Topological constraints for Stein fillings of tight structures on lens spaces

In this article we give a sharp upper bound on the possible values of the Euler characteristic for a minimal symplectic filling of a tight contact structure on a lens space. This estimate is obtained by looking at the topology of the spaces involved, extending this way what we already knew from the universally tight case to the virtually overtwisted one. As a lower bound, we prove that virtually overtwisted structures on lens spaces never bound Stein rational homology balls. Then we turn our attention to covering maps: since an overtwisted disk lifts to an overtwisted disk, all the coverings of a universally tight structure are themselves tight. The situation is less clear when we consider virtually overtwisted structures. By starting with such a structure on a lens space, we know that this lifts to an overtwisted structure on $S^3$, but what happens to all the other intermediate coverings? We give necessary conditions for these lifts to still be tight, and deduce some information about the fundamental groups of the possible Stein fillings of certain virtually overtwisted structures.