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

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

2010-01-04

arxiv.org/abs/1001.0450v1

Results in Mathematics, 57 (2010), 53-67

Mathematics

Algebraic Topology

14 pages, to appear in Results in Mathematics

Scientific paper

10.1007/s00025-009-0014-8

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.

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