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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Inversion of Sphere

I was ready 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 $f_0 = f $ ...
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36 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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38 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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50 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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393 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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“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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77 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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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
23 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
30 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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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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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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40 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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49 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
19 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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61 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 ...
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48 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 ...
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41 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 ...
3
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159 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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62 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
36 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 ...
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25 views

Homotopy sets for a pushout of spaces (Seifert-van-Kampen?)

my problem is the following: I have two bordisms $M : \Sigma_0 \to \Sigma_1$ and $M' : \Sigma_1 \to \Sigma_2$, so I can glue them along $\Sigma_1$ to get $M'\circ M$. The manifold $M'\circ M$ is the ...
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47 views

Homotopic type of $GL^+(n)$, $SL(n)$ and $SO(n)$

Question: Consider $GL^+(n) \supset SL(n) \supset SO(n)$ the groups of matrices $n \times n$ with positive determinant, determinant $1$ and orthogonal with positive determinant, respectively. Show ...
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156 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. It is true that if ...
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1answer
49 views

Do homotopy equivalences operate over discrete spaces?

My understanding is limited and I'm trying to learn more about how homotopy forms the notion of equivalence. I can grasp its definition as "continuous", but my understanding of homotopy falls away in ...
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15 views

Generalize equivalence of simply connected stated for subspaces of the complex plane

In complex theory we have the following proposition: Let $A \subset \mathbb{C}$ . Then A is simply connected (in the topological sense, i.e., that it's path connected and fundamental group is ...
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Existence of a (n-1)-connected map beween CW-spaces

I have two finite CW-spaces $K$ and $L$ (K is n-dimensional and L is (n-1)-dimensional), a topological space $X$ and two maps $\phi:K\to X$ and $\psi: L\to X$, while $\phi$ is n-connected and $\psi$ ...
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Homotopy continuations for solving systems of equations over a finite field

A way of solving systems of polynomial equations over $\mathbb{R}$ or $\mathbb{C}$ is using homotopy continuation. Roughly speaking this method uses a homotopy that starts from some system of ...
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60 views

Does $f^{\ast}$ homotopic to $g^{\ast}$ imply $\int f^{\ast} w = \int g^{\ast} w$?

Let $f,g: M^{k} \to N$ ($M$ and $N$ with out boundary ) such that they are homotopic then for $\omega$ a $k$-form on $N$ do we have that $$ \int_M f^{\ast} \omega = \int_M g^{\ast} \omega$$ as ...
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1answer
64 views

Why does this have to be $f(0)=g(0)$?

For the problem, I am not given any solution so no idea Prove that any two continuous maps $f,g; I \to X$ such that $$f(0)=g(0) \in X$$ are homotopic where $I=[0,1]$ is the unit line. ...
3
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1answer
48 views

What do paths have anything to do with homotopy equivalence?

I don't understand how to solve this problem, it seems disconnected from the definition of homootpy equivalence Let $X,Y$ be spaces with the underyling set $\{a,b\}$ for both but ...
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1answer
24 views

Contradictory; Homotopy equivalence and deformation retract problem

The question Show that $S^1$ is a deformation retract og $D^2\setminus\{(0,0)\}$ the unit punctured disc. The solution the inclusion map $i:S^2 \to D^2\setminus\{(0,0)\}$ and ...
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33 views

I don't understand what a Fundamental group is

I have been staring at the definition for days, drew diagrams but I don't understand as to what its elements are The fundamental group $\pi_1(X,x)$ at a base point $x$ is a set of rel $\{0,1\}$ ...
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14 views

Continuous deformation of loop to point.

Suppose I have a homotopy from a loop around the origin to a constant loop which is not the origin. Prove that the origin is in the image of the homotopy. Basically prove that if I deform a loop to ...
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35 views

“Reduction to finite case” arguments in algebraic topology

Hello I was studying the corollary to the excision property in Homotopy theory (Hatcher 4K.2) and the thing I can't understand is why the injectivity argument works when moving from an infinite ...
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Lifting paths in a fibration in families

Given a fibration (or a fiber bundle or whatever might make my question true) $f\colon E\rightarrow B$ and basepoints $e$ in $E$ and $b$ in $B$, such that $f$ preserves them. By the homotopy lifting ...
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1answer
22 views

Mind numbing homotopies; why is this a homotopy rel?

Proposition. every map $\alpha: [s_0,s_1] \to \mathbb{R}^n$ is homotopic rel $\{s_0,s_1\}$ to the linear map $$\beta=\frac{(s_1-s_0)\alpha(s_0)+(s-s_0)\alpha(s_1)}{s_1-s_0}:[s_0,s_1] \to ...
2
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0answers
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Smoothing a continuous section in a fibre bundle.

Let $\pi: X \rightarrow M$ be a smooth fibre bundle and let $p^{1}_{0} : X^{(1)} \rightarrow X$ be its 1-jet bundle. Suppose there is a $\mathcal{C}^{1}$ section $h: M \rightarrow X$ such that it is ...
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2answers
34 views

$f$ is null homotopy if and only if $f_*$ is the trivial induced homomorphism

I want to show that $f:S^1\to S^1$ is homotopic to the constant map if and only if $f_*:\pi_1(S^1,x_0)\to \pi_1(S^1,x)$ , $f_*([\gamma])=0$ This seems like it should be an obvious fact but I am having ...
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Abelianization in relative Hurewicz theorem for a non-simply connected subspace

I have found two (probably equivalent) versions of relative Hurewicz theorem. The first one is from Hatcher, and the second one is from a lecture note of MIT (page 9 of this link). $(\pi_n)_{ab}$ ...
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49 views

Showing $\mathbb{R}^3$ minus $n$ parallel lines is homotopic to $\mathbb{R}^2\setminus\{p_1,\dots,p_n\}$

I want to show that if I remove $n$ parallel lines from $\mathbb{R}^3$ then I get $\mathbb{R}^2\setminus \{p_1,\dots,p_n\}.$ There is also some underlying structure I wish to also understand. That ...
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1answer
59 views

Reference/Definition of Homotopy in an Abstract Category

Let $\mathscr{C}$ be a complete and cocomplete category, and let $W$ be the collection of weak equivalences relative to some model on $\mathscr{C}$. We can form the homotopy category by localizing at ...
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Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
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1answer
72 views

Unit sphere without a point is contractible

Let $a$ be a point on the unit sphere $S=\{(x,y,z)|x^2+y^2+z^2=1\}$. How do I show that $S\backslash\{a\}$ is contractible? How do I show that a non-surjective loop $\phi\in P(S,s)$ with ...
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5 views

Proof of decomposition of homotopy into elementary decompositions?

In my Complex Analysis notes, the following lemma is stated without proof: If $G$ is an open connected domain, and $C$ and $C'$ are homotopic in $G$, then the homotopy can be decomposed into a finite ...
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$\mathbb{CP}^2$ realizable as boundary of compact smooth $5$-manifold? [duplicate]

As the title says, can $\mathbb{CP}^2$ be realized as the boundary of a compact smooth $5$-manifold?
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Enriching categories of simplicial objects

Let $C$ be a cocomplete category and $Simp(C)=C^{\triangle^{op}}$ the category of simplicial objects in $C$. I want to show that $Simp(C)$ is simplicially enriched but I don't understand how the ...