On the homotopy class of 2-plane fields and its applications in contact topology

The goal of this thesis is to study homotopy classes of oriented 2-plane fields in 3-manifolds and apply them to 3-dimensional contact topology. Using a generalized version of the Pontryagin-Thom construction, we compute the homotopy class of a bypass attachment. Based on this computation, we give a new proof to the classification theorem of overtwisted contact structures, originally due to Eliashberg, using convex surface theory.