Orbit spaces of free involutions on the product of two projective spaces

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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 XZ₂ BZ₂. 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 Z₂-equivariant map from Sᵏ → X for k ≥ 2 (respectively k ≥ 3), where Sᵏ is equipped with the antipodal involution.