  This is the first of a series of papers devoted to the topology of symplectic
Calabi-Yau $4$-manifolds endowed with certain symplectic finite group actions.
We completely determine the fixed-point set structure of a finite cyclic action
on a symplectic Calabi-Yau $4$-manifold with $b_1>0$. As an outcome of this
fixed-point set analysis, the $4$-manifold is shown to be a $T^2$-bundle over
$T^2$ in some circumstances, e.g., in the case where the group action is an
involution which fixes a $2$-dimensional surface in the $4$-manifold. Our
project on symplectic Calabi-Yau $4$-manifolds is based on an analysis of the
existence and classification of disjoint embeddings of certain configurations
of symplectic surfaces in a rational $4$-manifold. This paper lays the ground
work for such an analysis at the homological level. Some other result which is
of independent interest, concerning the maximal number of disjointly embedded
symplectic $(-2)$-spheres in a rational $4$-manifold, is also obtained.
