Aspherical 4-manifolds with positive Euler characteristic and their geography

TL;DR

This paper constructs explicit examples of aspherical 4-manifolds with positive Euler characteristic, disproving a conjecture and revealing new insights into their geometric properties and relationships with 3-manifolds.

Contribution

It provides the first explicit constructions of aspherical 4-manifolds with arbitrary positive Euler characteristic, confirming a conjecture and exploring their geometric and topological implications.

Findings

Constructed aspherical 4-manifolds with Euler characteristic equal to their signature for all positive integers.

Showed the failure of the Bogomolov-Miyaoka-Yau inequality analogue for aspherical 4-manifolds.

Demonstrated that certain aspherical 3-manifolds are virtually boundary components of aspherical 4-manifolds with vanishing Euler characteristic.

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.