Math Conjectures

Singer Conjecture

OPENLandmarkConjectureProposed c. 1970 · Standard version

Canonical statement

Let MnM^n be a closed aspherical manifold and let M~\widetilde M be its universal cover. Then bi(2)(M~)=0b_i^{(2)}(\widetilde M)=0 for every i≠n/2i\ne n/2. In particular, when nn is odd, all L2L^2-Betti numbers vanish.
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Let \(M^n\) be a closed aspherical manifold and let \(\widetilde M\) be its universal cover. Then \(b_i^{(2)}(\widetilde M)=0\) for every \(i\ne n/2\). In particular, when \(n\) is odd, all \(L^2\)-Betti numbers vanish.
GEO-L2-022:[Lueck2002L2][DavisOkun2001Vanishing]
Geometry and topologyAnalysis and functional analysis

Notes

For a closed aspherical manifold MnM^n, the Singer conjecture predicts that every reduced L2L^2-homology group of its universal cover vanishes away from the middle dimension. Equivalently, all L2L^2-Betti numbers bi(2)(M~)b_i^{(2)}(\widetilde M) should be zero unless 2i=n2i=n; in odd dimensions this predicts total L2L^2-acyclicity. Lück develops the analytic and group-theoretic framework behind this formulation [Lueck2002L2].

L2L^2-Poincaré duality explains why the middle dimension is exceptional, and the conjecture is known for substantial geometric and combinatorial classes. Davis and Okun proved key cases for manifolds built from right-angled Coxeter groups and connected the problem to the combinatorics of flag complexes [DavisOkun2001Vanishing].

The missing step is a uniform vanishing theorem for arbitrary closed aspherical manifolds, especially in even dimensions where a middle-dimensional L2L^2-Betti number may survive. Results for special curvature hypotheses, reflection-group constructions, or individual dimensions do not settle the full conjecture.

References (2)

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