The Euler Characteristic of a Haken 4-Manifold

TL;DR

This paper introduces a numerical function related to the boundary pattern of Haken 4-manifolds, establishing an inequality that bounds their Euler characteristic from below, especially showing non-negativity for closed cases.

Contribution

The authors define a boundary-dependent numerical function for Haken 4-manifolds and prove it bounds the Euler characteristic, extending understanding of their topological properties.

Findings

Established a boundary-dependent function phi(X^4)

Proved chi(X^4) >= phi(X^4) for Haken 4-manifolds

Showed chi(X^4) >= 0 for closed Haken 4-manifolds

Abstract

Haken n-manifolds are aspherical manifolds, defined and studied by B. Foozwell and H. Rubinstein, that can be successively cut open along essential codimension-one submanifolds until a disjoint union of n-cells is obtained. Such manifolds come equipped with a boundary pattern, a particular kind of decomposition of the boundary into codimension-zero submanifolds. We prove that there is a certain numerical function phi(X^4) depending only on the boundary and boundary pattern of the compact Haken 4-manifold X^4 (and vanishing if X^4 has empty boundary), such that for any compact Haken 4-manifold X^4 the Euler characteristic satisfies the inequality chi(X^4) >= phi(X^4). In particular, if X^{4} is a closed Haken 4-manifold, then chi(X^4) >= 0.