# Questions tagged [retraction]

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• 8,641
1 vote
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### 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 ...
• 8,641
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### 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 $\mathbb R\mathbb P^2$ ...
• 831
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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 ...
• 51
1 vote
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### 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 ...
• 2,743
1 vote
93 views

• 2,865
1 vote
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 $X$ ...
• 139
### 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 ...
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 ...