  Let $X$ be a finitistic space having the mod 2 cohomology algebra of the
product of two projective spaces. We study free involutions on $X$ and
determine the possible mod 2 cohomology algebra of orbit space of any free
involution, using the Leray spectral sequence associated to the Borel fibration
$X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}$. We also
give an application of our result to show that if $X$ has the mod 2 cohomology
algebra of the product of two real projective spaces (respectively complex
projective spaces), then there does not exist any $\mathbb{Z}_2$-equivariant
map from $\mathbb{S}^k \to X$ for $k \geq 2$ (respectively $k \geq 3$), where
$\mathbb{S}^k$ is equipped with the antipodal involution.

Algebraic Topology