Bypass attachments and homotopy classes of 2-plane fields in contact topology

Yang Huang

Published 2011-05-12, updated 2011-05-13Version 2

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 3-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 Hopf invariant.