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Questions tagged [retraction]

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2
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1answer
10 views

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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2answers
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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
25 views

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 ...
1
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1answer
28 views

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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1answer
54 views

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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2answers
62 views

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 ...
1
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1answer
56 views

Deformation Retraction and Projection/Closest Vector

How do I compute the projection/closest vector to a subset? I have been thinking about this for far too long without any progress. If it helps, I am working in $\Bbb{R}^2$, but I would like formalue ...
0
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1answer
29 views

Is the orthogonal polar factor the unique retraction $\operatorname{GL}_n^+ \to \operatorname{SO}_n$?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
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0answers
22 views

Want a specific deformation retract of polygon with center deleted to boundary

Given a n-sided regular polygon $P$ centered at origin. I want to show that $P- \{0\}$ has a strong deformation retract the boundary of the polygon. My try: Every point of $P-\{0\}$ is of the form $...
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1answer
31 views

Questions about the proof sketch of “$\{x\}$ is a deformation retract of $\overline{St}(x)$”

Munkres Topology Section 83 First question: Is "obvious deformation" the straight line homotopy $F(b,t) = \overline{St}(x) \times I \to \overline{St}(x)$? Second question: Is the "result" in "This ...
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Is theta space with a hole in upper arc contractible?

Munkres Topology Example 70.1 Let $\theta$ be theta-space, $\theta_a := \theta \setminus \{a\}$ and $\theta_b := \theta \setminus \{b\}$. Let $\theta_{ab} := \theta_a \cap \theta_b = \theta \setminus ...
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1answer
33 views

In theta space, is the lower arc and line segment a deformation retract of punctured theta?

Munkres Topology Example 70.1 Let X be theta-space, U = $X \setminus \{a\}$ and V = $X \setminus \{b\}$. Let $U \cap V = X \setminus \{a,b\}$ be doubly punctured theta-space where $a,b$ are interior ...
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1answer
54 views

A topological space with the Universal Extension Property which is not homeomorphic to a retract of $\mathbb{R}^J$?

A topological space $Y$ has the universal extension property if for every normal space $X$, every closed subset $A$ of $X$, and every continuous function $f:A\rightarrow Y$, we can extend $f$ to a ...
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1answer
111 views

Can closed unit ball be a retract of unit sphere?

Let $B^3$ denote the closed unit ball in $\Bbb R^3$ and $S^2$ be the unit sphere. Does there exist a retraction $r$ from $B^3$ onto $S^2$? I cannot argue it using fundamental group since both have ...
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1answer
39 views

Proof of fixed point theorem on $D^1$ using the technique used in Brouwer's fixed point theorem.

Let $f : [-1,1] \longrightarrow [-1,1]$ be a continuous function. Then using IVT I have proved that it has a fixed point. Now my question is "Can I prove this result by the technique used in Brouwer's ...
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0answers
35 views

How to prove there exists a second-order cone that is retractive

Can you help me to prove there exists a second-order cone that is retractive. Second-order cones are sets of the form $\{(x,z)\in R^n \times R: ||x||_2\leq z\}$. Can you help me? Thanks
1
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1answer
58 views

$A\times Y$ deformation retract of $X\times Y$ if and only if $A$ deformation retract of $X$?

Let $X,Y$ topological spaces and $A$ subspace of $X$. I know that $A\times Y$ retract of $X\times Y$ if and only if $A$ retract of $X$. Because $r:X\times Y\to A\times Y$ retraction, then $R:X\to A$ ...
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0answers
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Fundamental group of the following figure: (half-full sphere with two “quotiented” holes)

Let $X$ be the following figure: I want to find the fundamental groups $\pi_{1}(X)$ and $\pi_{1}(X/\beta)$ (up to isomorphism), determine whether $B$, the part of the image including the "$\beta$-...
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1answer
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proving $(\mathbb{B}^2 \times \{0 \}) \bigcup (\mathbb{S}^1 \times [0, \infty))$ is a retract of $\mathbb{R}^3$

I don't know how to solve this. I tried constructing a retraction but nothing comes to mind. Can someone guide me through this and if possible explain the intuition behind solving this kind of problem?...
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2answers
33 views

If $X$ and $Y$ are subspaces of $Z$, $X \cong Y$ and $X$ is a retract of $Z$, is $Y$ also a retract of $Z$?

If $X$ and $Y$ are subspaces of $Z$, $X \cong Y$ and $X$ is a retract of $Z$, is $Y$ also a retract of $Z$? I think the answer is no, but I can't find a counterexample. Can anyone help me with this?
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0answers
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$\mathbb C^{\infty}\setminus\{0\}$ retracts on $S^{\infty}$

We want to show that $\mathbb C^{\infty}\setminus\{0\}$ retracts on $S^{\infty}$. $C^{\infty}\setminus\{0\}$ should be the same as $\mathbb R^{\infty}\setminus\{0\}$. Maybe the retraction could be $(...
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1answer
25 views

On the retract of product of two spaces

Let $X$ and $Y$ be two topological spaces. Let $R$ be a retract of $X\times Y$ with the retraction $r:X\times Y\longrightarrow R$. Assume that $\pi_1 :X\times Y\longrightarrow X$ and $\pi_2 :X\times Y\...
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2answers
47 views

The number of retracts of $\mathbb{R}^2 \setminus \{(0,0)\}$ up to homeomorphic

We know that $\mathbb{S}^1$ is a (deformation) retract of $\mathbb{R}^2 - \{ (0,0)\}$. Obviously, the number of retracts of $\mathbb{R}^2\setminus \{ (0,0)\}$ equals to 1 up to homotopy equivalence. ...
2
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0answers
48 views

Continuous surjective map on a path connected subset of $\mathbb R^n$ which induces a covering map on a deformation retract space

Let $A \subseteq B$ be path connected subsets of $\mathbb R^n$, $n\ge 2$. Let $i:A \to B$ be the inclusion map and $r:B \to A$ be a deformation retraction i.e. $r\circ i=id_A$ and $i\circ r$ is ...
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1answer
28 views

Construct retraction of $conv(X)$ onto $X=[0,1]\times\{0\}\cup\bigcup_{n\in\mathbb{N}}\{1/n\}\times[-1/n,1/n]$

Let $X=[0,1]\times\{0\}\cup\bigcup_{n\in\mathbb{N}}\{1/n\}\times[-1/n,1/n]$ be subset of $\mathbb{R^2}$. And let $Conv(X)$ be a convex hull of $X$. Construct retraction of this convex hull onto $X$. ...
2
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1answer
52 views

Local connectedness is preserved under retractions

I want to show that if $X$ is a locally connected topological space, $A\subseteq X$ is a subspace and $f:X \rightarrow A$ is continuous such that $f|_{A} = Id_{A}$, then $A$ must be locally connected ...
3
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1answer
60 views

About continuous extensions and retractions

I'm trying to prove the following: Let $X$ a topological space and $Z\subseteq X$ $\text{For every topological space } Y, \text{ and for every continuous function } f:Z\rightarrow Y \text{ exists } g:...
2
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1answer
70 views

strong deformation retracts & neighbourhood deformation retracts

Let $X$ be a topological space and $A \subset X$ be a closed subspace. Assume that $A$ is a strong deformation retract of $X$. Is it true that for every open subspace $U \subset X$, $A \subset U$, ...
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2answers
49 views

Retractions and local finiteness: Are these circles a retract of $\mathbb R^2$?

Let $N$=$\{1+\frac 1 n\;|\;n\in\mathbb N\}\cup\{1\}$. I'm interested in whether the space $$ A=\cup_{n\in N}A_n \text{ where } A_n:=\{(x,y)\in\mathbb R^2\;|\;(x-n)^2 + y^2 = n^2, x\leq 1\} $$ is a ...
1
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1answer
115 views

Topology: Determine whether a subset is a retract of R^2

1. The problem statement, all variables and given/known data Let $X=([1,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[0,1])$ and $Y=((0,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[...
2
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1answer
60 views

deformation retract from the standard $k$-cube to a nonempty, locally connected, closed, and contractible subset

Let $X$ be a topological space and $A$ be a subspace of $X$. A deformation retraction of $X$ onto $A$ is a continuous map $F: X\times [0,1]\longrightarrow X$ such that for any $x\in X$ and any $a\in ...
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0answers
139 views

deformation retract of contractible spaces

Let $X$ be a topological space and $A$ be a subspace of $X$. A deformation retraction of $X$ onto $A$ is a continuous map $F: X\times [0,1]\longrightarrow X$ such that for any $x\in X$ and any $a\in ...
3
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1answer
136 views

continuous, closed and surjective not open.

Above proof, [Topology, J.Munkres (Part 2 Algebraic topology)] I cannot show that the map $\pi: S^1\times I\to B^2$ given by $\pi(x,t)=(1-t)x$ is continuous, closed and surjective, but is not open. ...
2
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1answer
58 views

$A\times Y$ is retract of $X\times Y$ iff $A$ is retract of $X$

I have the following problem that I am stuck on. Let $A$ be a subspace of $X$ and let $Y$ be a non-empty topological space. Show that $A\times Y$ is a retract of $X\times Y$ if and only if $A$ is ...
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1answer
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Is retract of a finitely generated Hopfian group Hopfian?

A subgroup $H$ of a group $G$ is called retract of $G$ if there exists homomorphism $r:G\longrightarrow H$ so that $r\circ i=id_H$, where $i:H\hookrightarrow‎‎ G$ denotes the inclusion map. Also, ...
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2answers
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If $H$ is an induced subgraph $G$ and there is a homomorphism $G\to H$, then $H$ is a retract of $G$?

If $H$ is an induced subgraph of $G$ and there is a homomorphism $G\to H$, then $H$ is a retract of $G$? I ask because: A subgraph $H$ of $G$ is called a core of $G$ if there is a homomorphism $G \...
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1answer
84 views

A compact convex subset is a strong deformation retract

Show that every compact convex subset of $\mathbb{R}^n$ is strong deformation retract. I don't really know how to approach to this. Any help would be very appreciated. Thanks!
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0answers
41 views

Say if the following function is nulhomotopic

$f:\mathbb{R}^2 -\{0\} \to \mathbb{R}^2 -\{0\} $ such that $f(x,y)=(x^2, y)$ I believe this isn't nulhomotopic since I can't pass through 0, but I don't know how to justify this. I think I should ...
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1answer
81 views

Prove: If $n \ge 0$, then $S^n$ is not a retract of $D^{n+1}$ using homology functors.

Prove: If $n \ge 0$, then $S^n$ is not a retract of $D^{n+1}$ using homology functors. The book I'm reading says: For each $n \ge 0$, there is a homology functor $H_n$ with the following properties:...
3
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1answer
76 views

Is the intersection of a descending chain of group retracts a retract?

Let $G$ be a group. A subgroup $H\subset G$ is said to be a retract if there exists surjective homomorphism $\pi:G\to H$ s.t. $\pi\circ\iota=Id_{H}$ with $\iota$ being the inclusion of $H$ in $G$. So ...
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2answers
96 views

Retracts of CW-complexes

I remember hearing that (all) topological spaces are retracts of CW-complexes. Given a topological space $X$, I am trying to construct a CW-complex $Y$ with continuous morphisms $i:X\to Y$ and $r:Y\...
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2answers
63 views

A retract in normal space

Let $X$ be a normal space and $K$ be a closed subset of $X$ homeomorphic to $\mathbb R$. Can I always find a retract of $X$ onto $K$? From some examples, such as $X=\mathbb R^n$ and $K=\mathbb R$, I ...
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1answer
96 views

If $A$ is a retract of $X$ (via $r:X\to A$), then $H_q(X)\cong \text{im }i_*\oplus \ker r_*$

I need to prove the following statement: If $A$ is a retract of $X$ (via $r:X\to A$), then $H_q(X)\cong \text{im }i_*\oplus \ker r_*$ It is well know that $r\circ i=id_A$ implies that $r_*$ and $i_*$...
0
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1answer
58 views

Concerning a retraction $r : X\times I \to X\times \{0\} \cup A\times I$

Let $A \subset X$ be a subset, where $I =[0,1]$. I'd like to prove a suggestion that for any retraction $r : X\times I \to X\times \{0\} \cup A\times I$, if some $x\in X$ satisfies the condition $t=...
1
vote
1answer
113 views

Retract of $D^n \times [0,1] \times [0,1]$

Question: Show that $D^n \times [0,1] \times \{0\} \cup S^{n-1} \times [0,1] \times [0,1]$ is a retract of $D^n \times [0,1] \times [0,1]$. Attempt: I have proved a more intuitive retraction, ...
2
votes
1answer
65 views

Prove that two quotients of a bounded cylinder are not homeomorphic

Let $C$ be the cylinder $\{(x,y,z)\mid x^2+y^2\leq 1, |z|\leq 1\}$. Consider two relations $R$ and $R'$ on $C$ which identify a point such that $x^2+y^2=1$ with $(-x,-y,z)$ and $(-x,-y,-z)$ ...
0
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1answer
41 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 ...