Simplicial Volumes of Surface Bundles

Definitions and mathematical statement

A surface bundle with fiber $F$ and base $B$ is a locally trivial bundle $p\colon E\to B$ whose fibers are homeomorphic to $F$. The supplied record takes $F$ and $B$ to be fixed closed surfaces, but does not specify connectedness, orientability, genus restrictions, or the bundle category.

For a closed, connected, oriented $n$-manifold $M$, its simplicial volume is \[ \|M\|=\inf\left\{\sum_{j=1}^{r}|a_j|:\ c=\sum_{j=1}^{r}a_j\sigma_j\in C_n(M;{\mathbb{R}}),\ \partial c=0,\ [c]=[M]_{{\mathbb{R}}}\right\}, \] where $c$ ranges over finite real singular cycles and $[M]_{{\mathbb{R}}}$ is the real fundamental class. Write $\chi(X)$ for the Euler characteristic of $X$.

To express the mathematical target conditionally in this standard setting, fix closed, connected surfaces $F$ and $B$, and let $\mathcal B_{\mathrm{or}}(F,B)$ be the class of surface bundles $p\colon E\to B$ with fiber $F$ whose total space is a closed, connected, oriented $4$-manifold. Define \[ \mathcal V_{\mathrm{or}}(F,B)=\{\|E\|:p\colon E\to B\text{ belongs to }\mathcal B_{\mathrm{or}}(F,B)\}. \] The supplied questions, in this conditional setting, are:

1. Does there exist a finite-valued function $C\colon{\mathbb{Z}}^2\to[0,\infty)$ such that, for every pair $F,B$ and every bundle in the intended class, \[ \|E\|\le C\left(\chi(F),\chi(B)\right)? \] Here the bound must depend only on the two Euler characteristics, independently of the bundle.
2. For each fixed pair $F,B$, is $\mathcal V_{\mathrm{or}}(F,B)$ finite or infinite, and does the answer depend on the pair?

The connected, oriented-total-space formulation above is conditional, not a claim that these hypotheses were supplied. The intended orientation and genus assumptions, the bundle category, and any extension of simplicial volume to nonorientable or disconnected total spaces require clarification. The original statement refers to "the Euler characteristic," whereas the cleaned statement refers to "the Euler characteristics"; the displayed bound follows the latter wording. No particular functional form, sharpness requirement, or normalization of that bound is specified.

Short English statement

For bundles with fixed closed surface fiber $F$ and closed surface base $B$, ask whether their total spaces have simplicial volumes bounded using only Euler characteristics, and whether finitely or infinitely many distinct values occur. The precise class of bundles and the intended form of the bound remain unspecified; the displayed formulation is conditional on the stated connectedness and orientation conventions.

Sources

• Local UnsolvedMath registry, supplied complete record, permanent ID $30003288$, title Simplicial Volumes of Surface Bundles, exact source problem code catalogue entry. The record lists proposed year $2016$ and supplies no individual proposer.
• 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 background describes these reports as documenting mathematical workshops and their open-problem discussions. No exact problem page within the supplied range is identified.
• Supplied literature source: Oberwolfach Report 56 (2017). https://doi.org/10.4171/owr/2016/56.
• Supplied literature source: Bucher and Campagnolo (2021). https://doi.org/10.1007/s10711-020-00570-2. No article title or pinpoint reference is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports substantial bounds and special constructions for simplicial volumes, and states that no general theorem determines finiteness or infinitude of the value set for fixed fiber and base. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.