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Give an example of a continuous bijection $\phi: X \rightarrow Y$ between path-connected spaces such that the induced homomorphism $\phi_{*}: \pi_1(X, x) \rightarrow \pi_1(Y, \phi(x))$ on fundamental groups is not a bijection.

Attempt: I was thinking of taking a map $$ S^1 \times \mathbb{R} \rightarrow S^1 \times S^1 $$ but I'm not sure how to get a bijection, such that the induced homomorphism is not a bijection. Any help is appreciated.

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Let $X=[0,2\pi)$ and $Y=S^1$ and let $p:\mathbb{R}\to S^1$ be the quotient map. Then, $p|_X$ is continuous and bijective, but $X$ is contractible while $Y$ has a nontrivial fundamental group.

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  • $\begingroup$ What is $\pi$ precisely? What are the equivalence classes? $\endgroup$ – Kamil Jan 6 at 12:57
  • $\begingroup$ It's $\theta\mapsto \exp(i\theta)$ if you want a formula. $\endgroup$ – WoolierThanThou Jan 6 at 13:01
  • $\begingroup$ you might want to use a symbol other than $\pi$ for your map since your answer also contains the number $\pi$ $\endgroup$ – William Jan 6 at 14:16
  • $\begingroup$ ...Right you are. $\endgroup$ – WoolierThanThou Jan 6 at 17:46

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