I'm studying Fundamental groups and today I saw the follow theorem:
Theorem: Let be $X$ a topological space path-connected and $x,y\in X$. Then, the application $\psi:\pi_1(X,x)\to \pi_1(X,y)$ is a isomorphism of groups.
I understood the proof, but I would like to know if, under conditions of the previous theorem, for each patch $\alpha\in\pi_1(X,x)$ there is a homotopy between $\alpha$ and $\psi(\alpha)$? I know this result is positive when $X$ is simply connected, but not in this case.
Thank you!
Best Answer
The map $\psi$ is defined by $\psi(\alpha) = \gamma \alpha \gamma^{-1}$, where $\gamma:[0,1] \to X$ is a path with $\gamma(0) = y$, $\gamma(1) = x$. Then a homotopy can be given by "reeling in" $\gamma$.
A "reeling" homotopy is $$ h_s(t) = \begin{cases} \gamma(s) & t \in [0,s] \\ \gamma(t) & t \in [s,1]. \end{cases} $$ I'll leave you with the exercise to use this to get a homotopy from $\psi(\alpha)$ to $\alpha$.