A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell-complex which decomposes a closed 2-manifold without boudary, such that for every two vertices there is a 2-cell containing them. We prove that there are no w.n.p. maps on the orientable 2-manifold of genus 2. Genus 2 seems to be unique with respect to this property.