Kirby Problem 4.44

Definitions and mathematical statement

Let $X$ be a closed oriented simply connected four-manifold and $x\in H_2(X;\mathbb Z)$. For integers $g,d\ge0$, ask whether $x$ has a representative $f:\Sigma_g\to X$ by a generic immersion of a connected closed oriented genus-$g$ surface with exactly $d$ double points. Generic means that the map is locally an embedding except for transverse intersections of two sheets, with no triple points or other singularities; double points are counted without signs. Require $f_*[\Sigma_g]=x$.

Treat smooth immersions in a smooth $X$ and locally flat topological immersions in a topological $X$ separately, using the corresponding transverse local models. The problem is to determine existence for the prescribed data, not to assert that every triple $(x,g,d)$ is realizable.

For $d=0$, define the genus function as the minimum genus of a connected embedded representative of $x$, using the same category. Compute the locally flat genus function for manifolds other than $S^4$, and compute smooth genus functions for further four-manifolds.

The specified test is whether $(3,2)\in H_2(\mathbb{CP}^2\#\mathbb{CP}^2;\mathbb Z)$ is represented by a sphere. The coordinates refer to the standard positive generators in the two summands, so the square of the class is $13$. Ask the smooth and locally flat versions explicitly. An immersed sphere with double points would not settle this embedded-sphere question, and an answer in one category need not determine the other.

Short English statement

Determine which homology classes admit surfaces of prescribed genus and double-point count, including the sphere test in two positive projective planes.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2920 (KP-4.44), 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 4.44. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.