Duplicate Integral-Approximation Question

Definitions and mathematical statement

Let $M$ be an oriented, closed, connected manifold, and put $n=\dim M$. Here closed means compact with empty boundary. The manifold is aspherical if its universal covering space is contractible. Its fundamental group $\Gamma=\pi_1(M)$ is residually finite if, for every $g\in\Gamma\setminus\{1\}$, there is a homomorphism from $\Gamma$ to a finite group whose value on $g$ is nonidentity.

For $A\in\{{\mathbb{Z}},{\mathbb{R}}\}$, let $[M]_A\in H_n(M;A)$ be the fundamental class determined by the orientation. For a finite singular chain written using distinct singular simplices as $c=\sum_j a_j\sigma_j$, set $\lVert c\rVert_1=\sum_j|a_j|$. Define \[ \lVert M\rVert_A =\inf\left\{\lVert c\rVert_1 : c\in C_n(M;A),\ \partial c=0,\ [c]=[M]_A\right\}. \] The real simplicial volume is $\lVert M\rVert=\lVert M\rVert_{{\mathbb{R}}}$, and the integral simplicial volume is $\lVert M\rVert_{{\mathbb{Z}}}$. Define the stable integral simplicial volume by \[ \lVert M\rVert_{{\mathbb{Z}}}^{\infty} =\inf_{p:N\to M}\frac{\lVert N\rVert_{{\mathbb{Z}}}}{\deg p}, \] where the infimum ranges over all connected finite-sheeted covering maps, $N$ carries the orientation induced from $M$, and $\deg p$ is the positive number of sheets.

Question retained from the source record. Is it true that every oriented, closed, connected, aspherical manifold $M$ with residually finite fundamental group and $\lVert M\rVert=0$ satisfies \[ \lVert M\rVert_{{\mathbb{Z}}}^{\infty}=\lVert M\rVert=0? \] Equivalently, under these hypotheses, does every $\varepsilon>0$ admit a connected finite-sheeted covering $p:N\to M$ such that \[ \frac{\lVert N\rVert_{{\mathbb{Z}}}}{\deg p}<\varepsilon? \] No dimension restriction is supplied.

Source qualification. This is a restatement of the retained original question, not a separately verified clean statement. The registry marks this record as unrecoverable and says that it duplicates the second question represented canonically by OWR-17470-011; no separate clean statement is promoted. Its supplied open label is a curation status, not an independently verified mathematical conclusion.

Short English statement

Must every oriented, closed, connected, aspherical manifold with residually finite fundamental group and zero real simplicial volume have finite covers whose integral simplicial volumes, divided by their numbers of sheets, become arbitrarily small? The supplied record retains this question as a duplicate of the second question in OWR-17470-011, without promoting a separate verified statement.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004396$, exact source problem code catalogue entry, title "Duplicate Integral-Approximation Question," in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2020$ and no individual proposer. Its verification note identifies the second question of OWR-17470-011 as the canonical representation and retains this duplicate for provenance.
• Supplied report citation: Manifolds and Groups (2021), Oberwolfach Reports, pp. 417--463, DOI 10.4171/owr/2020/7. https://doi.org/10.4171/owr/2020/7. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied title year, DOI, page range, and proposed year are retained without independent reconciliation.
• Supplied literature source labeled "Oberwolfach report": https://doi.org/10.4171/owr/2020/7.
• Supplied literature source labeled "Fauser 2021": https://doi.org/10.1112/topo.12193. No fuller bibliographic description is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that integral approximation is proved for several families, especially aspherical $3$-manifolds, while no general theorem for all closed aspherical manifolds with zero simplicial volume was found. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims; hypotheses of the reported family-specific results are not supplied.