Tight and overtwisted contact structures

Summary

Contact structures on $3$-manifolds split into two classes. An overtwisted contact structure contains an overtwisted disc, a disc whose boundary circle is tangent to the contact planes in a special twisted way. A tight one contains no such disc. Eliashberg showed that overtwisted structures are classified by homotopy theory alone, so they are flexible. Tight structures are rigid and carry the deep information of three-dimensional contact topology.