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

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.