Consider this long exact sequence,

.

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.