Kirby Problem 2.32

Definitions and mathematical statement

Let $F=S_g$ be a closed oriented surface of genus $g\geq2$ and let $p:E\to B$ be a smooth oriented $F$-bundle. An Ehresmann connection is a subbundle $\mathcal H\subset TE$ complementary to the vertical tangent bundle $\ker(dp)$. It is flat if $\mathcal H$ is integrable, so its integral leaves form a foliation transverse to every fiber.

Equivalently, a flat bundle is a suspension \[ (\widetilde B\times F)/\pi_1(B),\qquad \gamma\cdot(b,x)=(\gamma b,\widetilde\rho(\gamma)x), \] for a homomorphism $\widetilde\rho:\pi_1(B)\to\operatorname{Diff}^+(F)$. The original bundle has monodromy $\rho:\pi_1(B)\to\operatorname{Mod}(F)$, and flatness asks for such a diffeomorphism-valued lift that realizes its bundle isomorphism class.

Does every oriented surface bundle over a closed oriented surface admit a flat connection? Does the same hold when the base is a closed three-manifold? These are existence questions for some connection on each bundle, not for flatness of an initially chosen connection. The higher-genus fiber setting is essential to the mapping-class formulation; sphere and torus bundles have additional classification issues and are not implicitly used to redefine it.

Short English statement

Can every higher-genus surface bundle over a surface, or over a three-manifold, be equipped with a flat connection?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2780 (KP-2.32), 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 2.32. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.