A topological obstruction to the removal of a degenerate complex tangent and some related homotopy and homology groups

Abstract

In this paper, we derive a topological obstruction to the removal of an isolated degenerate complex tangent to an embedding of a 3-manifold into ℂ3 (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient condition for the (local) removal of the isolated complex tangent. The obstruction is a certain homotopy class of the space 𝕐 consisting of totally real 3-planes in the Grassmannian of real 3-planes in ℂ3(= ℝ6). We further compute additional homotopy and homology groups for the space 𝕐 and of its complement 𝕎 consisting of "partially complex" 3-planes in ℂ3.