# degree of a map between Manifold and Hopf theorem

Let $$M$$, and $$N$$ be closed orientable manifolds with the same dimensions. Hopf Proved that if $$N=\mathbb{S}^n$$, then every two continuous maps from $$M$$ to $$N$$ with the same degree are homotopy equivalence. In particular, each degree zero map from $$M$$ to $$\mathbb{S}^n$$ is null-homotopic.

$$\textbf{Q})$$ When a degree zero map between closed orientable manifolds with the same dimensions is a null-homotopy? Or, is there any generalization of Hopf theorem for the case that the target space is not a sphere?

• Unfortunately there is no such generalization. – user98602 Dec 14 '18 at 15:31