Cohomology classification of spaces with free S^1 AND S^3-actions

Abstract

This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S 3 and the cohomology ring of the orbit space X/G is isomorphic to the integral cohomology quaternion projective space HP^n . We have proved that the integral cohomology ring of X is isomorphic either to S^{4n+3} or S^3 × HP^n . Similar results with other coefficient groups and for G = S^1 actions are also discussed. As an application, we determine a bound of the index and co-index of cohomology sphere S^{2n+1} (resp. S^{4n+3}) with respect to S^1 -actions (resp. S^3 -actions).