Cohomology classification of spaces with free S1 and S3-actions

This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S3 and the cohomology ring of the orbit space X/G is isomorphic to the integral cohomology quaternion projective space HPn. We have proved that the integral cohomology ring of X is isomorphic either to S4n+3 or S3 ? HPn. Similar results with other coefficient groups and for G = S1 actions are also discussed. As an application, we determine a bound of the index and co-index of cohomology sphere S2n+1 (resp. S4n+3) with respect to S1-actions (resp. S3-actions).