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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Richard I. Resch
Journal: Proc. Amer. Math. Soc. 49 (1975), 461-468
MSC: Primary 57E25; Secondary 55F25
DOI: https://doi.org/10.1090/S0002-9939-1975-0370633-5
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 <IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="${S^3}$">-action, principal <IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${S^3}$">-bundle, spin manifold, Postnikov system, second Stiefel-Whitney class, Sq<IMG WIDTH="15" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$^{2}$">
Article copyright: © Copyright 1975 American Mathematical Society