Question 2 in Chapter 1.B in Hatcher's Algebraic Topology:
Let $X$ be a connected CW complex and $G$ a group such that every homomorphism $\pi_1(X)\to G$ is trivial. Show that every map $X\to K(G, 1)$ is nullhomotopic.
A $K(G,1)$ space is a path connected spaces whose fundamental group is isomorphic to a given group $G$ and which has a contractible universal covering space.
A map is nullhomotopic if is homotopic to the constant map.
A theory I have been thinking of using is Theorem 1B.9 in the text:
Let $X$ be a connected CW complex and let $Y$ be a $K(G,1)$. Then every homomorphism $\pi_1(X, x_0)\to\pi_1(Y, y_0)$ is induced by a map $(X, x_0)\to(Y, y_0)$ that is unique up to homotopy fixing $x_0$ .