We ask about the simply connected compact smooth 6-manifolds which can support structures of Calabi-Yau threefolds. In particular, we study the interesting case of Calabi-Yau threefolds $X$ with second betti number 3. We have a cup-product cubic form on the second integral cohomology, a linear form given by the second Chern class, and the integral middle cohomology, and if the homology is torsion free this information determines precisely the diffeomorphism class of the underlying 6-manifold by a result of Wall. For simplicity, we assume that the cubic form defines a smooth real elliptic curve whose Hessian is also smooth. Under a further relatively mild assumption that there are no non-movable surfaces $E$ on $X$ with $1 \le E^3 \le 9$, we prove that the real elliptic curve must have two connected components rather than one, and that the K\"ahler cone is contained in the open positive cone on the bounded component; we show moreover that the second Chern class is also positive on this open cone. Using Wall's result, for any given third Betti number we therefore have an abundance of examples of smooth 6-manifolds which support no Calabi-Yau structures, both in the cases when the cubic defines a real elliptic curve with one or two connected components.