# Topological constraints for Stein fillings of tight structures on lens spaces

## TL;DR

This paper establishes topological bounds on Stein fillings of tight contact structures on lens spaces, distinguishing between universally tight and virtually overtwisted cases, and explores the effects of covering maps on these structures.

## Contribution

It provides sharp upper bounds on Euler characteristics for minimal symplectic fillings and analyzes the behavior of tightness under coverings for lens space contact structures.

## Key findings

- Virtually overtwisted structures on lens spaces do not bound Stein rational homology balls.
- Covering maps preserve tightness for universally tight structures, but the behavior for virtually overtwisted structures is more complex.
- Necessary conditions are identified for lifts of virtually overtwisted structures to remain tight, informing the fundamental groups of Stein fillings.

## Abstract

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.