Suppose $X$ is path connected, let $F:(X,x_0)\to (X,x_0)$ be a map such that $F_*: \Pi_1(X,x_0)\to \Pi_1(X,x_0)$ is identity, does it imply that $F$ is homotopic to identity?
Let $y_0$ be arbitrary, and choose any loop $\gamma:I\to X$ around $x_0$ such that $\gamma(1/2) = y_0$. Then there is a homotopy between $\gamma$ and $F\circ\gamma$, and when we restrict this homotopy to the point $1/2$ we will get a path from $y_0$ to $F(y_0)$. My idea is to show that this can be extended to a homotopy of $F$ to the identity map. But I don't know how to make this precise.
Why is it true if we assume $X$ to be $K(G,1)$?