Aspherical 4-manifolds of odd Euler characteristic

TL;DR

This paper constructs explicit examples of aspherical 4-manifolds with any odd Euler characteristic greater than 12, expanding the known landscape of such manifolds.

Contribution

It provides the first explicit constructions of aspherical 4-manifolds with odd Euler characteristics greater than 12, which were previously unknown.

Findings

Constructed aspherical 4-manifolds with odd Euler characteristic > 12

All constructed manifolds are Haken and reducible to balls

Examples with Euler characteristics 1, 5, 7, 11 remain unknown

Abstract

An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The manifolds constructed here are all Haken manifolds in the sense of B. Foozwell and H. Rubinstein and can be systematically reduced to balls by suitably cutting them open along essential codimension-one submanifolds. It is easy to construct examples with even Euler characteristic from products of surfaces. And Euler characteristics divisible by 3 are know to arise from complex algebraic geometry considerations. Examples with Euler characteristic 1, 5, 7, or 11 appear to be unknown.