Questions tagged [retraction]
161 questions
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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.
For the group $\mathbb{Z}\oplus\mathbb{Z}$,…
M.Ramana
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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, recall that a group $G$ is Hopfian if every…
user481657
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When does a contractible simple loop have a contraction where every intervening loop is simple?
If $\gamma:S^1\to X$ is a simple contractible loop, when can we say there must by a contraction, $H(s,t)$ such that $\gamma_t:s\mapsto H(s,t)$ is a simple loop for all $t<1?$
(1) It seems like you should be able to do this if $X$ is a manifold. (Of…
Thomas Andrews
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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 instead of each of $2, 4, 6, 8, 10,$ and $12,$ there…
casi
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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 space $X$ if $A\subseteq X$ and there exists a…
pyridoxal_trigeminus
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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 (this space is also called a handle).
(a) Prove that…
dorkichar
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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 subgroups.
Now my question is that a finitely…
M.Ramana
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Prove that the unit circle cannot be a retract of $\mathbb{R}^2$-Munkres sec 35 exercise 4
Munkres topology section 35(Tietze Extension Theorem) exercise 4-(c). The question is
Can you conjecture whether or not $S^1$ is a retract of $\mathbb{R}^2$?
I've read the answers in Is the unit circle $S^1$ a retract of $\mathbb{R}^2$?, but I…
Sphere
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Non contractible subspace of $\mathbb{R}^2$
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 $(1,0)$ and the line segments from the points…
user733335
4
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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. However, I couldn't prove there is no retraction…
afdsfasdf
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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 equipped with the supremum metric) and claim it…
Kasia
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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 as well.
My progress so far:
Take $U\subseteq A$…
Castor
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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 $a\in A$, $F(x,0)=x$, $F(x,1)\in A$, and $F(a,t)=a$ for…
Shiquan
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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.
Any help?
blowup
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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 $G(w,0)=f(w),\,G(\omega,t)=h(t)$, for all $w$ and $t$.…
FShrike
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