Mod-two degree obstruction for equal projective factors ($f_{\ast }=0:H_{2n}({\mathbb{RP}}^n\times {\mathbb{RP}}^n;{\mathbb{F}}_2)\to H_{2n}({\mathbb{RP}}^{2n};{\mathbb{F}}_2)(n\ge 1)$)

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 $\left(\genfrac{}{}{0ex}{}{2n}{n}\right){\alpha }^n{\beta }^n{x}^n{y}^n$, since all other terms exceed the truncation exponent of one factor. The coefficient is zero modulo two because $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)=2\left(\genfrac{}{}{0ex}{}{2n-1}{n-1}\right)$. 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.