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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 called a retract if $r|_A=id_A$. We say $A$ is a retraction of $B$.

Here $id_A$ stands for the identity on $A$. The definition is clear to me but I am lacking intuition maybe because the only example we have seen is the retract to a single point.

Can someone maybe give some more examples with explanation on what is going on, so that I can understand this concept better intuitively. Thank!

  • It's a map to a subspace that leaves every point in the subspace fixed. You can find retracts basically by finding maps to a subspace that don't move it around. This isn't always possible. If you like algebra, this is equivalent to finding a map that induces a homomorphism on your favorite invariant $H_i,\pi_i$ that has a section. – Andres Mejia Feb 11 '22 at 20:45

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Here is another example: the circle $S^1$ is a retract of $\Bbb R^2\setminus\{(0,0)\}$: consider the map$$\begin{array}{rccc}r\colon&\Bbb R^2\setminus\{(0,0)\}&\longrightarrow&S^1\\&v&\mapsto&\frac1{\|v\|}v.\end{array}$$The idea is that you can (in a continuous way) make the whole space $B$ collapse into $A$ in such a way that no element of $A$ is affected (that is, each element of $A$ is mapped into itself).

José Carlos Santos
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  • Great explanation! Thank u sir! One more question on the side is there like a strategy to find a retract? Or is that just experience? –  Feb 11 '22 at 15:33
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    It's mainly experience and geometric intuition. Here's another example: try to prove that ${(x,y)\in\Bbb R^2\mid x^2+y^2\geqslant1}$ is also a retract of $\Bbb R^2\setminus{0}$. – José Carlos Santos Feb 11 '22 at 16:22
  • Would that be $x \to \frac{x}{|x|^2}$? Thank u for the example and exercise! :) –  Feb 12 '22 at 10:26
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    No. That doesn't work since, for instance, it maps $(1,1)$ into $\left(\frac12,\frac12\right)$. What I had in mind was $x\mapsto\frac x{\min{|x|,1}}$. – José Carlos Santos Feb 12 '22 at 10:47
  • I see very helpful example thanks! :) –  Feb 12 '22 at 11:49