Kirby Problem 2.24

Definitions and mathematical statement

A complex surface is a connected complex manifold of complex dimension two. Consider a compact such surface $X$. A surface-bundle structure on its underlying smooth four-manifold is a smooth locally trivial map $p:X\to B$ whose base $B$ and fiber $F$ are closed connected oriented surfaces of genus at least two. Local triviality means that over each sufficiently small open set $U\subset B$ there is a diffeomorphism $p^{-1}(U)\cong U\times F$ commuting with projection.

Use the fiber-subgroup convention for distinct fiberings in the cited surface-bundle literature. A fibration determines the normal subgroup $K_p=\ker(p_*:\pi_1(X)\to\pi_1(B))$, identified with the fiber group. Two fiberings are equivalent in this convention when their fiber subgroups are equal inside $\pi_1(X)$. Normality removes basepoint-conjugacy ambiguity. Reparametrizing the base does not change the subgroup; arbitrary automorphisms of the total space are not used to identify different subgroups.

Does there exist a compact complex surface with at least three surface-bundle structures having pairwise different fiber subgroups? The same complex surface must support all the structures; constructing unrelated bundles with diffeomorphic fibers is insufficient. The genus restrictions are those in the source discussion: without them, torus constructions change the problem substantially. The question concerns bundles on a complex surface, without adding a requirement that their local trivializations be holomorphic.

Short English statement

Can one compact complex surface fiber over surfaces in at least three different ways, distinguished by their fiber subgroups, with both base and fiber of genus at least two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2772 (KP-2.24), 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 2.24. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 N. Salter, Surface bundles over surfaces with arbitrarily many fiberings, Section 2 and Proposition 2.2. Fiber-subgroup equivalence convention. https://nsalter.science.nd.edu/research/multifiberedsbs.pdf.