The four-dimensional Singer conjecture

Proves that the L²-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.

Manuscript

The Singer conjecture in dimension four

September 25, 2026 · source · cite as \cite{OAI:The-Singer-conjecture-in-dimension-four-September-25-2026}

We prove the four-dimensional Singer conjecture: the L²-Betti numbers of the universal cover of a closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same vanishing holds for finite connected aspherical integral Poincaré complexes of formal dimension four, including nonorientable ones.

Result 315 · Topology. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.