Kirby Problem 1.87

Definitions and mathematical statement

A link $L\subset S^3$ is fibered if its exterior fibers over $S^1$ with oriented surface fibers whose boundary is the preferred parallel copy of $L$. Let $f=(f_1,f_2):\mathbb R^4\to\mathbb R^2$ be a polynomial map with $f(0)=0$. An isolated singular point at the origin means that $Df(0)$ has rank less than two, while $Df(x)$ has rank two for every nonzero $x$ in some neighborhood of the origin. For sufficiently small transverse spheres, define its link by \[ L_f=\epsilon^{-1}\left(f^{-1}(0)\cap S^3_\epsilon\right)\subset S^3. \] The small-sphere link type is independent of the sufficiently small radius.

Does every fibered link in $S^3$ occur as $L_f$ for such a real polynomial map? The rank condition is required on a whole punctured neighborhood, including points outside the zero fiber. Requiring it only on $f^{-1}(0)\setminus\{0\}$ is the weaker notion of a weakly isolated singularity and gives a different realization problem. The polynomials are real; they need not arise from a holomorphic map $\mathbb C^2\to\mathbb C$.

Short English statement

Is every fibered link the link of an isolated critical point of a real polynomial map from four-space to two-space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2746 (KP-1.87), 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 1.87. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.