Kirby Problem 2.25

Definitions and mathematical statement

Let $p:E\to B$ be a smooth oriented bundle with fiber $F$, where $B$ and $F$ are closed connected surfaces of genus at least two. Transport around loops in $B$ gives its monodromy homomorphism $\rho:\pi_1(B)\to\operatorname{Mod}(F)$, defined up to conjugacy, where $\operatorname{Mod}(F)$ is the group of orientation-preserving diffeomorphisms of $F$ modulo isotopy. Assume that $\rho$ is injective.

An unbranched multisection of degree $k\geq1$ is an embedded surface $C\subset E$ such that $p|_C:C\to B$ is a $k$-sheeted covering map. The surface $C$ need not be connected. Equivalently, there is a continuously varying unordered configuration of $k$ distinct points in every fiber, with no collisions. In configuration-space language it is a section of the associated bundle whose fiber is \[ \operatorname{UConf}_k(F)= \{(x_1,\ldots,x_k)\in F^k:x_i\ne x_j\ (i\ne j)\}/\mathfrak S_k. \] Does every such bundle admit an unbranched multisection for some finite $k$? The degree may depend on the bundle. A branched multisection, where points can merge over exceptional base points, does not meet the requirement.

Short English statement

Does every surface bundle with injective monodromy admit a finite, continuously varying set of distinct points in each fiber?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2773 (KP-2.25), 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 2.25. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.