  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.
