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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.

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No. There are examples of maps $f$, $g:(X,a)\to(Y,b)$ between simply connected spaces, whose $\pi_1$s are trivial, but which don't induce the same map on higher $\pi_n$s, so aren't homotopic.

The classic example is the Hopf fibration $f$ from $S^3$ to $S^2$. It induces a non trivial element of $\pi_3(S^2)$ so a nontrivial map from $\pi_3(S^3)\cong\Bbb Z$ to $\pi_3(S^2)$. But it induces a trivial map from $\pi_1(S^3)$ to $\pi_1(S^2)$ as these groups are trivial. Now take $g$ to be the map collapsing $S^3$ to the basepoint in $S^2$.

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