

Principal bundles with compact simply connected structure group over 2-manifolds

I'm reading Thomas Friedrich's "Dirac Operators in Riemannian Geometry," where the following is stated (in the Remark on page 42 before section 2.2 begins, if anyone is following along with the ...
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28 views

When is a fibration (canonically) a principal fibration over its group of automorphisms?

The question is inspired by the following observation: Let $p: X'\to X$ be a connected covering space where both spaces are suitably nice (say they are CW complexes), then $p: X' \to X$ is a principal ...
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52 views

commutivity of suspension and quotient space

Let $(A,A_0)$ be a $CW$-pair such that $A_0$ is a $CW$-subcomplex of $A$. Let $\Sigma$ be the suspension. Is $\Sigma(A/A_0)$ homotopy equivalent to $\Sigma A/\Sigma A_0$ or not?
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1answer
40 views

Direct Limit of Grassmannians

Let $X$ be a topological space and $G_n(\mathbb{C}^m)$ be the space of vector subspaces of $\mathbb{C}^m$ of codimension $n$. Let $G_n(\mathbb{C}^\infty):=\bigcup_{m=n}^{\infty}G_n(\mathbb{C}^m)$ ...
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1answer
50 views

Finding an explicit homotopy to prove inverses exist in fundamental group

The problem statement is: Suppose $X$ is a topological space with base point $x$. Let $\gamma_0:I\to X$ be the constant map $\gamma_0(s)=x, \forall s\in I$. Suppose $\gamma:I\to X$ is a continuous map ...
2
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1answer
57 views

Is there a rational homotopy equivalence between $\Omega S^3$ and infinite complex projective space?

The singular cohomology of the loop space $\Omega S^3$ of the 3-sphere is a divided power algebra $\Gamma_{\mathbb Z}[s]$ on one generator $s$ of degree 2, so the rational cohomology is a the ...
2
votes
1answer
91 views

Moore Spaces: explicit CW-complex for $M(\mathbb{Z}_m, n)$

Given an abelian group $G$ and an integer $n \ge 1$ we can construct a $CW$ complex such that $H_n(X) \cong G$ and $\tilde{H}_i(X)=0$ for all $i \neq n$. We call this $CW$ complex a Moore space and ...
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112 views

Does a group homomorphism up to homotopy induce a map between classifying spaces?

Let $H$ and $G$ be topological groups and denote by $BH$ and $BG$ their classifying spaces. If $$f\colon H\rightarrow G$$ is a continuous group homomorphism, we get an induced map of spaces ...
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1answer
79 views

Homotopy commutative diagrams and homotopy equivalent spaces

The question is fairly general. Suppose I have a homotopy commutative diagram of the form \begin{equation} \require{AMScd} \begin{CD} A @>{f}>> B\\ @V{h}VV @V{i}VV \\ C @>{g}>> D ...
4
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59 views

Curves Knotted in the Torus

I've been struggling with the following problem. I suspect it's actually not so hard, but I'm missing something relatively obvious about the proof. The attached image will help. Suppose $K$ and $L$ ...
3
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2answers
90 views

One point union, second homotopy group is not finitely generated?

Let $X$ be the one-point (wedge sum) union of the circle $S^1$ and the sphere $S^2$. What is the easiest way to see that the abelian group $\pi_2(X)$ is not finitely generated?