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?