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 \({X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}}\). 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 ≥ 2 (respectively, k ≥ 3), where \({\mathbb{S}^k}\) is equipped with the antipodal involution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allday C., Puppe V.: Cohomological methods in transformation groups. Cambridge Studies in Advanced Mathematics, vol. 32. Cambridge University Press, Cambridge (1993)
Bredon G.E.: Introduction to Compact Transformation Groups. Academic Press, New York (1972)
Borel A. et al.: Seminar on transformation groups. Annals of Mathematics Studies, vol. 46. Princeton University Press, Princeton (1960)
Chang C.N., Su J.C.: Group actions on a product of two projective spaces. Am. J. Math. 101, 1063–1081 (1979)
Conner P.E., Floyd E.E.: Fixed point free involutions and equivariant maps-I. Bull. Am. Math. Soc. 66, 416–441 (1960)
Deo S., Tripathi H.S.: Compact Lie group actions on finitistic spaces. Topology 21, 393–399 (1982)
Deo S., Singh T.B.: On the converse of some theorems about orbit spaces. J. Lond. Math. Soc. 25, 162–170 (1982)
Dotzel R.M., Singh T.B., Tripathi S.P.: The cohomology rings of the orbit spaces of free transformation groups on the product of two spheres. Proc. Am. Math. Soc. 129, 921–930 (2000)
Livesay G.R.: Fixed point free involutions on the 3-sphere. Ann. Math. 72, 603–611 (1960)
McCleary J.: A user’s guide to spectral sequences Cambridge Studies in Advanced Mathematics, vol. 58, 2nd edn. Cambridge University Press, Cambridge (2001)
Myers R.: Free involutions on lens spaces. Topology 20, 313–318 (1981)
Rice P.M.: Free actions of \({\mathbb{Z}_4}\) on \({\mathbb{S}^3}\). Duke Math. J. 36, 749–751 (1969)
Ritter G.X.: Free \({\mathbb{Z}_8}\) actions on \({\mathbb{S}^3}\). Trans. Am. Math. Soc. 181, 195–212 (1973)
Ritter G.X.: Free actions of cyclic groups of order 2n on \({\mathbb{S}^1 \times \mathbb{S}^2}\). Proc. Am. Math. Soc. 46, 137–140 (1974)
Rubinstein J.H.: Free actions of some finite groups on \({\mathbb{S}^3}\) —I. Math. Ann. 240, 165–175 (1979)
Swan R.G.: A new method in fixed point theory. Comment. Math. Helv. 34, 1–16 (1960)
Tao Y.: On fixed point free involutions on \({\mathbb{S}^1 \times \mathbb{S}^2}\). Osaka J. Math. 14, 145–152 (1962)
Yang C.T.: On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobo and Dyson-I. Ann. Math. 60, 262–282 (1954)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Singh, M. Orbit Spaces of Free Involutions on the Product of Two Projective Spaces. Results. Math. 57, 53–67 (2010). https://doi.org/10.1007/s00025-009-0014-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-009-0014-8