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

Abstract

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 $X\hookrightarrow X_{{\mathbb{Z}}_2}⟶B_{{\mathbb{Z}}_2}$. 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 $k\ge 2$ (respectively $k\ge 3$), where ${\mathbb{S}}^k$ is equipped with the antipodal involution.

arXiv comment14 pages, to appear in Results in Mathematics

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