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

Tight contact structures on some plumbed 3-manifolds