= Factorial degree obstruction for products of two-spheres
{title2=$\deg f=m!\prod_{i=1}^m a_i$}
Give $(S^2)^m$ its product <orientation> and $\mathbb{CP}^m$ its complex <orientation>. If $u_i$ are the degree-two sphere classes and $h$ the normalized projective class, write $f^*h=\sum_i a_i u_i$. In the source <cohomology ring>, $u_i^2=0$ and the generators commute. Hence $f^*h^m=m!(\prod_i a_i)u_1\cdots u_m$. Evaluation on the <fundamental class> gives the displayed <mapping degree>. Every positive degree is therefore at least $m!$, and the <polynomial multiplication map to complex projective space> attains that bound.
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