315. The four-dimensional Singer conjecture

The four-dimensional Singer conjecture

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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.

Papers 1

The Singer conjecture in dimension four

OpenAI · 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.

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@misc{OAI:The-Singer-conjecture-in-dimension-four-September-25-2026,
  author = {{OpenAI}},
  title = {{The Singer conjecture in dimension four}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf}{OAI:The-Singer-conjecture-in-dimension-four-September-25-2026}},
  year = {2026}
}

The four-dimensional Singer conjecture

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The Singer conjecture in dimension four

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