Questions tagged [retraction]
194 questions
21
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0 answers
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
- 2,915
12
votes
2 answers
Surface of genus $g$ does not retract to circle (Hatcher exercise)
I'm trying exercise 9 on page 53 in Hatcher but I need some help with it. The exercise is:
In the surface $M_g$ of genus $g$, let $C$ be a circle that separates $M_g$ into two compact subsurfaces $M_h^\prime $ and $M_k^\prime$ obtained from the…
Rudy the Reindeer
- 48,301
10
votes
1 answer
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
9
votes
3 answers
Every retraction is a quotient map?
I have to proof that every retraction is a quotient map.. I have no idea where to start or what to use! A retraction $r:X \rightarrow A$ is a continuous map s.t. $r(a)=a$ for every $a\in A$.
user57012
- 118
6
votes
1 answer
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
- 186,215
5
votes
2 answers
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 necessarily have $f \circ g = \Id_B$, but it might be…
Sambo
- 7,469
5
votes
1 answer
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
- 51
5
votes
1 answer
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…
R. H. Vellstra
- 3,263
5
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0 answers
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
- 353
5
votes
1 answer
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
- 2,915
5
votes
1 answer
Bing's house with two rooms: an attempt for an explicit deformation retraction and a question on homotopy type
In the first pages of his "Algebraic Topology" book, Allen Hatcher describes a 2-dimensional subspace of $\mathbb{R^3}$, a box divided horizontally by a rectangle in two chambers, where the south chamber is accessible by a vertical tunnel (in green…
latelrn
- 592
5
votes
2 answers
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
- 729
4
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1 answer
Distinguishing locally contractible (in the sense of Borsuk) at $x$ and to $x$ for every $x$.
(As I researched the question further, this became an ask-and-answer, with (I think) a rather basic answer. I hope it may clarify confusion for any future learners. Please note that self-answering is allowed on StackExchange sites, as asserted…
Geoffrey Sangston
- 2,310
4
votes
1 answer
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
votes
1 answer
Intuition behind retracts
I am studying algebraic topology at the moment and we just started with introducing a bunch of definitions that we will use throughout the course. One of those definitions is:
Definition: Let $A\subset B$ be topological spaces, a map $r:B\to A$ is…
user974406