On the topological structure of complex tangencies to embeddings of S3 into C3

摘要

In the mid-1980's, M. Gromov used his machinery of the h-principle to prove that there exists totally real embeddings of S3 into C3. Subsequently, Patrick Ahern and Walter Rudin explicitlydemonstrated such a totally real embedding. In this paper, we consider the generic situation for such embeddings, namely where complex tangents arise as codimension-2 subspaces. We first consider the Heisenberg group H and generate some interesting results therein. Then, by using the biholomorphism of H with the 3-sphere minus a point, wedemonstrate that every homeomorphism-type of knot in S3 may arise precisely as the set of complex tangents to an embedding S3 ⟶ C3. We also make note of the (nongeneric) situation where complex tangents arise along surfaces.