  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.
