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- W1489621829 abstract "Let X be a PL homotopy Cp2k+1 corresponding by Sullivan's classification to the element (N1, a2, N2, ak, Nk) of ZeZ2eZ @ ... eZ2eZ. THEOREM 1. The topological circle action on S4k+3 with orbit space X is the restriction of an S3 action with a triangulable orbit space iff ai =O, i=2, , k; and N1=O mod 2; and j (-1)iN =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 53 action. In this note we establish some necessary conditions for extending smooth free circle actions on homotopy (4k+3)-spheres to free S3 actions (SI is the group of unit quaternions). It is known that the orbit space of a free circle action on a homotopy sphere 414k+3 is a manifold X4k+2 with the homotopy type of complex projective space Cp2k+1, while the orbit space of an SI action is a homotopy quaternionic projective space Y4k. If the circle action is the restriction of the S3 action (by the maximal torus theorem, the restriction is unique), then X4k+2 is an S2 bundle over Y4k; in fact this bundle is induced by a homotopy equivalence Y4k->QPk from the natural fibering Cp2k+l -->Qpk. In terms of D. Sullivan's classification up to PL isomorphism of PL homotopy complex projective spaces [7] we are able to state in Theorem 1 necessary and sufficient conditions for a homotopy complex projective space X4k+2 to fiber in such a way over some PL homotopy quaternionic projective space. We also show that if k>1 the PL isomorphism class of the orbit space Y4k of an S3 action on 414k+3 is determined by the restricted circle action (the case k = 1 is of course equivalent to the four dimensional Poincare conjecture; in this case Theorem 1 implies that only the standard circle action on S7 extends to an S3 action). A smooth circle action satisfying the hypotheses of Theorem 1 will thus extend to a topological S3 action with a triangulable orbit space. Presented to the Society, August 20, 1970; received by the editors April 2, 1970. AMS 1968 subject classifications. Primary 5710, 5732, 5747." @default.
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- W1489621829 date "1971-01-01" @default.
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- W1489621829 title "Extending free circle actions on spheres to $Ssp{3}$ actions" @default.
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- W1489621829 doi "https://doi.org/10.1090/s0002-9939-1971-0275470-4" @default.
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