Browse › Topology › 315 The four-dimensional Singer conjecture
The Singer conjecture in dimension four
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- B. Bekka and A. Valette, Group cohomology, harmonic functions and the first L²-Betti number (Potential Analysis, 1997). For non-amenable finitely generated groups, it identifies first cohomology with coefficients in the regular representation with harmonic functions of finite Dirichlet sum on a Cayley graph, modulo constants; this relates to the paper's first step, which turns positive first L²-Betti number into a finite-energy function on the Cayley graph.
- Wiktor J. Mogilski, The weighted Singer conjecture for Coxeter groups in dimensions three and four (arXiv:1503.02518, 2015-03-09). Establishes weighted Singer vanishing in dimension four under additional restrictions on Coxeter nerves, for comparison with the manuscript's four-dimensional vanishing theorem, which has no such restrictions.
Digests and expositions
- Latent Seminar, The four-dimensional Singer conjecture: finite energy and boundary paths (Latent Seminar, 2026-10-07). Explains the duality reduction, finite-energy boundary, cut-point reduction and uniform path threshold, explicitly limiting its review to statements and selected proof mechanisms rather than independent verification of the complete proof. [family-level]
- Open Mathematics Courses, The finite-energy boundary of a group (Open Mathematics Courses, undated). Presents ten lessons on the group-theoretic boundary argument used in this manuscript, with proofs and exercises written and self-checked by Claude Opus 5.5 and independent review described as incomplete. [AI-generated per source]
- Gradient Damage, Kervaire Invariant Problem at Prime 3, The Grothendieck Homotopy Hypothesis and 8 More (video compilation, 2026-10-15). The chapter beginning at 3:09 introduces Singer vanishing and links this manuscript, describing the animation as an illustration rather than a proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 315
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 315: 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.
This is the only paper in this family.