By the Hurewicz theorem, the first homology group of a torus is isomorphic to the fundamental group of a torus . By applying the Mayer-Vietoris sequence, we have and .
Now we see that is an isomorphism in the above long exact sequence. Since is isomorphic to , the quotient map collapsing to a point is not null homotopic.
Let . The universal cover for is a such that is a covering map. By using the lifting criterion (Hatcher p61), we have a lifting map such that . Since is contractible, h is nulhomotopic. It follows that is nullhomotopic.On the other hand, show via covering spaces that any map is nullhomotopic.