The Singer conjecture in dimension four: L2-Betti numbers of closed aspherical four-manifolds vanish outside degree two

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The Singer conjecture in dimension four: L2-Betti numbers of closed aspherical four-manifolds vanish outside degree two parent of this event

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Theorem 1.1: if $Q$ is a finite connected aspherical integral Poincaré complex of formal dimension four, with any orientation character, then $b_p^{(2)}(\widetilde Q)=0$ for $p\ne2$. Corollary 1.2: the same holds for every closed connected aspherical topological four-manifold, so the Singer conjecture holds in dimension four. Stated consequences: $\chi(Q)=b_2^{(2)}(\widetilde Q)\ge0$, so closed aspherical four-manifolds have nonnegative Euler characteristic; $\chi(M)\ge|\sigma(M)|$ for oriented $M$; limits of normalized Betti numbers along residual towers of covers. Not shown: the Singer conjecture in any other dimension, or any computation of $b_2^{(2)}$. parent of this event

Davis and Okun, Vanishing theorems and conjectures for the l2-homology of right-angled Coxeter groups (2001) evidence for this event

VibeMathed record: The Singer conjecture in dimension four: L2-Betti numbers of closed aspherical four-manifolds vanish outside degree two evidence for this event

Dodziuk, L2 harmonic forms on rotationally symmetric Riemannian manifolds (1979) evidence for this event

The Singer conjecture in dimension four: L2-Betti numbers of closed aspherical four-manifolds vanish outside degree two evidence for this event