A question concerning fundamental groups and whether a map is null-homotopic. Is it true that if $X$ and $Y$ are topological spaces, and $f:X \rightarrow Y$ is a continuous map and the induced group homomorphism $\pi_1(f):\pi_1(X) \rightarrow \pi_1(Y)$ is the trivial homomorphism, then we have that $f$ is null-homotopic?

Not at all.
For example, the identity $Id_ {S^2}: S^2\to S^2 $ induces trivial homomorphism in the fundamental groups - but it is not null homotopic.
In fact, this identity is the generator of the second homotopy group of the sphere, which is not trivial.
