Unbounded Simplicial Volume in Surface-Bundle Families

Definitions and mathematical statement

For a closed, connected, oriented $n$-manifold $M$, its simplicial volume is \[ \|M\|=\inf\left\{\sum_{j=1}^{k}|a_j|:\ c=\sum_{j=1}^{k}a_j\sigma_j\text{ is a real singular cycle representing }[M]_{{\mathbb{R}}}\right\}, \] where $a_j\in{\mathbb{R}}$, the $\sigma_j$ are singular $n$-simplices, and $[M]_{{\mathbb{R}}}$ is the real fundamental class. This definition states the usual closed oriented manifold convention; the supplied record does not specify whether all its hypotheses hold for the total spaces in question or whether another version of simplicial volume is intended.

The source refers to surface bundles \[ p_R:E_R\longrightarrow\Sigma_2 \] "constructed with pullback simplicial volumes at least $R$." Conventionally, $\Sigma_2$ denotes a closed, connected, oriented surface of genus $2$, but this meaning is not explicitly defined in the supplied record. Neither the construction of $E_R$ nor the pullback maps and spaces whose simplicial volumes satisfy the stated lower bound are supplied. Consequently, that lower bound cannot be formulated precisely from this record alone.

Intended question, conditional on specifying the missing construction and conventions. For the particular family $(E_R)$ referred to in the source, is \[ \lim_{R\to\infty}\|E_R\|=+\infty? \] More explicitly, if $I$ is the family's parameter set, ordered by its numerical parameter $R$ and unbounded above, the requested assertion is \[ \forall A>0\ \exists R_0\ \forall R\in I, \qquad R\ge R_0\ \Longrightarrow\ \|E_R\|>A. \] Here $I$ records an unspecified parameter domain rather than an additional supplied hypothesis about a known construction. The source does not determine whether $R$ ranges over integers, real numbers, or another unbounded numerical set, or how choices of bundles affect the intended quantification. The question concerns divergence of this family, which is stronger than merely having examples with arbitrarily large simplicial volume.

Short English statement

The question asks whether the simplicial volumes of the specified surface-bundle total spaces eventually exceed every fixed bound as $R$ grows. A precise unconditional statement requires the missing construction, the meaning of the pullback-volume bound, and the family and manifold conventions. The supplied record labels the problem open, but that label is dated curation rather than an independently verified mathematical assessment.

Sources

• Local UnsolvedMath registry, permanent ID $30003291$, source problem code catalogue entry, title "Unbounded Simplicial Volume in Surface-Bundle Families." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2016$; no individual proposer is identified.
• Supplied source citation: Surface Bundles (2017), Oberwolfach Reports, pp. 3149--3195, DOI 10.4171/owr/2016/56. https://doi.org/10.4171/owr/2016/56. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The literature-source list labels this reference "Oberwolfach Report 56 (2017)."
• Supplied literature reference: Bucher and Campagnolo, https://arxiv.org/abs/1605.03226. No title, theorem number, or precise connection to the family $E_R$ is supplied.
• Literature assessment checked on 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem open under a triage rule because the extracted record does not define $E_R$ sufficiently to identify a theorem proving the limit. It further reports that related surface-bundle constructions give unbounded examples without establishing this exact assertion. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the mathematical target. That supplied verification claim does not resolve the missing construction identified in the same record's literature assessment.