The signature of geometrically decomposable aspherical 4-manifolds

TL;DR

This paper constructs and analyzes geometrically decomposable aspherical 4-manifolds with non-zero signature, establishing inequalities relating their Euler characteristic and signature, and exploring special cases with complex-hyperbolic vertices.

Contribution

It introduces new examples of aspherical 4-manifolds with non-zero signature and proves inequalities relating their topological invariants, extending known results to broader classes.

Findings

All such 4-manifolds satisfy a Bogomolov--Miyaoka--Yau type inequality χ ≥ 3|σ|.

Examples attaining equality are non-geometric with non-zero signature.

For higher graph 4-manifolds with complex-hyperbolic vertices, the inequality is always strict.

Abstract

We construct examples of geometrically decomposable aspherical 4-manifolds with non-zero signature. We show that all such 4-manifolds satisfy the inequality (of Bogomolov--Miyaoka--Yau type) $\chi\geq 3|\sigma|$. We also construct examples attaining the equality that are non-geometric and have non-zero signature. Finally, we prove that for higher graph 4-manifolds, with complex-hyperbolic vertices, the strict inequality always holds. Moreover, we construct infinitely many examples of higher graph 4-manifolds with non-zero signature and prove that the inequality is strict and sharp in this class.