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Definition:

Let $A$ be a subspace of $X$ with an inclusion $i:A \to X$. Then $r:X \to A$ is called a retraction if $r \circ i = id_A$ that is $r(a)=a$ $\forall a \in A$.

I read the question Is the unit circle $S^1$ a retract of $\mathbb{R}^2$? in whose answer it is stated that the induced homomorphism $i_*$ on the injection $i$ has to be injective as well.

Let $x_0 = (1,0) \in S^1$, then $r:S^1 \to \{x_0\}$ is clearly continuous and a retraction according to the definition. However, the induced homomorphism $i_*:\pi_1(S^1) \to \pi_1(\{x_0\})$ cannot be injective since $\pi_1(S^1)=\mathbb{Z}$ and $\pi_1({\{x_0}\})=0$.

What is wrong with my reasoning?

Community
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Dávid Natingga
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    The map is induced by the inclusion, so it is actually $i_*: \pi_1({x_0}) \rightarrow \pi_1(S^1)$. So there is no problem. – Potato May 07 '13 at 04:50
  • Thanks, how come I did not notice this? – Dávid Natingga May 07 '13 at 04:51
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    It's easy to get confused. I have done it many times. Wait until you get to cohomology, where all the induced homomorphisms are opposite the direction of the maps... – Potato May 07 '13 at 04:52

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The map is induced by the inclusion, so it is actually $i_*: \pi_1(\{x_0\}) \rightarrow \pi_1(S^1)$. So there is no problem.

Potato
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  • @B11b: It is inappropriate to contact other users just to ask them to help you. Take a look at this meta thread for example. – Zev Chonoles May 16 '13 at 14:42
  • Sorry @Zevchonoles I have already deleted my question when nobody helps me:( – user3911 May 16 '13 at 16:32
  • But some users solve more instructive and understandable. Thus I ask them @ZevChonoles – user3911 May 16 '13 at 16:33
  • @B11b: Do you think any of those users are interested in getting thousands of pings a day by everyone on math.SE who wants their help? No. If anything that sort of behavior will drive them away. Do you contact every mathematician in the world who you think is a good expositor whenever you have a question? Or do you realize that that is inappropriate? Participating on math.SE is not an open invitation to be bombarde with personal requests. – Zev Chonoles May 16 '13 at 17:12