Calabi-Yau threefolds of quotient type

First, we classify Calabi-Yau threefolds with infinite fundamental group by means of their minimal splitting coverings introduced by Beauville, and deduce that the nef cone is a rational simplicial cone and any rational nef divisor is semi-ample if the second Chern class is identically zero. We also derive a sufficient condition for the fundamental group to be finite in terms of the Picard number in an optimal form. Next, we give a concrete structure Theorem concerning $c_{2}$-contractions of Calabi-Yau threefolds as a generalisation and also a correction of our earlier works for simply connected ones. Finally, as an application, we show the finiteness of the isomorphism classes of $c_{2}$-contractions of each Calabi-Yau threefold.