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

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 $b_2^{(2)}$.

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 $b_2^{(2)}$.