Extending free circle actions on spheres to 𝑆³ actions

Abstract

Let X X be a PL homotopy C P 2 k + 1 C{P^{2k + 1}} corresponding by Sullivan’s classification to the element ( N 1 , α 2 , N 2 , ⋯ , α k , N k ) ({N_1},{\alpha _2},{N_2}, \cdots ,{\alpha _k},{N_k}) of Z ⊕ Z 2 ⊕ Z ⊕ ⋯ ⊕ Z 2 ⊕ Z Z \oplus {Z_2} \oplus Z \oplus \cdots \oplus {Z_2} \oplus Z . Theorem 1. The topological circle action on S 4 k + 3 {S^{4k + 3}} with orbit space X X is the restriction of an S 3 {S^3} action with a triangulable orbit space iff α i = 0 , i = 2 , ⋯ , k {\alpha _i} = 0,i = 2, \cdots ,k ; and N 1 ≡ 0 mod 2 {N_1} \equiv 0\bmod 2 ; and ∑ ( − 1 ) i N i = 0 \sum {( - 1)^i}{N_i} = 0 . If X 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 {S^3} action.