Questions tagged [retraction]

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Continuity of a weak deformation retraction of the zigzag comb space

This quesiton is related to Why does the "zig-zag comb" weakly deformation retract onto a point? and How to prove "zigzag comb" is contractible (Hatcher ex. 0.6(b)). In both links, ...
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Local Contractibility of CW Complexes

Let $X$ be a CW complex and $X^n$ the $n$-skeleton of $X$ for all $n$. Hatcher demonstrates that $X$ is locally contractible by constructing a deformation retract from a certain open neighborhood $N_\...
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Continuity of an Unusual Deformation Retraction

Suppose we are given a countable chain of topological spaces $X_0 \subset X_1 \subset X_2 \subset \cdots$ and let $X = \bigcup_n X_n$; and suppose further that for each $n$ we have a deformation ...
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Does retracting commute with taking quotients?

Oftentimes I see that when you have a quotient $Y=X/{\sim}$ and you want to retract it to some subspace $B\subset Y,$ you first retract $X$ to $A \subset X$ and then observe that $B=A/{\sim}$. I have ...
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prove that $i_{*} :\ \Pi_{1}(A, a) \rightarrow \Pi_{1}(X, a)$ induced a isomorphism while X has a deformation retraction to A.

$X$ is a topological space which has a deformation retraction to $A\subset X$, $a$ is an arbitrary element in $A$ and $i$ is an inclusion function. A subspace $A$ of $X$ is called a deformation ...
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Is $\mathbb{D} = [-1,1]^3$ a compact manifold?

Today I read about a generalization of the no-retraction theorem here which states the following: Then there is no smooth mapping $f:M\to \partial M$ such that the restriction $f_{|\partial M}: \...
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Hatcher Theorem 2.13 - is the subspace $X$ of its cone $CX$ a deformation retract of some neighborhood in $CX$?

Hatcher's Theorem 2.13 says If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence $$\cdots \to \widetilde{...
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understanding a statement in the proof of a retract of a contractible space is contractible.

Here is the proof of the question: My question is: Why we can assume a contraction of $X$ to a point which lies in $A$? Can not this point be outside $A$? if so, how we will deal with the ...
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Does a deformation retraction of $X$ onto a subspace $A\subset X$ induce an isomorphism $\pi_n(X) \to \pi_n(A)$?

Let's say we have a topological space $X$ and a subspace $A\subset X$. Assume $A$ is a deformation retraction of $X$. Does that imply that the induced homomorphism of the deformation retraction is an ...
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Comparing 2 solutions of problem 2 chapter 0 of Allen Hatcher.

The question is to construct an explicit deformation retraction of $\mathbb{R^n} - \{0\}$ onto $S^{n-1}.$ Here is the answers I found online so far: The first solution: The second solution ...
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Retract spaces- Material or book recomendations

Good afternoon to all! I have just done a course on Algebraic Topology and I came across the definition of a retract space. Not much more is mentioned with regards to the topological retraction ...
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Deformation Retract is an isomorphism [duplicate]

I wanna proof that if A is a derformation retract of $X$, then $(j)_*: \pi_1(A,x_0) \to \pi_1(X,x_0)$ which is induced by the inclusion $j:A \to X$ is an isomorphism for all $x_0 \in A$. I already ...
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1answer
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Retraction formulation of the homotopy extension property [duplicate]

Hatcher's algebraic topology (p. 14) defines that a pair $(X, A)$ where $X$ is a topological space and $A$ an arbitrary subset has the homotopy extension property (HEP) if: Given a continuous map $...
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Use the function $f$ to show that $S^1$ is a retract of $\mathbb{R}^2.$

This is my main question: Use the function $f$ to show that $S^1$ is a retract of $\mathbb{R}^2.$ Deduce that $\partial U_{i} = J.$ Where $J$ is a Jordan curve. Here is the function $f$: Let $U_{i}...
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Show that the function $f$ is well-defined and continuous.

Let $U_{i}$ be a bounded path component of $X = \mathbb{R}^2 \setminus J,$ and assume $\partial U_{i} \neq J.$ Choose a point $c \in U_{i}.$ Use the retraction $r,$ we define the function $f: \mathbb{...
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Find a function that makes the following diagram commutes.

Find a function that makes the following diagram commutes. Here is the diagram: For $n \geq 0.$ Find a function $r: \mathbb{R}^{n+1} - \{0\} \rightarrow S^n$ filling in the dotted arrow in the ...
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Proving that $S^1$ is not a retract of the disk $D^2$ or of the plane $\mathbb{R}^2$ using a problem that I already solved.

I've proven that "If $A$ is a retract of $X$ and $X$ is contractible, then $A$ is also contractible." Can I use this to prove the circle $S^1$ isn't a retract of the disk $D^2$ or of the ...
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Proving that every arc in a normal space $X$ is a retract of it using commutative diagrams and a different version of Tietze extension theorem.

Here is the question: Let $X$ be a normal space and let $A \subseteq X$ be an arc, with inclusion function $i: A \rightarrow X,$ Since $A$ is an arc, we have a homeomorphism $\bar{\alpha} : I \...
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Universal covering via spines

Suppose you have a 2-manifold with boundary $M$. Let $S\subseteq M$ be a spine of $M$, i.e. $S$ is a connected subspace of $M$ s.t. $M \backslash S \cong \partial M \times [0,1)$. I am trying to proof ...
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Show that a retract of a cofibration is also a cofibration.

Here is the question: Suppose that $g: A \rightarrow C $ is a retract of $f: B \rightarrow D.$ Show that if $f$ is a cofibration, then so is $g.$ Could anyone help me in answering this question, ...
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Is there any closed embedding which is not cofibration?

Is there any closed embedding which is not cofibration? I firstly think that if $X$ is Topologist's sine curve and $A$ is $(0,0)$, then embedding $i:A\rightarrow X$ might satisfy this condition. ...
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If $X \times Y$ is contractible, then $X$ and $Y$ are contractible.

The aim of this problem is to guess whether the affirmation is true or not. I have proved the oposite implication by using the fact that if $X$ is contractible, then $X \times Y$ has the same homotopy ...
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Is the 3d model of a Klein bottle a cross-section of its 4d embedding?

(or homeomorphic to it?) Here's what prompts this question. The way we draw a 3d cube on 2d paper is essentially the image of the cube's skeleton under a retract to a plane cross section of the 3d ...
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What does it mean to show that r is retraction? I am confused about what the problem wants.

This problem follows the previous problem where I had to prove that there exist no $C^2$ retraction on a unit 2-ball to its boundary. Here, the problem is actually asking to prove Brouwer's fixed ...
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1answer
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Union of a sequence of increasing CW-complexes and Retract

I have updated my question based on the comments. I think I can solve the problem. But I hope you can help me verify it. I am reviewing Algebraic Topology to prepare for my Qual. I have this problem ...
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Showing $S^1$ is not a retract of $\mathbb{D}^2$

I am trying to show that $\mathbb{D}^2$ doesn't retract to $S^1$. I am using the fact that if $X$ is a subspace of $Y$ then if there is a retract $r:X\rightarrow{Y}$ with $r\circ{i}=Id_{X}$, with $i$ ...
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Doubt in the proof of Lemma to prove No Retraction Theorem

I have been reading Munkres, Topology. In Section 55, he goes on to prove No Retraction Theorem. To prove it, he uses the following lemma: $$\text{If A is a retract of X, then the homomorphism of } \...
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Subspace of $[0,1] \times [0,1]$ that deformation retracts to any point in $[0,1] \times \{0\}$, but not any other point.

This is an exercise that I'm doing, and I would like some comment on my solution. This is the picture I have in my head, where $(p,0)$ is the point $X$ deformation retracts to, and $(x_1,x_2)$ ...
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Are there any $\mathbf{Z}[x]$-algebra maps of the following form?

An exercise I am computing reduces to the following question: Is there a map $f$ of $\mathbf{Z}[x]$-algebras such that the composition $$ \mathbf{Z}[x,x^{-1}]\xrightarrow{f}\mathbf{Z}[x]\to\...
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A counter-example showing the inclusion $\{x_{0}\}\hookrightarrow X$ is a homotopy equivalence but $\{x_{0}\}$ is not a deformation retract of X

This is the exercise from Munkres' section 58 Exercise 8. Find a space $X$ and a point $x_{0}$ of $X$ such that the inclusion $\{x_{0}\}\hookrightarrow X$ is a homotopy equivalence but $\{x_{0}\}$ ...
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A compact Riemann surface with boundary strongly deformation retracting onto its boundary

Let $X$ be an open Riemann surface and $K$ an open, relatively compact, connected subset of $X$. It seems to me that it is impossible for $\overline{K}$ to strongly deformation retract onto its ...
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Intuitively, why doesn't a ball retract to a sphere?

I'm a studying an intro to topology course and we have just learnt about retractions. As a side note, the lecture notes state that the ball $\mathbb{B}^{n}$ does not retract to $S^{n-1}$ in $\mathbb{R}...
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There is no retraction map from unit disk to its boundary.(2)

There is no retraction map from unit disk to its boundary. I was reading this proof: It is found here in this link Retraction map from unit disk to its boundary But I do not understand why the ...
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For a differentiable retraction $f$, there is a local coordinate system in which $f$ is the canonical projection.

Let $M$ be a n-dimensional manifold and $f: M \to M$ be a differentiable retraction, that is $f\circ f=f $. Let $p\in f(M)$. Show that there is a chart $(\phi, U)$ of $M$, around $p$, such that $\phi \...
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Does there exist two h-equal but not h-equivalent topological spaces?

In his "Theory of retracts", Borsuk defines the concept of h-dominance (h is for homotopically), that is: given two (in this post i consider only Hausdorff spaces) topological spaces $X,Y$ we say that ...
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deformation retraction of the torus with 1 pt. deleted onto 2 circles intersecting at a pt. and that of $\mathbb R^n \setminus \{0\}$ onto $S^{n-1}$.

Here are two questions with their answers: My question is: Why in the second term of $H(x,t)$, in the first problem we have the term $t (g^{-1} \circ f)$ while, in the second term of $H(x,t)$...
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A small discussion about Hatcher Q.1, chapter 0

The question and its solution are given below: My question is: Should the deleted point of the torus be the intersection point of the two circles? If yes why? and if no, why also?
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Changing the codomain in the definition of the retraction map.

In Allen Hatcher, Algebraic topology, the following paragraph exists: And I am wondering if in the sentence starting with "From a more formal viewpoint .....", I changed the codomain of the ...
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Deformation retraction arising from a mapping cylinder.

In Allen Hatcher Algebraic topology, chapter zero, I had the following: My questions are: 1- Why the mapping cylinder is the quotient space, what makes it specifically a quotient space? 2-...
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Are open local embeddings equivalent to local diffeomorphisms? (Do not use immersions)

Open immersions are equivalent to local diffeomorphisms, and immersions are equivalent to local embeddings, so obviously yes. I would like to understand why open local embeddings equivalent to local ...
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Dimension of domain is greater than/less than/equal to dimension of range for a smooth surjection/injection/submersion/immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Let $N$ and $M$ be smooth manifolds with dimensions. Let $p \in N$. Let $F: N \to M$ be a smooth map. Question 1. Are these correct? A. If $...
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How to show that a mapping is a 2-Lipschitz retraction from $l_\infty$ to $c_0$?

In the book "Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss, I came across an example (Example 1.5) in which the authors construct a retraction from $l_\infty$ to $c_0$ (both ...
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Deformation retraction onto an open subset of manifold boundary

I'd like to prove the following result, perhaps with additional assumption if needed -- I don't know whether the claim holds. Let $M$ be a compact connected manifold with boundary $\partial M$. Let $U'...
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1answer
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$\mathbb{S}^n$ without two points

In "An Introduction to Algebraic Topology" of Rotman, Exercise 1.31 asks to show that the equator $\mathbb{S}^{n-1}$ is a deformation retract of $\mathbb{S}^n\setminus\{a, b\}$. I thought that if ...
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1answer
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Problem explanation regarding subspaces of $\mathbb{R}$

Does the word "subspace" here imply "linear subspace"? A subspace $Y$ of $\mathbb{R}$ is said to be a retract of $\mathbb{R}$ if there exists a continuous map $r: \mathbb{R} \to Y$ such that $r(y)= ...
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Why is $S^1\times \{1\}$ homotopy equivalent to the solid torus $T^2 = D^2 \times S^1$? (see attached picture)

I am currently self-studying the basics of algebraic topology and i just learned the definitions of retract, deformationretract and homotopy equivalence. Now in my book there is an example of a ...
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1answer
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If $A$ has a neighbourhood in $X$ that deformation retracts onto $A$ then does $(X,A)$ have HEP?

Let $(X,A)$ be a topological pair. Assume that $A$ has a neighbourhood $V$ such that $V$ deformation retracts onto $A$. Then can we say that $(X,A)$ has the homotopy extension property? Edit: I know ...
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Suppose $A \subset V$. How does a deformation retraction of $V$ onto $A$ induces a deformation retraction of $V/A$ onto $A/A$?

Suppose $A \subset V$. If there is a deformation of $V$ onto $A$, then there exists maps $i: A \hookrightarrow V$ and $r: V \to A$ such that $ri=Id_A$ and $ir \simeq Id_V$. How does this ...
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Fundamental group as a product of normal subgroups

Let $A$ be a retract of a non-empty topological space $X$ and let $a \in A$. Let's denote $r : X → A$ the retraction and $i : A → X$ the inclusion. prove that $i_∗ (π_1 (A, a))\triangleleft π_1 (X, ...
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Retract of noncompact surface to its boundary?

Suppose $M$ is a connected, noncompact 2-manifold, and its boundary $\partial M$ is a circle. What's the simplest way to show there is a retraction $r: M\rightarrow \partial M$? Here are some ...