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.