I'm really struggling with following exercise:
Let $f:X \to Y$ be a continuous map between topological space. Let $f_*:\pi_1(X,x) \to \pi_1(Y,f(x))$ be the induced homomorphism between the fundamental groups. Prove/disprove:
- If $f$ is injective, then $f_*$ is injective.
- If $f$ is surjective, then $f_*$ is surjective.
I'm not even sure how to begin with the first question, but I might have a start for the second question - Take the function $f: [0,1] \to S^1$, $f(t) = (\cos(2\pi t), \sin(2\pi t))$ which is surjective and the path $\gamma : [0,1] \to S^1$, $\gamma = f$. Choose $x = 0, f(x)=(0,1)$. If $f \circ \delta \sim \gamma$ for some $\delta:[0,1] \to [0,1]$, since $\gamma = f$ we will have $f \circ \delta \sim f$. I'm not really sure how to proceed from here - I think the issue will arise from either $\delta$ being a closed path or $S^1$ not being simply-connected, but I'm not sure.