  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.
