Show that if a path-connected, locally path-connected space X has a finite fundamental group , then every map $X$ to $S^1 \times S^1$ is nullhomotopic (i.e. homotopic to a constant map) .
Is the same true if we replace the torus with the wedge sum of two circles?
I was able to solve the first part using that the fundamental group of the covering space: $R \times R$ is trivial, but regarding the second part, about the wedge sum of two circles, I think the issue is that the fundamental group of the covering space of the wedge sum is not trivial,is my intuition true? or I am missing something?