Narrow Results

The notion of a complex tangent arises for embeddings of real manifolds into complex spaces. It is of particular interest when studying embeddings of real n-dimensional manifolds into ℂⁿ. The generic topological structure of the set complex tangents to such embeddings Mⁿ ↪ ℂⁿ takes the form of a (stratified) (n-2)-dimensional submanifold of Mⁿ. In this paper, we generalize our results from our previous work for the 3-dimensional sphere and the Heisenberg group to obtain results regarding the possible topological configurations of the sets of complex tangents to embeddings of odd-dimensional spheres S^2n-1 ↪ ℂ^2n-1 by first considering the situation for the higher-dimensional analogues of the Heisenberg group.

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.