Let $f:X\rightarrow Y$ be a map that induces the zero map in the fundamental groups if $f$ nullhomotopic?

Question

Let $f:X\rightarrow Y$ be a map that induces the zero map in the fundamental groups is $f$ nullhomotopic? It seems intuitively correct, however I cant come up with a proof or a counterexample
Thanks in advance

Answer 1

It's not true. For example, take the identity map of the 2 sphere.

The induced map on fundamental groups is zero because the fundamental group is zero. However it is not null homotopic because it induces the identity map on the second homology.