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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Chain Homotopy in abelian category

When dealing with complexes of modules or groups, the following lemma is pretty easy: If $f,g :E\rightarrow E'$ are homotopic, i.e. $f-g=d'h+hd$ for some h, then $f,g$ induce the same homomorphism ...
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46 views

How to prove that the Lefschetz number is invariante under homotopy?

How to prove that the Lefschetz number is invariante under homotopy? We define the Lefschetz number as the number of $f : M \to M$ as the number of intersection of the map $g(x) = (x,f(x))$ with the ...
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Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then:

Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then: $$\int_{\gamma_1(I)} \omega = \int_{\gamma_2(I)}\omega.$$ Then, conclude ...
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17 views

How do I see if the induced homomorphism from the inclusion map $S^{1'}\to S^1\times S^3$ is injective

Let $S^1\times S^3$ be parametrised as $\{(\alpha,\beta, \gamma)\in \mathbb{C}^3||\alpha|^2+|\beta|^2=1, |\gamma|=1\}$ and let $S^{1'}=\{e^{i\theta}(1,0,1)|\theta\in [0,2\pi]\}$. I would like to see ...
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18 views

Proving weak homotopy equivalence of a map

This is a modification of the question I previously asked here. Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting ...
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38 views

Which theorem of homotopy theory states that if two objects have different genus then they are not homotopy equivalent?

I'm quite new inexperienced in the field but from what I see two objects with different genus are not homotopy equivalent. Question: Which theorem of homotopy theory states that if two objects have ...
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17 views

Isomorphism of Homotopy groups across a filtration

Let $X$ and $Y$ be CW complexes. Let $sk_{\bullet}(X)$ and $sk_{\bullet}(Y)$ denote the canonical skeleta filtrations of $X$ and $Y$, respectively. Suppose that we have isomorphisms on homotopy groups ...
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32 views

Compositions of homotopic maps are homotopic

I'm reading some lecture notes on homotopy, and the author has just proved the theorem: If $f_{0} \simeq f_{1}$ and $g_{0} \simeq g_{1}$, then $g_{0} \circ f_{0} \simeq g_{1} \circ f_{1}$ ...
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69 views

How can we assume the first homology group of the complement is zero when constructing a Casson handle?

I am currently working through Scorpan's Wild World of 4-Manifolds specifically the section on Casson Handles. On page 78, he says if $D$ is the core of the handle after $n$ stages we may assume $\...
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54 views

Proving a function isn't homotopic to a map to the boundary

Let $X$ be a compact Hausdorff space and $S$ a finite dimensional, convex, Hausdorff space. Moreover, let $f:X \to S$ be a closed, surjective map with $\tilde{H}^q(f^{-1}(s))=0$ for all $q\ge 0$ and $...
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40 views

Proving that a map is a weak homotopy equivalence

Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting diagrams where all the morphisms are serre fibrations $$\begin{...
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1answer
18 views

What is special about retraction mapping?

What is special about the retraction mapping? Can't we always find such a mapping, namely identity map of $X$. Then every space $A$ will be a retract of $X$. EDIT: Do we need retraction to be ...
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1answer
44 views

A modern approach to homotopy theory in $\mathbf{SSet}$

I'm currently trying to understand the basics of homotopy theory for simplicial sets. However, my current sources (Peter Mays "Simplicial objects in algebraic topology" and Kans original "on c.s.s. ...
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33 views

Proving that Emb($D^m,N$) is homotopy equivalent to $V_m(TN)$

I am reading online lecture notes by John Francis on h-principle. I want to prove that Emb($D^m,N$) is homotopy equivalent to $V_m(TN)$ where $V_m(TN)$ is stiefel bundle of the tangent bundle on $N$....
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67 views

Topology on $\mathcal{C}(X,Y)$ to work with homotopy.

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...
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26 views

Derived functors in abelian categories and homotopy theory

For two Abelian categories $\mathcal A,\mathcal B$ and a right exact additive function $F\colon\mathcal A\to\mathcal B$, there is a left derived functor $LF$ acts on chain complexes $K_+(\mathcal A)$ ...
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1answer
27 views

Proving that a map is Null Homotopic

Suppose $X$ is a manifold of dimension $n$ and $f:Y \to Z$ is an $n-$connected map. Then I want to show that given any map from $g:X \to Y$, the composite map $f \circ g$ is nullhomotopic. Definition ...
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72 views

Infinite sequence of distinct spaces, all with same homology

Using the following fact, we get infinitely many non-homotopic maps $f_k:S^{2n-1}\to S^n\vee S^n$. Fact: $\pi_{2n-1}(S^n\vee S^n)$ contains a $\Bbb Z$-summand. So we can consider the spaces $X_k=...
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Can Path Connectedness be Defined without Using the Unit Interval?

Can path connectedness be defined without using the unit interval or more generally the real numbers? I.e., do we need Dedekind cuts or Cauchy convergence equivalence classes of the rational ...
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55 views

Is every map between connected CW-complexes homotopic to a map sending $x$ to $y$?

Let $f \colon X \to Y$ be a continuous map between connected CW-complexes $X$ and $Y$. Choose points $x \in X$ and $y \in Y$. Is $f$ homotopic to a map sending $x$ to $y$?
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Fundamental group of a topological group [duplicate]

Given $(G,\cdot)$ a topological group with identity $e$, is it always true that $\pi_1(G,e)$ is abelian? For what it's worth: I have already shown that if $f,g\in\Omega(X,e)$, then $[f \times g]=[f\...
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19 views

Equivalent definitions for simple connectivity

For a path connected space $X$, it is simply connected iff any two paths sharing endpoints are homotopically equivalent. Here simple connectivity means it has trivial fundamental group. I'm doing ...
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38 views

The complement $\mathbb{R^3}-A$ of a single circle $A$ deformation retracts onto the wedge of a Circle and a sphere. 2$

Here is my intuitive idea. We can pull all the points lying outside a sphere of some fixed radius containing the circle onto the sphere without disturbing the points inside the sphere. Now we have a ...
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33 views

Hopf map visualization (animation request)

Let $\phi:D^3\to S^2$ be the composition $D^3\to S^3\to S^2$, the first map being the quotient by the boundary and the second map being the Hopf map. Then: $$f_t:x\mapsto(1-t)x+t\phi(x)$$ is a ...
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1answer
71 views

The identification $G=\Omega^\infty \Sigma^\infty S_0$ of the stable group of self homotopy equivalences of spheres with the suspension spectrum

My topology professor told me in a discussion that the suspension spectrum $colim \Omega_n \Sigma_n S_0$ is the same as the monoid $G$ where $G=colim G_n$ where $G_n$ are self homotopy equivalences of ...
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1answer
31 views

Every continuous function $\mathbb{R}^n\rightarrow X$ is null-homotopic

Let $f:\mathbb{R}^n\rightarrow X$ be continuous and $g:\mathbb{R}^n\rightarrow X$ a constant map with image $\{x_0\}$. For the proof, I thought that we can define $F:[0,1]\times\mathbb{R}^n$ with $F(s,...
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1answer
51 views

Inversion of Sphere

I was reading about inversion of sphere. Wikipedia defines it as: Let $f: S^2\to R^3$ be the standard embedding; then there is a regular homotopy of immersions $f_t\colon S^2\to R^3$ such that $...
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1answer
41 views

Vector field on n-manifold whose sum of indexes is equal to Euler charasteristic

For 2-manifolds and 3-manifolds such a tangent field (whose singular points indexes sum to manifold's Euler chracteristic) construction can be done visually. For example, for triangulated 2-manifold ...
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68 views

when a space is a product of eilenberg mclane spaces for $\pi_1(X)$ is not abelian

In When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$. , I discuss in my answer when an Abelian cw complex $X$ is a product of Eilenberg-Maclane spaces, and show that it ...
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1answer
57 views

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products?

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products? I know it preserves products, but it seems to actually create them.
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408 views

Why is the homotopy category actually important?

Since the homotopy category (of whatever) generally has very few limits and colimits, and these don't coincide with homotopy limits and colimits in the lifted model structure, why do we care about the ...
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1answer
90 views

“Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
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1answer
80 views

Is $(0,1)$ homotopic to $(0,1) \times (0,1)$? [closed]

Is there a homotopy between a map with $(0,1) \in R^2$ as image and one with $(0,1) \times (0,1)\in R^2$ as image? Both have domain $(0,1)$. Or the same question for closed intervals.
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104 views

What is wrong with this proof that the identity map of $S^1$ is nullhomotopic?

I have read that the identity map of the unit circle $S^1$ is not nullhomotopic. In fact, I am very new to the subject, so I wonder what is wrong with the following reasoning (that seems to suggest ...
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1answer
43 views

Homotopy 'diagrams' for Klein bottle and projective plane

Background: I recently discovered that the complement to the circle and vertical axis shown below is homotopy equivalent to a torus Also complement to three infinite straight non-intersecting ...
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1answer
34 views

Definition of G-crossed complex.

I was reading about crossed complexes following R.Brown. I was wondering how one define G-crossed complexes for a topological group G? Is it just dimension wise action of the group?
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41 views

How do I get these equations ($n$th-order deformation equation and its initial/boundary conditions)?

Earlier, I asked a question What does 'equating the like-power of $q$ ' mean? and I already got the answer about the meaning of a particular phrase in a certain context. However, my real question ...
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1answer
29 views

What does 'equating the like-power of $q$' mean?

I am reading a book "Homotopy Analysis Method in Nonlinear Differential Equations" by Shijun Liao chapter 13 Applications in Finance: American Put Options. It is stated there that Substituting ...
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2answers
42 views

Homotopic retract vs deformation retract

Let's say that $A \subset X$ is a deformation retract. It follows that $A$ is both a retract and a space homotopically equivalent to $X$. Is the converse true? Probably not, but I couldn't find any ...
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1answer
59 views

Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence.

Let $M,N$ be smooth manifolds. It seems to be well known that if the sets $C^0(M,N)$ and $C^\infty(M,N)$ are equipped with the appropriate topologies (I suppose the weak/strong Whitney topology), then ...
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1answer
25 views

principal bundle morphism preserves fundamental group

Two related questions. What is the morphism for principal bundles? Does it "preserve" fundamental groups? Fibre bundle morphisms usually preserve "the structure on the fibre". I am not sure how to ...
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63 views

A Natural Question When Reading Van Kampen Theorem

Let $A$ and $B$ be path connected open subspaces of a topological space $X$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x$ and $y$ be two points in $A\cap B$. Let $\gamma$ and $\...
2
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1answer
50 views

Trying to Understand Van Kampen Theorem

Theorem. Let $X$ be the union of two path-connected open sets $A$ and $B$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x_0$ be a point in $A\cap B$ and all fundamental groups will ...
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2answers
47 views

Is path-connected a homotopy property of toplogical spaces?

$X$ and $Y$ are homotopy equivalent so there are maps $\alpha: X \rightarrow Y$ and $\beta : Y \rightarrow X$ whose composites satisfy : $\beta\alpha \simeq id_X$ and $\alpha\beta \simeq id_Y$ $X$ is ...
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40 views

Question about proving Cauchy's integral theorem

In my course we were given several proofs of Cauchy's theorem, each at various points in the course, each version stronger than the previous. I'd like to learn a proof of the theorem, so naturally ...
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1answer
33 views

Fundamental group of the complement of $S^1 \cup Z \subset \mathbb{R^3}$

I want to calculate the fundamental group of the complement of this space: This is the complement of $S^1 \cup Z \subset \mathbb{R^3}$ where $S^1$ is the unit circle in the $xy$-plane and $Z$ is ...
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2answers
163 views

Homotopy Poincaré conjecture - no map inducing the isomorphism on homology

$\newcommand{\Z}{\mathbb{Z}}$ In Terence Tao's notes on page 18, concerning the Poincaré conjecture, he gave the following sketchy proof of the homotopy Poincaré conjecture. Given $M^3$ a 3-manifold ...
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1answer
65 views

Why does the homotopy lifting property imply that fibers are homotopy equivalent if the base is path connected?

Suppose that $\pi:E \to B$ has the homotopy lifting property, so that for any space Y with a map $f:Y \to E$ and a homotopy $G$ of $g = \pi \circ f$, we have a homotopy $F: Y \times I \to E$ that ...
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1answer
37 views

Are two equivariant maps between aspherical topological spaces homotopic?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial hight homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...
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12 views

Continuous maps to $S^n$ without antipodal pairs are homotopic [duplicate]

Let $S^n$ denote the unit sphere in the Euclidean space $\Bbb R^{n+1}$, $X$ a topological space, $f,g:X\to S^n$ are both continuous and there doesn't exist $x\in X$ such that $f(x)=-g(x)$, show that $...