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Proceedings of the American Mathematical Society

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Extending free circle actions on spheres to $ S\sp{3}$ actions


Author: Bruce Conrad
Journal: Proc. Amer. Math. Soc. 27 (1971), 168-174
MSC: Primary 57.47
DOI: https://doi.org/10.1090/S0002-9939-1971-0275470-4
MathSciNet review: 0275470
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Abstract: Let $ X$ be a PL homotopy $ C{P^{2k + 1}}$ corresponding by Sullivan's classification to the element $ ({N_1},{\alpha _2},{N_2}, \cdots ,{\alpha _k},{N_k})$ of $ Z \oplus {Z_2} \oplus Z \oplus \cdots \oplus {Z_2} \oplus Z$.

Theorem 1. The topological circle action on $ {S^{4k + 3}}$ with orbit space $ X$ is the restriction of an $ {S^3}$ action with a triangulable orbit space iff $ {\alpha _i} = 0,i = 2, \cdots ,k$; and $ {N_1} \equiv 0\bmod 2$; and $ \sum {( - 1)^i}{N_i} = 0$.

If $ X$ admits a smooth structure and satisfies the hypotheses of Theorem 1, a certain smoothing obstruction arising from the integrality theorems vanishes for the corresponding $ {S^3}$ action.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0275470-4
Keywords: Complex protective space, quaternionic projective space, $ h$-smoothing, $ h$-triangulation, triangulated vector bundle, index, Pontrjagin classes, spin manifold, $ \hat A$-genus
Article copyright: © Copyright 1971 American Mathematical Society

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