Tight contact structures on some plumbed 3-manifolds

TL;DR

This paper classifies tight contact structures on specific plumbed 3-manifolds bounding non-simply connected 4-manifolds, extending previous theorems and introducing new methods for distinguishing Stein cobordisms.

Contribution

It generalizes Lisca-Matic's theorem to Stein cobordisms and develops a novel approach using rotation numbers for their classification.

Findings

Classified tight contact structures on certain plumbed 3-manifolds.

Provided descriptions of Stein fillings for Stein fillable structures.

Extended theoretical framework for Stein cobordisms and contact topology.

Abstract

In this article, we prove a generalization of a theorem of Lisca-Matic to Stein cobordisms and develop a method for distinguishing certain Stein cobordisms using rotation numbers. Using these results along with standard techniques from convex surface theory and classifications of tight contact structures on certain 3-manifolds due to Honda, we classify the tight contact structures on a certain class of plumbed 3-manifolds that bound non-simply connected 4-manifolds. Moreover, we give descriptions of the Stein fillings of the Stein fillable contact structures.