= Nonzero degree from a sphere obstructs manifold products
{title2=$\deg f\ne0\Longrightarrow M\not\cong X^k\times Y^{m-k}$}
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 $\alpha\smile\beta$ 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.
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