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Free $ S\sp{3}$-actions on simply connected eight-manifolds

Author: Richard I. Resch
Journal: Proc. Amer. Math. Soc. 49 (1975), 461-468
MSC: Primary 57E25; Secondary 55F25
MathSciNet review: 0370633
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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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Keywords: Free $ {S^3}$-action, principal $ {S^3}$-bundle, spin manifold, Postnikov system, second Stiefel-Whitney class, Sq$ ^{2}$
Article copyright: © Copyright 1975 American Mathematical Society