Simplicial-Volume-Generated Seminorms in Third Homology

Definitions and mathematical statement

For a topological space $X$, let $H_3(X;{\mathbb{R}})$ denote ordinary singular homology with real coefficients. For a finite singular chain $c=\sum_i a_i\sigma_i$, written with distinct singular simplices, set $\lVert c\rVert_1=\sum_i|a_i|$. The $\ell^1$-seminorm of $\alpha\in H_3(X;{\mathbb{R}})$ is \[ \lVert\alpha\rVert_1=\inf\{\lVert c\rVert_1:c\text{ is a singular }3\text{-cycle representing }\alpha\}. \] For a closed, connected, oriented $3$-manifold $M$, write $[M]_{{\mathbb{R}}}$ for its real fundamental class and define its simplicial volume by $\lVert M\rVert=\lVert[M]_{{\mathbb{R}}}\rVert_1$. Here closed means compact and without boundary.

A functorial seminorm assigns a seminorm to each $H_3(X;{\mathbb{R}})$ such that every continuous map $f:X\to Y$ satisfies $p_Y(f_*\alpha)\leq p_X(\alpha)$. Under the usual interpretation of generation from manifold fundamental classes, the seminorm generated by simplicial volume is \[ |\alpha|_{\mathrm{SV}}=\inf\left\{\sum_{j=1}^{k}|a_j|\,\lVert M_j\rVert:\ \alpha=\sum_{j=1}^{k}a_j(f_j)_*[M_j]_{{\mathbb{R}}}\right\}, \] where $k$ is finite, $a_j\in{\mathbb{R}}$, each $M_j$ is a closed, connected, oriented $3$-manifold, and each $f_j:M_j\to X$ is continuous. Allow the empty sum for the zero class and use $\inf\varnothing=+\infty$.

Under this interpretation, the question is whether \[ \forall X\ \forall\alpha\in H_3(X;{\mathbb{R}}),\qquad |\alpha|_{\mathrm{SV}}=\lVert\alpha\rVert_1. \] The supplied record does not explicitly specify the category of spaces or the construction meant by "generated by simplicial volume." The displayed construction and quantification over topological spaces give the usual interpretation, which requires confirmation against the report. The supplied background states that equality is known outside dimension three and identifies dimension three as the remaining question; this is supplied source context, not an independently verified assertion here.

Short English statement

Does the seminorm on real third homology generated by the simplicial volumes of closed oriented $3$-manifolds equal the singular $\ell^1$-seminorm for every space and every homology class? The precise category and generation convention require source confirmation. The supplied record labels the problem open.

Sources

• Local UnsolvedMath registry, permanent ID $30001812$, source problem code catalogue entry, title: "Simplicial-Volume-Generated Seminorms in Third Homology." The record belongs to the Oberwolfach Reports Open Problems collection and gives the proposed year as 2011; no individual proposer is supplied.
• Arithmetic Groups vs. Mapping Class Groups: Similarities, Analogies and Differences (2011), Oberwolfach Reports, pp. 1637--1708, DOI 10.4171/owr/2011/30. Supplied source and literature reference, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/30.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports equality outside dimension three, with the dimension-three case explicitly posed as a question. This is supplied dated curation, not independently verified. The registry also labels the statement corrected_verified and reports checking its mathematical content against the official report; that verification claim is supplied metadata and has not been independently repeated here.