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.

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What the papers state

Theorem 1.1

In The Singer conjecture in dimension four, 25 September 2026

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Let Q be a finite connected aspherical integral Poincaré complex of formal dimension four, with arbitrary orientation character. Then

b_p^{(2)}(\widetilde Q)=0\qquad(p\ge0,\ p\ne2).

Corollary 1.2

In The Singer conjecture in dimension four, 25 September 2026

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Let M be a closed connected aspherical topological four-manifold. Then

b_p^{(2)}(\widetilde M)=0\qquad(p\ge0,\ p\ne2).

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Papers

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The Singer conjecture in dimension four

25 September 2026, current edition.

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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.

Lean statements

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Independent comparisons and proof-verification reports

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What these papers cite in the release, and what cites them

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Papers elsewhere in the release that this family’s papers cite

A PD4 group without an aspherical manifold modelCurrent edition; no accompanying Lean statement Family Four-dimensional disk embedding and Wall’s conjectureCited by The Singer conjecture in dimension four.
Global Spherical Shells on Minimal Surfaces of Class VIICurrent edition; no accompanying Lean statement Family The Global Spherical Shell conjectureCited by The Singer conjecture in dimension four.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

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29 references in The Singer conjecture in dimension four

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What people have said

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