Nontorus Abelian Duality Manifolds

Definitions and mathematical statement

For a connected space $X$ of finite CW homotopy type, let $G=\pi_1(X)$, let $G_{\mathrm{ab}}=G/[G,G]\cong H_1(X;{\mathbb{Z}})$, and let ${\mathbb{Z}}[G_{\mathrm{ab}}]$ denote its integral group ring. Regard this ring as a local coefficient system on $X$ through the quotient map $G\to G_{\mathrm{ab}}$ and the regular action. Under the standard integral convention, $X$ is an abelian duality space of dimension $n$ if $n\geq 0$, \[ H^q\left(X;{\mathbb{Z}}[G_{\mathrm{ab}}]\right)=0\quad\text{for every }q\geq 0\text{ with }q\neq n, \] and $H^n(X;{\mathbb{Z}}[G_{\mathrm{ab}}])$ is a nonzero torsion-free abelian group. Torsion-freeness here refers to the underlying ${\mathbb{Z}}$-module.

A closed manifold is compact and has empty boundary. A real $m$-torus is $T^m=(S^1)^m$, with $T^0$ a point. Using connected manifolds and interpreting "is not a real torus" as "is not homeomorphic to a real torus," the question is: \[ \begin{gathered} \text{Do there exist integers }m,n\geq 0\text{ and a connected closed orientable }m\text{-manifold }M\\ \text{such that }M\text{ is an abelian duality space of dimension }n \quad\text{and}\quad M\not\cong T^m\text{ by a homeomorphism}? \end{gathered} \] The source does not explicitly specify connectedness, the manifold category, the intended equivalence with a torus, or the coefficient convention for abelian duality. Thus the displayed formulation records explicit working conventions; a requirement involving diffeomorphism or homotopy equivalence would be a different formulation and requires source confirmation.

Short English statement

Does a connected compact orientable manifold without boundary satisfy integral abelian duality while failing to be a real torus? The displayed formulation uses homeomorphism to interpret the torus exclusion; the source leaves this equivalence and the manifold category unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30003629$, title "Nontorus Abelian Duality Manifolds," source problem code catalogue entry; supplied complete source record. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied source citation: Mini-Workshop: Interactions between Low-dimensional Topology and Complex Algebraic Geometry (2018), Oberwolfach Reports, pp. 2987--3033, DOI 10.4171/owr/2017/50. https://doi.org/10.4171/owr/2017/50. The literature-source label is "Oberwolfach Report 50/2017." The citation's 2018 date and the report's 2017 designation are retained as supplied.
• Supplied literature source: Liu--Maxim--Wang. https://arxiv.org/abs/1709.02870.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the question open and reports that compact Kähler abelian-duality manifolds must be complex tori, that the Heisenberg nilmanifold is explicitly ruled out, and that no closed orientable non-torus example is known. The supplied background describes this as literature triage derived from the Oberwolfach curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that notation, typography, and sentence structure were repaired while mathematical content was preserved. These are supplied editorial labels, not independent mathematical verification of this draft.