Let Z be a compact, connected 3-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from Z onto any 2-dimensional complex space. Combining this with a result of Campana-Demailly-Peternell, a corollary is that any holomorphic mapping from the 6-dimensional sphere S^6, endowed with any hypothetical complex structure, to a strictly lower-dimensional complex space must be constant. In other words, there does not exist any holomorphic fibration on S^6. This is joint work with Nobuhiro Honda.