Free 𝑆³-actions on simply connected eight-manifolds

Resch, Richard I.

doi:10.1090/S0002-9939-1975-0370633-5

Free $S^{3}$-actions on simply connected eight-manifolds
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by Richard I. Resch PDF

Proc. Amer. Math. Soc. 49 (1975), 461-468 Request permission

Abstract:

In this paper the canonical equivalence between free actions of a compact Lie group $G$ and principal $G$-bundles is used to apply the theory of fiber bundles to the problem of classifying free differentiable ${S^3}$-actions. The orbit spaces that may occur are determined and a calculation of homotopy classes of maps from these spaces into the classifying space for principal ${S^3}$-bundles is made with the aid of the Postnikov system for ${S^4}$. The bundles corresponding to these classes of maps are then studied to prove that for each positive integer $k$ there exist exactly three simply connected $8$-manifolds which admit free differentiable ${S^3}$-actions and have second homology group free of rank $k$, and that the action on each of these manifolds is unique. It is also proved that even if the second homology group of the $8$-manifold has torsion, it can admit at most one action.

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