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When answering this question I used the fact that when we have a retract $r:X \rightarrow Y$ the induced homomorphism $r_\ast: \pi_1(X) \rightarrow \pi_1(Y)$ is surjective. I can recall how to prove this fact, but I can't seem to remember any examples where the induced map wasn't an isomorphism. So I was wondering if there are some fairly simple examples of retractions where the induced map is not injective.

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up vote 4 down vote accepted

Take $X$ and $Y=\{x\}$ with $x\in X$. Therefore the obivous map $r:X\to Y$ is continuous. So $r_*:\pi_1(X)\to\{e\}$ is surjective but, if $X$ is not simply connected, it isn't injective.

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I do like this example. But I would also like an example where $\pi_1(Y)$ is not trivial. Which I suppose can easily be constructed by looking at $X \times \mathbb S^1$ and $Y \times \mathbb S^1$. – JSchlather Jan 7 '13 at 2:41
You answered on your own :) or by a wedge product if you want it more spicy ... – wisefool Jan 7 '13 at 2:46

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