Questions tagged [retraction]

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Retraction onto subgroup?

Let $N\subset G$ be a nontrivial subgroup of finite group $G$ , is there a "retraction" onto $N$ , i.e. a homomorphism $\varphi : G \rightarrow N$ s.t. $ Im(\varphi) = N $ , $\varphi |_N = ...
Anton Shcherbina's user avatar
2 votes
0 answers
15 views

Morse flow: cancelling handle pairs away from deformation retract

Given a smooth manifold (not closed, maybe with boundary) $M$ in $R^n$, take a section with a hyperplane $H$ of some dimension $d$. Assume that $M$ has $M\cap H$ as deformation retract. For example, a ...
MathBug's user avatar
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1 answer
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No homotopy with the identity of the unit circle

Let $Y\subset S^{1}$ be a proper subspace of the unit circle. Prove that does not exists any homotopy $F$ between $\operatorname{id}_{S^{1}}$ and $i\circ f$, where $f: S^{1}\rightarrow Y$ is any ...
Dungessio's user avatar
  • 137
1 vote
1 answer
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Arcwise connected components under a retraction on two points

Let $X$ be an Hausdorff space, and let $p,q$ two distinct points. Then if $X$ retracts on $\{ p,q\}$, establish if $X$ has at least two arcwise connected components, or if $X$ has at most two arcwise ...
Dungessio's user avatar
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3 votes
1 answer
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Computing the fundamental group of a union of subspaces

If $x, y \in \Bbb{R}^n$, we denote by $[x,y] := \{ (1-t) \, x + t \, y : t \in [0,1] \subset \Bbb{R} \}$ Let denote the following subspaces of $\Bbb{R}^2$: $$ X_1 := \{ x \in \Bbb{R}^2 : ||x||=1 \} \\ ...
Superdivinidad's user avatar
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2 answers
58 views

Fundamental group of logarithmic surface

I am trying to determine the fundamental group of the following parametrized surface: $$ X(r,\theta) = (r\cos\theta,r\sin\theta, ln(r^2)),$$ where $r\in(0,+\infty)$ and $\theta \in [0,2\pi)$. It can ...
Rainbow's user avatar
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Homotopy equivalence vs deformation retract for analytic spaces

I know that there are many examples of spaces $Y\subset X$ such that $X$ and $Y$ are homotopy equivalent but there is no deformation retract of $X$ to $Y$ (e.g., Does homotopy equivalence to a ...
MathBug's user avatar
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2 votes
2 answers
85 views

$S^1$ has no deformation retracts other than itself

I am trying to prove that if $A \subset S^1$ is a deformation retract of $S^1$, then $A=S^1$. $A$ being a deformation retract of $S^1$ means that there exists a continuous map $r \colon S^1 \to A$ ...
David's user avatar
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1 answer
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What are some examples of bad pairs $(A,X)$?

A good pair $(A,X)$ is such that $A\subset X$ and there is a neighborhood deformation retract of A in X. What are some examples of bad pairs? I think $(A,I)$ where $A = ${$0,1,1/2,1/3...$} is a bad ...
user13121312's user avatar
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Optimizing linear least square subjected to manifold constraints

I am trying to optimize a function of a matrix $X \in \mathbb{R}^{m \times n}$ $$ f(X) = \frac{1}{2} \left\lVert AX - B \right\rVert_F^2 $$ s.t. $X^T X = I_n$ (identity matrix). As I've been watching ...
user8469759's user avatar
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39 views

Retracts and Closed Subspaces in Discrete Spaces

Prove that any closed subspace of the space 𝐵 = $𝐷^{ℵ0}$, where 𝐷 is a discrete space of any power, is a retract of the space 𝐵. I think we can try to reduce the problem to this: Show that a ...
Alex's user avatar
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1 answer
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Can compact, non-hollow subsets of Euclidean space retract onto their topological boundary?

Following discussion in the comments under this question, I have decided to reframe the question and post it here. The main motivation for this is that I am told there is an answer to at least one of ...
FShrike's user avatar
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2 votes
0 answers
40 views

Wedge for non-good pairs

Call a pair of topological spaces $(X,A)$ a good pair if $A\subseteq X$ is a closed subspace and there exists an open neighborhood $A\subseteq U \subseteq X$ such that $A$ is a deformation retract of $...
Peter's user avatar
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3 votes
1 answer
68 views

Understanding 3-manifold retraction to graph

I am reading On Fibering Certain 3-Manifolds by Stallings. It's a short paper, and I think I understand at a high level what's happening, but there are a few technical details I don't quite get. This ...
Hempelicious's user avatar
1 vote
2 answers
76 views

When is a freely contractible space also based contractible?

There are spaces, for example the cone of the Hawaiian earring, which are contractible but that have a basepoint such that no contraction fixes that basepoint. Are there any good sufficient conditions ...
subrosar's user avatar
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5 votes
2 answers
82 views

Is there a name for a morphism which makes a left inverse act like a two-sided inverse?

$\newcommand{\Id}{\operatorname{Id}}$Consider a morphism $f : A \rightarrow B$ which has a left inverse $g$, i.e. $g \circ f = \Id_A$. (That is, $f$ is a split monomorphism.) Of course, we don't ...
Sambo's user avatar
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Retraction on non-compact Stiefel manifold

I have been looking online for the retraction of the non-compact Stiefel manifold (where $k <m$): $R_*^{m \times k} = \{ M \in \mathbb{R}^{m \times k} : \operatorname{rk}(M)=k \}$ as seen here (...
Hugo B's user avatar
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1 vote
1 answer
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Retracting $\mathbb{R}^3$ minus the wedge sum of two circumferences to a double torus.

I am trying to prove that there is a deformation retraction from $\mathbb{R}^3-\left(S^1\vee S^1\right)$ to a double torus. I don't need a super rigurous proof, but I am not able to picture it.
jaime jdtrechuelo's user avatar
4 votes
1 answer
93 views

Non contractible subspace of $\mathbb{R}^2$ [duplicate]

I'm having trouble proving that the subspace $X$ of $\mathbb{R}^2$ such that $X$ is the union of $[-1,1] \times \{ 0 \}$ and the line segments that join the points $(0,\frac{1}{n})$ with the point $(1,...
user avatar
1 vote
2 answers
56 views

Deformation retract on subspace of $S^3$

Let $M = \{(x, y, 0, 0) \in \mathbb{R}^4 ~|~ x^2 + y^2 = 1\}$ and $N = \{(0, 0, z, w) \in \mathbb{R}^4 ~|~ z^2 + w^2 = 1\}$ be subspaces of $S^3$. Construct a deformation retract of $S^3 \setminus M$ ...
Abced Decba's user avatar
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1 answer
79 views

Contractible and deformation retract

I know there are many questions on this but I ask a proof verification and a clarification. I am trying to prove that given $X$ topological space, $X$ is contractible if and only if it is a ...
Crash Bandicoot's user avatar
1 vote
1 answer
90 views

Homotopy equivalence iff weak deformation retract

This theorem is from page 29 in spanier's book. It is not detailed enough for me. So, In the first side, if $X$ and $Y$ are embedded as weak deformation retracts of $Z$, then we have the inclusion ...
user avatar
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0 answers
47 views

Can any compact smooth $n$-dimensional submanifold of the open $n$-ball be deformed into a finite (n-1)-dimensional simplicial complex?

Please, forgive me if I say something silly. I would like to prove that for any smooth compact $n$-dimensional sub-manifold $M$ of the open $n$-ball $B$ there is a finite $(n-1)$-dimensional ...
asymmetriad's user avatar
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0 answers
100 views

proving that the inclusions $i:X \to CX$, $j:Y \to C_f$, $j':Y \to M_f$ have the homotopy extension property

Let $X$ be a topological space, and $f:X \to Y$ a continuous map. I want to show that the inclusions $i: X \to CX, \: j:Y \to C_f, \: j':Y \to M_f$ all have the homotopy extension property. Where $CX$...
Paul Joh's user avatar
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3 votes
1 answer
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If $(W;\omega)$ is a well-pointed space that weakly contracts onto $\omega$, does it strongly contract onto $\omega$?

We are given a well-pointed space $(W;\omega)$, by which I mean, for every pair of maps $f:W\to Y$, $h:I\to Y$, there exists at least one extension $G:W\times I\to Y$ which satisfies $G(w,0)=f(w),\,G(\...
FShrike's user avatar
  • 39k
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0 answers
57 views

Deformation retraction definition – why need the requirement $f_t|A = \mathrm{Id}$ for all $t$?

There's a part of the definition of deformation retraction that I keep forgetting, and that is because I cannot understand why it is required. The definition states (from Hatcher's Algebraic Toplogy): ...
Anon's user avatar
  • 1,713
3 votes
2 answers
180 views

Proof of Brouwer's fixed point theorem without direct computation

Brouwer's Fixed Point theorem says - If $f:\Bbb{D}\to\Bbb{D}$ be a continuous map, then $f$ has a fixed point. Suppose not, then we look at the line segment joining $f(x)$ and $x$ and extend it till ...
DeltaEpsilon's user avatar
  • 1,086
0 votes
1 answer
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A question related to an induced homomorphism between two groups

Suppose $X$ is obtained by gluing two tori at a single point and let $r:\sum_2\to X$ be the retraction given by collapsing a circle around the middle of $\sum_2$ (surface of genus $2$) to a single ...
neophyte's user avatar
  • 510
1 vote
1 answer
57 views

Prove or give counter example: $G \cong N \times G/N \Rightarrow N$ has a normal complement.

My original question is the following: Given a short exact sequence $1 \to N \xrightarrow{\iota} G \xrightarrow{\pi} Q \to 1$, we have: \begin{equation*} G \cong N \times Q \ \Longrightarrow \ \...
Metin Ersin Arıcan's user avatar
1 vote
1 answer
56 views

Is this homotopy equivalence a deformation retract?

Suppose I have a Serre fibration of smooth manifolds $f:X\to Y$ (one may assume $Y$ is an open ball) with a section $s:Y\to X$ of $f\,,$ and furthermore assume the fibers of $f$ are contractible. This ...
JLA's user avatar
  • 6,422
19 votes
0 answers
253 views

Does every finitely generated group have finitely many retracts up to isomorphism?

The infinite dihedral group $D_\infty = \langle a,b \mid a^2 = b^2 = \text{Id}\rangle $ is a finitely generated group with infinitely many cyclic subgroups of order 2, every one of which is a retract. ...
M.Ramana's user avatar
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3 votes
1 answer
53 views

Is every splitting-simple group $G$ Hopfian?

Let $H$ be a subgroup of $G$. Then a homomorphism $r : G \to H$ is said to be a retraction if $r(x) = x$ for all elements $x\in H$. Then $H$ is called a retract of $G$. A nontrivial group is said to ...
M.Ramana's user avatar
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1 vote
1 answer
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Proving a given circle is not a retract of the Klein bottle -- is my solution correct?

I am currently self-studying Algebraic Topology: A First Course by Greenberg and Harper. I have "solved" the following exercise: Exercise: Given the polygonal presentation $aba^{-1}b$ of the ...
While I Am's user avatar
  • 2,381
2 votes
1 answer
225 views

Deformation retraction of Möbius strip minus a point

Let the Möbius strip be $\mathcal M=[0,1]\times [0,1]/\sim$ where $(0,t)\sim (1-t,1)$. Let $$A=\{(x,y)|(x-1/2)^2+(y-1/2)^2=1/9\}\cup (\{1/2\}\times [0,1/6])\cup (\{1/2\}\times [5/6,1])$$ Show that $A/ ...
Sayan Dutta's user avatar
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1 vote
0 answers
104 views

Show that $\Theta=S^1\cup (\{0\}\times [-1,1])$ is homotopy equivalent to $\infty=(S^1+(1,0))\cup (S^1+(-1,0))$

Show that $\Theta=S^1\cup (\{0\}\times [-1,1])$ is homotopy equivalent to $\infty=(S^1+(1,0))\cup (S^1+(-1,0))$ where $(S^1 + (a, b))$ is the circle described by $(x - a)^2 + (y - b)^2 = 1$. I can ...
Sayan Dutta's user avatar
  • 8,641
0 votes
0 answers
58 views

Compute the fundamental group of the real projective plane $\mathbb R\mathbb P^2$ minus a point $p$

I am trying to solve the following exercise by two different ways: Compute the fundamental group $\pi_1(\mathbb R\mathbb P^2-\{p\})$ For do so, first we need to remember that $\mathbb R\mathbb P^2$ ...
FUUNK1000's user avatar
  • 831
5 votes
1 answer
233 views

What would be some retracts of this graph?

I'm not fully understanding the concept of retracts, I think. For example, does this graph have any possible retracts? It seems like it doesn't to me, but I'm not sure how to check. Also, what if ...
casi's user avatar
  • 51
1 vote
1 answer
51 views

Is there a finite bound on the indexes of infinite retracts of an infinite virtually cyclic group $G$?

Let $G$ be an infinite, virtually cyclic group, i.e., $G$ has an infinite cyclic subgroup $H$ of finite index (or equivalently, it contains a finite index infinite cyclic normal subgroup $N$). By a ...
M.Ramana's user avatar
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1 vote
1 answer
93 views

The number of retracts of $G\ltimes \mathbb{Z}$

I asked here that: does $\mathbb{Z}\oplus G$, where $G$ is a finite abelian group, has only finitely many retracts? The answer was yes. Now I've tried to prove that the number of retracts of $\mathbb{...
M.Ramana's user avatar
  • 2,743
1 vote
1 answer
103 views

Does a free group $F$ of finite rank $n$ have finitely many retracts (as a subgroup)?

A subgroup $H$ of a group $G$ is called a retract of $G$ if there exists an epimorphism $r:G\to H$ such that $r(h)=h$ for all $h\in H$. Does a free group $F$ of finite rank $n$ have finitely many ...
M.Ramana's user avatar
  • 2,743
5 votes
1 answer
92 views

Are there any topological spaces containing no proper retracts? i.e. $A\subset X$ s.t. $\exists r:X\to A$ continuous w/ $r(a)=a,\forall a\in A$

Are there any topological spaces containing no proper retracts? i.e. $A\subset X$ s.t. $\exists r:X\to A$ continuous w/ $r(a)=a,\forall a\in A$ Definition: Say that $A$ is a retract of a topological ...
pyridoxal_trigeminus's user avatar
5 votes
0 answers
220 views

Fundamental group of the boundary of a torus with a point removed

The question below is from an old topology qualifying exam. I am mostly stuck on parts (c) and (d). Let $X$ be a 2-dimensional torus $T^2$ with the interior of a small disk $D \subset T^2$ removed (...
dorkichar's user avatar
  • 353
1 vote
1 answer
364 views

$\Bbb{R}^3 - S^1$ deformation retracts to $S^2 \vee S^1$?

I'm reading this blog post where they state that $\Bbb{R}^3 - S^1$ deformation retracts to $S^2 \vee S^1$ and then they proceed to give the following sketch with the explanation You can see this in ...
Emil Grönberg's user avatar
2 votes
1 answer
71 views

If $Y \subseteq Q$ is a retract of the Hilbert cube $Q$ and $Z \subseteq Q$ is homeomorphic to $Y$, is $Z$ a retract of $Q$?

For a topological space $X$ and its subspace $A$, we say that $A$ is a retract of $X$ if there is a continuous map $f \colon X \to A$ such that $f(x)=x$ for every $x \in A$. The Hilbert cube $Q$ is ...
jenda358's user avatar
  • 501
5 votes
1 answer
166 views

Does finitely generated groups have finitely many finite retracts?

A group $H$ is called a retract of a group $G$ if there exists homomorphisms $f:H\to G$ and $g:G\to H$ such that $gf=id_H$. We know that a group $G$ is finite if and only if $G$ has finitely many ...
M.Ramana's user avatar
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0 votes
0 answers
200 views

Explicit retraction of $\mathbb{R}^n$ minus $k$ points to wedge sum of $k$ copies of $S^{(n-1)}$

In proving $\mathbb{R}^n$ minus finitely many points is simply connected for $n\geq 3$, we use the fact above. This is intuitively very much clear but it would be very helpful if someone can help me ...
User11111's user avatar
  • 160
2 votes
1 answer
83 views

Show that retraction is coequalizer.

Let $f:A\rightarrow B$ be a retraction (i.e. a morphism that is a left inverse) and let $g:B\rightarrow A$ be the one such that $f\circ g = \operatorname{id}_B$. I'd like to show that in fact $f:A\...
Michal Dvořák's user avatar
1 vote
0 answers
35 views

Question about homotopy equivalence (Hatcher $0.4$)

A deformation retraction in the weak sense of a space $X$ to a subspace $A$ is a homotopy $f_t:X \to X$ such that $f_0=1_X, f_1(X) \subset A$ and $f_t(A)\subset A$ for all $t$. Show that if $X$ ...
Tien Nguyen's user avatar
2 votes
1 answer
138 views

Showing there is a retraction from $\Bbb R^2$ to the closed disk $D$

I have seen a number of proofs on this site about the existence or lack of a retraction from $\Bbb R^2$ to $S^1$. A common point for these discussions is about the relationship between the unit circle ...
Edward Peterson's user avatar
2 votes
1 answer
193 views

Almost retracting a ball to its boundary [duplicate]

One cannot continuously retract a closed ball $B_n$ to it boundary $S_n$, that is one cannot find a continuous map $f:B_n\to S_n$ such that $f|_{S_n} = \mathrm{id}_{S_n}$. What if we drop the latter ...
SBF's user avatar
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