Aspherical 4-manifolds with positive Euler characteristic and their geography

Abstract

We present an explicit construction of closed oriented aspherical smooth 4-manifolds with $\chi = \sigma = n$ for every positive integer $n$. This proves a conjecture of Edmonds by providing a closed oriented aspherical 4-manifold with Euler characteristic 1, and it shows that the real analogue of the Bogomolov-Miyaoka-Yau inequality fails for aspherical 4-manifolds. By the Hitchin-Thorpe inequality, these manifolds do not admit Einstein metrics. As a further consequence of our construction, we show that every closed aspherical 3-manifold with amenable fundamental group is virtually the $\pi_1$-injective boundary of an aspherical 4-manifold with vanishing Euler characteristic and vanishing simplicial volume, thereby answering questions of Edmonds and of L\"oh-Moraschini-Raptis up to finite covers.