Free $S^{3}$-actions on $2$-connected nine-manifolds

In this paper a classification of free ${S^3}$-actions on 2-connected 9-manifolds is obtained by examining the corresponding principal ${S^3}$-bundles. The orbit spaces that may occur are determined and it is proved that there are exactly two homotopy classes of maps from each of these spaces into the classifying space for principal ${S^3}$-bundles. It is shown that the total spaces of the corresponding bundles are distinct, yielding the main result that for each nonnegative integer k, there exist exactly two 2-connected 9-manifolds which admit free ${S^3}$-actions and, furthermore, the actions on each of these manifolds are unique.