= Mod-two degree obstruction for equal projective factors
{title2=$f_*=0:H_{2n}(\mathbb{RP}^n\times\mathbb{RP}^n;\mathbb F_2)\to H_{2n}(\mathbb{RP}^{2n};\mathbb F_2)\quad(n\ge1)$}
For any continuous map between the displayed <Real projective spaces>, the <induced map on cohomology> sends the degree-one generator to $\alpha x+\beta y$. Its $2n$th power is $\binom{2n}{n}\alpha^n\beta^n x^ny^n$, since all other terms exceed the truncation exponent of one factor. The coefficient is zero modulo two because $\binom{2n}{n}=2\binom{2n-1}{n-1}$. Naturality of the evaluation pairing then makes the top <induced map on homology> zero. This obstruction does not apply to $n=0$, when both spaces are points.
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