Nonzero degree from a sphere obstructs manifold products ($\mathrm{deg}f\ne 0⟹M\ncong X^k\times Y^{m-k}$)

Suppose a map $S^m\to M$ of closed connected oriented manifolds has nonzero mapping degree. A product decomposition with $0<k<m$ would have oriented closed factors: an orientation-reversing loop in either factor, with the other coordinate fixed, would reverse the product orientation. Pull back their top orientation classes to obtain $\alpha \in H^k(M;\mathbb{Z})$ and $\beta \in H^{m-k}(M;\mathbb{Z})$ with $\alpha ⌣\beta$ a generator of $H^m(M;\mathbb{Z})$. Both pullbacks to the sphere vanish in their intermediate degrees, contradicting the nonzero-degree pullback of their product.