Kirby Problem 5.7

Definitions and mathematical statement

Let $S^5\subset\mathbb R^6$ be the standard smooth sphere. A smooth circle action is a smooth map $a:S^1\times S^5\to S^5$ satisfying the identity and multiplication laws. The stabilizer of a point $x$ is $G_x=\{u\in S^1:a(u,x)=x\}$. An orbit is multiple or exceptional if $G_x$ is a nontrivial finite subgroup of the circle, hence a cyclic group of some order $m>1$.

The action is pseudo-free if there are no fixed points of the whole circle and the exceptional orbits are isolated: each has an invariant neighborhood containing no other exceptional orbit. Use effective actions, so no nonidentity element fixes all points. Away from the exceptional orbits the action is free, with trivial stabilizers.

Does there exist such a smooth pseudo-free action on $S^5$ with more than three exceptional orbits? Equivalently, is four or more possible in this dimension? Different exceptional orbits are counted separately, even if their stabilizers have the same order. The source poses the existence question underlying the Montgomery--Yang problem.

The sphere must have its standard smooth structure. An action on a homology sphere or another $5$-manifold would not suffice without identifying the manifold with $S^5$. Actions with fixed points or positive-dimensional families of exceptional orbits are outside the pseudo-free hypothesis; their orbit counts do not answer the question.

Short English statement

Can a smooth pseudo-free circle action on the standard five-sphere have more than three exceptional orbits?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3014 (KP-5.7), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 5.7. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.