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On the Topological Structure of Complex Tangencies to Embeddings of S3 Into C3

出版	University of Chicago, Division of the Physical Sciences, Department of Mathematics, 2011
ISBN	11247175959781124717593
URL	http://books.google.com.hk/books?id=aRQUnQAACAAJ&hl=&source=gbs_api

註釋In the early 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 explicitly demonstrated 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. Using some analysis on the Heisenberg group H and its biholomorphism with the 3-sphere minus a point, we demonstrate that every homeomorphism-type of knot in S3 may arise precisely as the set of complex tangents to an embedding S3 & rarrhk;C3 . We also note that every algebraic surface in S3 (and in H) may arise as the set of complex tangents to some embedding. Further, we investigate Bishop invariants along curves of complex tangencies, and make a note of the numerous configurations that are possible. We also investigate the behavior of the tangential Cauchy-Riemann operator to the Heisenberg group when it acts on polynomials. We then extend some of our results to the higher dimensional analogues of the Heisenberg group, and use the natural biholomorphism to extend to all odd-dimensional spheres. Finally, we compute some homology and homotopy groups of interesting manifolds.