Let $X$ be a simply connected topological space. Let $x_0,x_1\in X$ and let $\gamma$ be a path in $X$ from $x_0$ to $x_1$. Let $g$ be a homeomorphism of $X$ with itself. Then $g\circ\gamma$ is a path in $X$ from $g(x_0)$ to $g(x_1)$.
Is $\gamma$ homotopic to $g\circ\gamma$? (NOT path homotopic, just homotopic)
I think it should be true but I can't seem to give a homotopy from $\gamma$ to $g\circ\gamma$. That is I need a continuous map $F:I\times I\rightarrow X$ such that $F(s,0)=\gamma(s)$ and $F(s,1)=g\circ\gamma(s)$ for all $s\in I$.
Thanks!
Since $X$ is simply connected, every two paths are homotopic (almost) by definition.