This question is the opposite side of Exercise 3.3, page 20 of Forster's Lectures on Riemann Surfaces.
Let $(X,a)$ and $(Y,b)$ be topological spaces with base points $a\in X$ and $b\in Y$. Let $f$, $g:X\to Y$ be two continuous maps with $f(a)=g(a)=b$. Then $f$ and $g$ are called homotopic if there exists a continuous map
$F:X\times [0,1]\to Y$
such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for every $x\in X$ and $F(a,t)=b$ for every $t\in [0,1]$. Consider the induced maps
$f_*$, $g_* :\pi _1 (X,a)\to \pi _1 (Y,b)$.
My question is, if $f_*=g_*$, are $f$ and $g$ necessarily homotopic?
The point is we cannot find a common homotopy, so it may be hard to say yes directly. And I haven't find a counterexample yet. So I post the question here.