Non-Liftable Monodromy of Surface Bundles

Definitions and mathematical statement

Under the conventional interpretation of the surface notation, let $\Sigma_k$ denote a closed, connected, oriented smooth surface of genus $k$. The supplied background specifies closed fibers; closedness and orientability of the base are understood here from the notation and require confirmation. The record gives no explicit permitted ranges for the genera $g$ and $h$.

Let $\operatorname{Diff}^{+}(\Sigma_g)$ be the group of orientation-preserving smooth diffeomorphisms, and let \[ \operatorname{Mod}_g=\pi_0\left(\operatorname{Diff}^{+}(\Sigma_g)\right) \] be its mapping class group. Write $q:\operatorname{Diff}^{+}(\Sigma_g)\to\operatorname{Mod}_g$ for the homomorphism taking a diffeomorphism to its isotopy class. An oriented smooth surface bundle $p:E\to\Sigma_h$ with fiber $\Sigma_g$ determines, after choosing a base point and identifying its fiber with $\Sigma_g$, a monodromy homomorphism \[ \rho:\pi_1(\Sigma_h)\longrightarrow\operatorname{Mod}_g. \] A lift in the category of groups is a group homomorphism $\widetilde\rho:\pi_1(\Sigma_h)\to\operatorname{Diff}^{+}(\Sigma_g)$ satisfying $q\circ\widetilde\rho=\rho$. Here the lifting question concerns abstract groups, with no additional continuity requirement.

A smooth bundle is called flat here if it admits local trivializations whose transition diffeomorphisms are locally constant on overlaps, equivalently a flat smooth bundle connection. It is nonflat if no such structure exists on that bundle.

Existence question. Do there exist genera $g,h$ and an oriented smooth surface bundle \[ \Sigma_g\longrightarrow E\xrightarrow{\ p\ }\Sigma_h \] which is nonflat and whose monodromy satisfies \[ \nexists\,\widetilde\rho:\pi_1(\Sigma_h)\longrightarrow\operatorname{Diff}^{+}(\Sigma_g) \quad\text{a group homomorphism with}\quad q\circ\widetilde\rho=\rho\,? \] The intended genus restrictions remain unspecified. Both requirements from the supplied statement are retained; no equivalence between them is asserted here. Nonflatness concerns the bundle itself, without imposing preservation of a distinguished section.

Short English statement

The question asks whether a surface bundle over a surface can have no flat structure and have monodromy that cannot be lifted from mapping classes to actual orientation-preserving diffeomorphisms by a group homomorphism. The intended genus ranges and the conventional interpretation of the base surface require confirmation. Failure to find a flat connection preserving a chosen section does not by itself meet the stated target.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30003294$, title Non-Liftable Monodromy of Surface Bundles, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2016; no proposer is named.
• Supplied report citation: Surface Bundles (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/56, pp. 3149--3195. https://doi.org/10.4171/owr/2016/56. The supplied literature-source label is "Oberwolfach Report 56 (2017)." The citation's year and the year appearing in the DOI are retained as supplied.
• Supplied literature source: Baykur, Flat bundles and commutator lengths. https://arxiv.org/abs/1206.3512.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem open and reports that Baykur establishes only an approximate statement: certain bundles admit no flat connection making a distinguished section parallel, although the bundles themselves are flat. The supplied assessment says the full closed-fiber lifting problem remains open and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied editorial qualifications, not an independent mathematical verification of this draft or of the literature assessment.