Equalities Between Induced Homomorphisms of Fundamental Groups

Question

Let $f$ be a continuous map from a topological space $X$ to the unit circle, $S^1$, such that $f$ is NOT homotopic to a constant map. It is known that two homotopic maps induce the same homomorphisms on the fundamental group. If $f$ was homotopic to a constant map, it would induce a trivial homomorphism. My question is, in this situation, since $f$ is not homotopic to a constant map, can I conclude that $f$ induces a non-trivial homomorphism?

Answer 1

Homotopy classes of maps from a cw complex $X$ to $S^1$ are in bijection with $H^1(X, \mathbb{Z}) \cong hom(\pi_1(X), \pi_1(S^1)$. So the answer to your question is yes.

More generaly, any map $f\colon X\to Y$ that induces a trivial map on fundamental groups can be lifted to the universal cover of $Y$, hence if $Y$ has contractible universal cover, then any such map factors through a contractible space, and therefore is nullhomotopic.