Examples of non-K\"{a}hler Calabi-Yau 3-folds with arbitrarily large $b_2$.

We construct non-K\"{a}hler simply connected Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties with trivial dualizing sheaves. We also give an example of a family of K3 surfaces with involutions which do not lift biregularly over the total space.