Kirby Problem 4.12

Definitions and mathematical statement

For a closed oriented smooth four-manifold $X$, define $g_X(x)$, $x\in H_2(X;\mathbb Z)$, as the smallest genus of a connected oriented smoothly embedded closed surface representing $x$. Include genus zero. If $X$ has boundary and $K\subset\partial X$ is an oriented knot, define $g_{X,K}(x)$ similarly using connected properly embedded oriented surfaces with boundary exactly $K$ and relative class $x\in H_2(X,\partial X;\mathbb Z)$. Set the value to $\infty$ when no such surface exists; in particular, compatibility requires $\partial x=[K]$.

(a) Suppose $X_1,X_2$ are closed simply connected smooth four-manifolds and a homeomorphism $f:X_1\to X_2$ satisfies \[ g_{X_1}(x)=g_{X_2}(f_*x)\quad\text{for every }x. \] Must $X_1$ and $X_2$ be diffeomorphic?

(b) Suppose instead that they are compact and simply connected with the same identified boundary, and $f$ is the identity there. If \[ g_{X_1,K}(x)=g_{X_2,K}(f_*x) \quad\text{for every boundary knot }K\text{ and every relative class }x, \] must they be diffeomorphic relative to the boundary?

The conclusion requires some diffeomorphism, not one inducing the specific map $f_*$ or isotopic to $f$. Equality must hold for the full genus functions, rather than only selected classes or their intersection numbers. The infinity convention completes the relative formulation on classes that cannot have the prescribed boundary.

Short English statement

Do the complete closed or relative minimum-genus functions determine a simply connected four-manifold's smooth type?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2888 (KP-4.12), 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 4.12. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.