On the Existence of Free Action of S3 on Certain Finitistic Mod P Cohomology Spaces

In this paper we investigate the possibility of free actions of G = S³ on a finitistic mod p cohomology (sphere, real projective space or lens space) X. If X is a mod 2 cohomology k-sphere, then it is observed that G can act freely on X only if k = 4n − 1. In this case, with the canonical free G-action on S^4m−1, we prove that there exist no equivariant map from S⁴ⁿ⁻¹ to X if m > n. If X is a mod 2 cohomology real projective space or mod p cohomology lens space, p an odd prime then we prove that G can not act freely on X.