Aspherical 4-manifolds with positive Euler characteristic and their geography

2025/11/19 by Capovilla, Pietro
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · doi:10.48550/arxiv.2511.15577

openalex publication_date 2025/11/19 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28

Abstract

We present an explicit construction of closed oriented aspherical smooth 4-manifolds with χ= σ= 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 π₁-injective boundary of an aspherical 4-manifold with vanishing Euler characteristic and vanishing simplicial volume, thereby answering questions of Edmonds and of Löh-Moraschini-Raptis up to finite covers.

Citations

• The signature of geometrically decomposable aspherical 4-manifolds
• On the (super)additivity of simplicial volume
• Signature of surface bundles and bounded cohomology
• Aspherical 4-manifolds of odd Euler characteristic
• Bounded cohomology of discrete groups
• Isometric Embeddings in Bounded Cohomology
• 3-manifold groups
• Describing the platycosms

Related

• Aspherical 4-manifolds of odd Euler characteristic
• Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics
• Exotic aspherical 4-manifolds
• Some examples of aspherical 4-manifolds that are homology 4-spheres
• Survey on aspherical manifolds