Kirby Problem 3.38

Definitions and mathematical statement

Use the finitely presented convention for Poincaré duality groups in the source. A group $G$ is a $\mathrm{PD}_3$ group over $\mathbb Z$ if it is finitely presented, the trivial module $\mathbb Z$ admits a finite-length resolution by finitely generated projective $\mathbb ZG$-modules, its cohomological dimension over $\mathbb Z$ is three, and \[ H^i(G;\mathbb ZG)=0\ (i\ne3),\qquad H^3(G;\mathbb ZG)\cong\mathbb Z \quad\text{as abelian groups}. \] The right $G$-action on the last group may be nontrivial; it records the orientation character $G\to\{\pm1\}$. These conditions express integral Poincaré duality with the corresponding orientation module, rather than requiring orientability.

Is every such group isomorphic to the fundamental group of a closed connected aspherical three-manifold? Aspherical means that the universal cover is contractible, or equivalently all higher homotopy groups vanish. Both orientable and nonorientable manifolds are allowed as dictated by the orientation character.

A realization by a three-dimensional complex satisfying duality alone is not the conclusion: the complex must be realizable by a genuine closed manifold. Finite presentation is stated explicitly so that this formulation is not confused with broader finiteness conventions for duality groups.

Short English statement

Must every finitely presented group satisfying integral three-dimensional Poincaré duality come from a closed aspherical three-manifold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2836 (KP-3.38), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.38. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.