Let $f$ be a map from the real projective plane to the torus. Show that $f$ must be homotopic to a constant map.

Question

Let $f$ be a map from the real projective plane to the torus. Show that $f$ must be homotopic to a constant map. This is a qual problem. Any help would be appreciated. Thank you.

Answer 1

Since the universal cover of the torus is contractible, it's enough to show that such a map must lift to the universal cover. When does that happen?