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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42 views

How to show the failure of the homotopy extension property for the pair $I=[0,1]$ and $A=\left\{ 0,1,\frac{1}{2},\frac{1}{3}…\right\}$?

I'm facing difficulties regarding a statement which Hatcher makes in page 14 of his 'Algebraic Topology' book-that there's no continuous retraction from $I\times I\rightarrow ...
2
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1answer
36 views

Does a cofiber sequence of CW complexes induce a cofiber sequence of skeleta?

Suppose $X\to Y \to Z$ is a cofiber sequence of CW complexes. We can replace the maps with homotopic cellular maps $X_n\to Y_n \to Z_n$ taking the $n$-skeleton of $X$ to the $n$-skeleton of $Y$ and ...
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243 views

A question about Homotopy (Michael Harris's recent book)

In the recent book "Mathematics without Apologies: Portrait of a Problematic Vocation" by Michael Harris there is some passage I want to call your attention on. Specifically, pages 211-212. Could ...
2
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76 views

H-space and fundamental group

I have a certain pointed path-connected topological space $X$, and a natural map $\mu : X \times X \to X$. I want to prove it is $H$-space. So I need to prove that maps $\mu(x_0,-)$ and $\mu(-,x_0)$ ...
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52 views

Question on the relation between the $n$-coskeleton functor and the internal hom-adjunction

(First, let me apologize that I asked an unanswered and related question in the stable case.) The category $\operatorname{sSet_+}$ of pointed simplicial sets is symmetric monoidal closed, i.e. there ...
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32 views

A question on the relation of truncation $\tau_{\leq n}$ to the internal hom adjunction of spectra

Let $$f\colon X\wedge Y\to Z$$ a morphism of (symmetric) spectra with adjoint morphism $$g\colon X\to [Y,Z]$$ where the brackets denote the internal hom of spectra. Let $\tau_{\leq n}$ be the standard ...
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149 views

Jeffrey Strom, ''Modern Classical Homotopy Theory'', prerequesites and recommended knowledge

This is mostly adressed to those who has studied the book. I've heard a lot about this particular book and browsed it's contents. However, I would like to be certain whether I will encounter some ...
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1answer
87 views

How do I show homotopy equivalence between two topological spaces?

Are two topological spaces homotopically equivalent if they are homeomorphic? If one is a deform retract of the other? Is there a way to use quotient spaces here or is that something else?
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1answer
118 views

Why is the identity map of the circle not straight-line homotopic to a constant map?

According to the book 'Introduction to Topology. Pure and Applied' by C Adams & R Franzosa : $\boldsymbol{\sf THEOREM\ 9.9.}$ A circle function $f : S^1 \rightarrow S^1$ has degree $0$ if and ...
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43 views

Checking continuity for homotopy on glued space — is it necessary?

These days, I'm working through Bredon's Topology and Geometry in order to get a better grasp on topology and fill in the gap of never having seen homology theory. In chapter 14, Bredon proves: ...
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1answer
19 views

What is the degree of $A(\theta)=\theta+\pi$?

What is the degree of the circle function $A: S^1 \rightarrow S^1$, $A(\theta)=\theta+\pi$? Considering coefficient of $\theta$ to be $1$ and ignoring translation $\pi$ so $deg (A)=1$. On the other ...
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56 views

Homotopy of maps to the product topology

Let $RP$ denote the real projective plane and let $S$ denote the circle. I recently encountered a question of the form: Prove every map $f:RP \to S\times S$ is null-homotopic. I know that every map ...
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1answer
86 views

Loop group of U(1)

The loop group of $U(1)$, $LU(1)$ is the space of maps from the circle, $S^1$ to $U(1)$. The based loop group of $U(1)$, $\Omega U(1)$ is the space of based maps from the circle, $S^1$ to $U(1)$. It ...
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1answer
26 views

Prove that there is only one homotopy class of continuous functions from $X$ to $D$

Question: Show that if $X$ is a topological space, and $D$ is the disk in the plane, then there is only one homotopy class of continuous functions from $X$ to $D$. Answer: Only one case I know ...
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1answer
36 views

Let f and g be paths in R. Show that f is homotopic to g.

Question: Let $f$ and $g$ be paths in $\mathbb R$. Show that $f$ is homotopic to $g$. Answer: Does the homotopy $F(x,t)=tf(x)+(1-t)g(x)$ is a proof for the statement?
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2answers
51 views

Why $f$ and $g$ are still homotopic in $\mathbb R^2 - {\{O}\}$?

According to Topology by C Adams : and according to Exercise 9.4. : I understand the Example but I don't understand the case of $\mathbb R^2 - {\{O}\}$: 1- The only way to 'deform' $f$ to ...
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22 views

Notation in H-spaces from Homotopy point of view

This book doesn't have a notation index. What do m|e, and rel(e,e) likely mean? Many thanks!
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1answer
193 views

Algebraic Topology and Homotopy Theory prerequisites

I would like to know what are the prerequisites for studying algebraic topology (with a homotopical viewpoint). The books I have in mind are Selick ("Introduction To Homotopy Theory") May ("A ...
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1answer
184 views

Based space, commuting in diagram up to homotopy, dual Barratt-Puppe sequence.

For a based map $f : X \to Y$, define the "homotopy fiber" $Ff$ to be$$Fd = X \times_f PY = \{(x, \chi) : f(x) = \chi(1)\} \subset X \times PY.$$Equivalently, $Ff$ is the pullback displayed in the ...
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1answer
86 views

Why is the unit circle, $\mathbf{S}^1$, a deformation retract of $\mathbf{R}^2$ minus any point?

It is clear that $\mathbf{S}^1$ is a deformation retract of $\mathbf{R}^2\setminus\{0\}$ since we can consider the straight line deformation retract $H\colon (\mathbf{R}^2\setminus\{0\}) \times [0, 1] ...
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38 views

inclusion of homotopy fiber and induced map on homology group

Given a fibration $F \to E \to B$, under what circumstances does the inclusion of the homotopy fiber into $E$, $F \to E$, induce injections on homology? The specific case I'm dealing with involves the ...
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22 views

Lifting a homotopy class $S^k\to X$ into a simplicial set $X$ which is not fibrant but satisfies some weaker horn filling condition

Let $X$ be a connected simplicial set. If $X$ is an Kan complex and $k\geq 0$, then every element $$ \tilde f\in\operatorname{Hom}_{Ho(sSet)}(S^k,X) $$ of the homotopy classes from $S^k$ to $X$ lifts ...
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1answer
30 views

Intersection preserves homotopy equivalence

Let $Z$ be a topological space with subspaces $X$, $Y$, $X'$ and $Y'$. Suppose that $X$ is homotopy equivalent to $X'$ and $Y$ is homotopy equivalent to $Y'$ do we have that $X\cap Y$ is homotopy ...
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55 views

Nth Homotopy Group Isomorphic to [T^n, X]

Following Spanier's book on algebraic topology chapter 1, section 6 about suspensions, I'm wondering about the following questions: 1) We know that $S^n$ is an H co-group for all $n\geq1$ because ...
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0answers
52 views

Homotopic family of curves

I stumbled over the following question. Imagine we have a two homotopic curves on the sphere $\mathbb{S}^1$ namely $\gamma_1,\gamma_2$. Then we can write them as $\gamma_{i}(t) = e^{i \alpha_i (t)}$ ...
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1answer
76 views

Action of $S^1$ on homotopy groups of an $S^1$-space

I am interested in the following question : Let $S^1$ be the $1$-sphere, seen as a topological group by being the unit sphere in the complexe plane $\mathbb{C}$. Let $X$ be a (good, ... etc) pointed ...
2
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1answer
56 views

Question about maps to $K(G,1)$

I have an unfortunately basic confusion about two results in Hatcher's Algebraic Topology. Let $G$ be an abelian group. Theorem 4.57 specialized to $n=1$ says that there is a bijection $$\langle X, ...
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1answer
27 views

How to check F:AxI->B is continuous

A and B are topological spaces.Let f and f' are continuous maps from A to B and homotopic.Then we need F:AxI->B,continuous,where F(s,0)=f(s) and F(s,1)=f'(s). Now my question is if we want to ...
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66 views

Convexity of space of integrals of homotopic paths

Let $B$ be a Banach space. Consider a set $Q \subset B$, a path $\gamma_0:[0,1] \to Q$ and the set of paths $\Gamma_0$ of paths $[0,1] \to Q$ homotopic (with ends fixed) to the path $\gamma_0$. Let ...
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0answers
68 views

How is the delooping of a groupoid constructed explicitly?

Let $G$ be a group, nlab's "delooping" page says that $G$ can be considered as a discrete groupoid in the $(\infty,1)$-topos $\infty$Grpd of $\infty$-grupoids, the delooping of $BG$ is then the ...
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19 views

show that exists exactly one n such that $\gamma_1$ is homotopic to $\gamma_2 = e^{2\pi int}$

Let $\gamma_1 $ be a closed path in $\Bbb{C}\setminus0$ such that $0 \in int (\gamma_1)$ show that exists exactly one $n \in \Bbb{Z}\setminus 0$ such that $\gamma_1$ is homotopic to $\gamma_2 : ...
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6 views

Homothetic Transformation for a Jordan closed set

The set $T$ is a closed Jordan set, and there exists a limit $$ T^{\infty}=\mathbf{t}_{0}+ \lim_{a \rightarrow \infty } h_{a}(T-\mathbf{t}_{0}) $$ of non-zero measure, where $h_{a}$ is a homothety ...
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1answer
22 views

Is the homotopy category of modules over a quasi-Frobenius ring (pre)additive?

Let $R$ be a quasi-Frobenius ring and $Mod_R$ the category of $R$-modules. One can prove that it admits a model structure whose weak-equivalences are stable equivalences, whose cofibrations are monos ...
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33 views

If weak-equivalences in a model category are generated by an equivalence relation $\sim$ on hom-sets…

Suppose that $C$ is a bicomplete category, and $\sim$ is an equivalence relation defined on every hom-set of $C$, compatible with composition. Then we can define the quotient category $C/\sim$ whose ...
3
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1answer
67 views

Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N\to M$ between closed oriented connected manifolds. Let $X$ and $Y$ be diffeomorphic submanifolds of $M$, and assume $h$ to be ...
2
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0answers
76 views

Is the Quillen Injective Model Structure on the category of positive cochain complexes of R-modules (co)fibrantly generated?

Denoted with $Ch^+_R$ the category of positive cochain complexes of R-modules (for a commutative ring $R$), it admits a model structure where: weak-equivalences are quasi-isomorphisms; cofibrations ...
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35 views

Intuition for homotopy (co)limits in triangulated categories

The following definition is taken from Daniel Murfet's Triangulated Categories Part I notes. Let $\mathcal T$ be a triangulated category with countable coproducts. Suppose we are given a ...
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1answer
41 views

Hypothesis in homotopy equivalence inducing isomorphism in the fundamental groups

Let $X$ and $Y$ be topological spaces. If $f\colon X\to Y$ is a homeomorphism, then it induces an isomorphism $f_\sharp\colon\pi_1(X,x_0)\to \pi_1(Y,f(x_0))$. All good. As far as I know, the result ...
2
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0answers
50 views

Topological Boundary Map

In May, Concise Algebraic Topology, p. 108-109, for a cofibration $A \rightarrow X$ a "topological boundary map" is defined as the composite: $X/A \xrightarrow{\psi^{-1}} Ci \xrightarrow{\pi} \Sigma ...
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1answer
43 views

Is the reflective localization $L_WC$ of a category $C$ equivalent to $C$? What am I missing?

This is probably a dumb question but this is going over my head at the moment, I came here from nlab's entry on localization (http://ncatlab.org/nlab/show/localization). Let $C$ be a category, let $W ...
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1answer
103 views

Composition of homotopy classes with self-maps of spheres

Are there some general rules/formulas on the relation between the homotopy class $[f]\in \pi_i(S^n)$ and the homotopy class of the composition $S^i\stackrel{a}{\to} ...
3
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1answer
70 views

Choosing projective replacement to be functorial

A basic result of homological algebra says that if $\mathsf A$ is an abelian category with enough projectives, then the mapping $P:\mathsf{Obj}(\mathsf A)\rightarrow \mathsf{Obj}(\mathsf{K} ^+(\mathsf ...
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1answer
81 views

Proving that the tangent vector of a simple closed curve rotates by $ 2 \pi$

I am trying to prove that if $\gamma(t)=(x(t),y(t))$ ,a function from the closed interval $[0,1]$ to $\mathbb{R^2}$ is a simple closed unit speed curve such that $\gamma '(0)=\gamma '(1)$. Then the ...
2
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1answer
61 views

$S^n$ is not a retract of the disk $D^{n+1}$ and Brouwer's Fixed Point Theorem.

I was trying to understand Hirsch's proof of this fact: "There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by ...
2
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1answer
66 views

Does symplectic K-theory $KSp$ have products?

The real and unitary topological $K$-theories are cohomology theories defined by the $\Omega$-spectra $KO$ and $K$ respectively. These are multiplicative theories with products deriving from the ...
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2answers
62 views

Another clarification about Thom-Pontrjagin construction

This is the second part of the following solved question. [I'm following Bredon's Book]. After explaining the idea behind the "desired" bijection we want to build, Bredon start dealing with the ...
2
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1answer
82 views

If the product of two homotopy equivalences is a homotopy equivalence are the factors homotopy equivalences?

The question says it all: Given two maps $f\colon A\rightarrow B$ and $f'\colon A'\rightarrow B'$, such that their product $$f\times f'\colon A\times A'\rightarrow B\times B'$$ is a homotopy ...
2
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0answers
22 views

How stable is the top cell of Lie group?

It is well know fundamental class of a compact lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable equivalence ...
9
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1answer
149 views

Gap between “fibration” and “fiber bundle”.

There are fibrations $E \rightarrow B$ which are not fiber bundles. Example: $E = [0,1]^2 / \text{middle vertical line segment}$ and $B=[0,1]$. In this example, $E$ has the homotopy type of a total ...
4
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1answer
74 views

Clarification about the Thom-Pontrjagin construction as explained in Bredon's book

In Bredon's book, at page 118-119, there is a little chapter about the Thom-Pontrjagin construction, and I'm trying to follow the reason depicted there. He starts with a map $f \colon R^{n+k}\to ...