Strong surjections from two-complexes with trivial top-cohomology onto nonorientable surfaces

For every nonorientable closed surface $\U$, we present a strong surjection $f\colon X\to\U$, where $X$ is a finite two-dimensional {\sc cw}-complex with trivial second integer cohomology group. This provides an answer, for all nonorientable closed surfaces, to a problem in topological root theory for which we have hitherto known solutions only for the sphere, the torus, the projective plane and the Klein bottle.