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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?

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

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