Kirby Problem 3.33

Definitions and mathematical statement

For a closed connected orientable three-manifold $M$, a Heegaard splitting is a decomposition $M=H_1\cup_\Sigma H_2$ into two handlebodies with common boundary $\Sigma$. A genus-$g$ handlebody is a regular neighborhood of a connected graph with first Betti number $g$, and its boundary is a closed surface of genus $g$. The Heegaard genus $g(M)$ is the smallest genus of such a splitting.

Choose orientations on $M$ and another closed connected three-manifold $N$. A continuous map $f:M\to N$ has degree one when \[ f_*[M]=[N]\quad\text{in }H_3(N;\mathbb Z), \] where $[M]$ and $[N]$ are their fundamental classes. The conjecture is \[ \deg(f)=1\quad\Longrightarrow\quad g(M)\geq g(N). \] No assumption is made that $f$ is a covering, a bundle map, or injective on fundamental groups. Although degree-one maps induce surjections on fundamental groups, the number of group generators need not by itself determine Heegaard genus. The task is to prove the displayed monotonicity for all such maps or produce a degree-one map reversing it. The genera are the minimal genera, rather than the genera of arbitrarily chosen splittings.

Short English statement

Can a degree-one map between closed oriented three-manifolds ever increase the minimum genus of a Heegaard splitting?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2831 (KP-3.33), 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 3.33. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.