Kirby Problem 4.87

Definitions and mathematical statement

For an integer $n\ge2$, let $C_n={\mathbb{Z}}/n{\mathbb{Z}}$. A smooth effective action on the standard sphere $S^4\subset{\mathbb{R}}^5$ is an injective homomorphism $\alpha:C_n\to\operatorname{Diff}(S^4)$. Its global fixed set is \[ F(\alpha)=\{x\in S^4:\alpha(g)x=x\text{ for every }g\in C_n\}, \] equivalently the fixed set of a generator. For each $d\in\{0,1,2\}$ the source asks for an exotic action with a nonempty fixed set of dimension $d$. The zero-dimensional question concerns a finite nonempty fixed set, not an empty fixed set. No orientation-preserving restriction is stated.

Two actions are topologically conjugate if a homeomorphism $h:S^4\to S^4$ satisfies $h\alpha(g)h^{-1}=\beta(g)$ for every $g$, and smoothly conjugate if such an $h$ can be a diffeomorphism. A common smooth-exotic question would ask for actions topologically but not smoothly conjugate to a specified standard action, for example one induced by an orthogonal representation $C_n\to O(5)$. However, the record never explicitly defines "exotic," fixes a reference action, or says whether merely being nonconjugate to a linear action suffices. These choices are material: knotted fixed-set examples detected topologically need not be exotic in the stronger topologically-standard/smoothly-nonstandard sense. The draft therefore gives this distinction rather than choosing one definition silently.

The scope of $n$ is also unstated: for each requested dimension, one might ask for some $n\ge2$, or for a fixed or every specified $n$. A precise existential template is \[ \text{for each }d\in\{0,1,2\},\quad \exists n\ge2\ \exists\alpha:C_n\hookrightarrow\operatorname{Diff}(S^4), \quad F(\alpha)\ne\varnothing,\ \dim F(\alpha)=d, \] together with a confirmed exoticness predicate. This is a template pending those source choices, not a completed replacement theorem.

The source's free-action context is separate. Free means that every nonidentity element has no fixed point, stronger than just empty global fixed set when $n$ is composite. For a free cyclic action on $S^4$, every nonidentity element would have to reverse orientation, so the cyclic group has order $2$. The background records Cappell--Shaneson's exotic smooth $\mathbb{RP}^4$ examples with standard $S^4$ double cover, as established by Akbulut--Kirby and Gompf, and Fintushel--Stern's other free-involution construction. It says it is unknown whether those examples are diffeomorphic, or whether more than one exotic smooth $\mathbb{RP}^4$ exists. These free cases do not answer any of the requested nonempty fixed-set questions.

In the topological category the background reports exactly two homeomorphism classes of manifolds homotopy equivalent to $\mathbb{RP}^4$, distinguished by the Kirby--Siebenmann invariant, citing Wall's surgery calculation. It points to Ruberman's explicit nonsmoothable representative. This homeomorphism classification is not a count of exotic smooth structures.

For nonempty fixed sets, the background says little is known smoothly in dimensions $0$ and $1$. It records actions with knotted two-dimensional fixed sets due to Giffen, Gordon and Sumners, but says their distinguishing tools are topological. It separately notes substantial work for three-dimensional fixed sets, including examples from gluing two copies of a cork along the boundary and Mazur's work; that dimension is outside the three requested cases.

The literature triage checked 17 August 2026 classifies partial progress from exotic free involutions, while reporting no verified exotic cyclic actions in the intended sense for each requested nonempty fixed-set dimension. It asks for construction or obstruction in dimensions $0,1,2$ and retains open status. This dated assessment does not discard the known knotted-fixed-set actions or redefine them as smooth-exotic solutions.

Short English statement

Do exotic smooth cyclic actions on $S^4$ exist with nonempty fixed sets of dimension $0$, $1$, or $2$? The record needs a precise meaning of exotic and a quantifier on the cyclic order; known free involutions and topologically knotted fixed sets do not automatically settle those intended cases.

Sources

• Local UnsolvedMath registry, permanent ID $2963$, exact source code catalogue entry, "Kirby Problem 4.87," including its literature triage checked 17 August 2026.
• K3: A New Problem List in Low-Dimensional Topology (2026), Problem 4.87, https://aimath.org/pastworkshops/kirbylistrep.pdf, as identified in the supplied triage.
• [CS76] Sylvain E. Cappell and Julius L. Shaneson, Some new four-manifolds, Annals of Mathematics (2) 104(1) (1976), 61--72, DOI 10.2307/1971056.
• [AK79a] Selman Akbulut and Robion Kirby, An exotic involution of $S^4$, Topology 18(1) (1979), 75--81, DOI 10.1016/0040-9383(79)90015-6.
• [Gom91b] Robert E. Gompf, On Cappell-Shaneson 4-spheres, Topology and its Applications 38(2) (1991), 123--136, DOI 10.1016/0166-8641(91)90079-2.
• [FS81] Ronald Fintushel and Ronald J. Stern, An exotic free involution on $S^4$, Annals of Mathematics (2) 113(2) (1981), 357--365, DOI 10.2307/2006987.
• [Wal99] C. T. C. Wall, Surgery on compact manifolds, Mathematical Surveys and Monographs 69, second edition, AMS, Providence, RI, 1999, Chapter 14; edited with a foreword by A. A. Ranicki, DOI 10.1090/surv/069.
• [Rub84] Daniel Ruberman, Invariant knots of free involutions of $S^4$, Topology and its Applications 18(2--3) (1984), 217--224, Section 2, DOI 10.1016/0166-8641(84)90011-7.
• [Gif66] Charles H. Giffen, The generalized Smith conjecture, American Journal of Mathematics 88 (1966), 187--198, DOI 10.2307/2373054.
• [Gor74] C. McA. Gordon, On the higher-dimensional Smith conjecture, Proceedings of the London Mathematical Society (3) 29 (1974), 98--110, DOI 10.1112/plms/s3-29.1.98.
• [Sum75] D. W. Sumners, Smooth ${\mathbb{Z}}_p$-actions on spheres which leave knots pointwise fixed, Transactions of the American Mathematical Society 205 (1975), 193--203, DOI 10.2307/1997199.
• [Maz61] Barry Mazur, A note on some contractible 4-manifolds, Annals of Mathematics (2) 73 (1961), 221--228, DOI 10.2307/1970288.