Kirby Problem 4.88

Definitions and mathematical statement

Let $\tau:S^4\to S^4$ be a smooth free involution: $\tau^2=\operatorname{id}$ and $\tau(x)\ne x$ for every point. Its quotient $Q=S^4/\langle\tau\rangle$ is a smooth four-manifold covered by $S^4$ with deck group $\mathbb Z/2$. The involution reverses orientation, and $Q$ has the homotopy type of $\mathbb{RP}^4$.

Must there be a smoothly embedded sphere $S\cong S^2$ with $\tau(S)=S$? The restriction is a free involution of the sphere, so its quotient is an embedded $\mathbb{RP}^2$ in $Q$. Equivalently, ask for an embedded projective plane $P\subset Q$ representing the nonzero class in $H_2(Q;\mathbb Z/2)$. This condition ensures the essential projective-plane lift, rather than a null-homologous local embedding.

The analogous topological question uses a free topological involution and a locally flat embedded projective plane with a normal disk bundle. It must be considered separately from the smooth question. In this category the quotient need not be identified with the standard projective four-space; the statement is naturally made directly in $Q$.

The invariant sphere need not be a standard equatorial sphere or unknotted, and no conjugacy of the entire involution to a linear action is requested. Conversely, merely finding some embedded projective plane in the quotient without checking its mod-two class does not establish the desired equivalent formulation.

Short English statement

Does every free involution of the four-sphere preserve an embedded two-sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2964 (KP-4.88), 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 4.88. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.