Strong surjections from two-complexes with trivial top-cohomology onto the torus

We build a countable collection of two-dimensional CW complexes with trivial second integer cohomology group and, from each of them, a strong surjection onto the torus. Furthermore, we prove that such two-complexes are the simplest with these properties. This answers, for dimension two, a problem originally proposed in the 2000's for dimension three.