Immersions of 3-manifolds into the 5-space, almost contact structures, and complex tangents

The Wu invariant is a regular homotopy invariant for immersions of oriented 3-manifolds into the 5-space. In this paper, we present new expressions and vanishing theorems for the invariant, from the viewpoint of almost contact structures and complex tangents. As an application, we determine the regular homotopy classes of inclusion maps of the surface singularity links of type ADE into the 5-sphere.