Journal of Symplectic Geometry
Volume 12 (2014)
Bypass attachments and homotopy classes of 2-plane fields in contact topology
Pages: 599 – 617
We use the generalized Pontryagin-Thom construction to analyze the effect of attaching a bypass on the homotopy class of the contact structure. In particular, given a three-dimensional contact manifold with convex boundary, we show that the bypass triangle attachment changes the homotopy class of the contact structure relative to the boundary, and the difference is measured by the homotopy group $\pi_3(S^2)$.