Kirby Problem 4.37

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A projective plane in $S^4$ is an embedding of $\mathbb{RP}^2$. A two-knot is an embedding $S^2\hookrightarrow S^4$, with no assumption that it is unknotted. Connected sum of embedded surfaces is formed by removing a small standard disk from each in separate four-balls and joining the boundary circles by a tube in the connected sum $S^4\#S^4\cong S^4$.

For orientable genus $g\ge0$, the standard unknotted surface is the boundary of a standard genus-$g$ handlebody in an equatorial $S^3\subset S^4$. For the nonorientable models, write $P_+$ and $P_-$ for the standard unknotted projective planes with normal Euler numbers $+2$ and $-2$. One explicit model is the normalized Veronese embedding \[ \mathbb{RP}^2\longrightarrow S^4\subset\operatorname{Sym}_0(3,\mathbb R),\qquad [v]\longmapsto\sqrt{\frac32}\left(vv^{\mathsf T}-\frac13I_3\right),\quad |v|=1, \] where $\operatorname{Sym}_0(3,\mathbb R)$ is the five-dimensional space of real symmetric trace-zero matrices with its Euclidean matrix norm; reflection gives the other sign. The standard nonorientable genus-$h$ surfaces are connected sums of $h$ copies of $P_+$ or $P_-$, formed by joining standard local disks by tubes. Their normal Euler number is the sum of the signs $\pm2$. An embedding is topologically or smoothly unknotted if it is ambiently isotopic, in that category, to a corresponding standard model.

Statement recorded in the supplied source.

In each of the smooth and the locally flat categories, is every embedded projective plane $P\subset S^4$ ambiently isotopic to \[ S\#P_+\quad\text{or}\quad S\#P_- \] for some two-knot $S\subset S^4$ in the same category? This is the Kinoshita decomposition question; the sphere summand is allowed to be knotted. The source also records these subsidiary questions:

1. (i) Does there exist an embedded Klein bottle $\Sigma\subset S^4$ with $|e(\Sigma)|=4$ that has no unknotted projective-plane connected-sum summand? The normal Euler number $e(\Sigma)$ is the Euler number of its normal two-plane bundle, with local orientation coefficients.
2. (ii) For a projective plane $P\subset S^4$ and a tubular neighborhood $\nu P$, must the kernel of the inclusion-induced map \[ \pi_1\left(\partial(S^4\setminus\operatorname{int}\nu P)\right) \longrightarrow\pi_1(S^4\setminus P) \] have order four? Use a boundary basepoint and its image; changing it by a path only conjugates the kernel.

As with the main decomposition question, these embedding questions can be considered in the smooth or locally flat category. [S2]

Short English statement

Ask whether every projective plane splits off an unknotted projective plane, whether an Euler-four Klein bottle can avoid such a summand, and whether each projective-plane peripheral inclusion has kernel of order four.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2913 (KP-4.37). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.37 and Remarks, PDF page 220. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.