A Topological Obstruction to the Removal of a Degenerate Complex Tangent and Some Related Homotopy and Homology Groups

TL;DR

This paper identifies a topological obstruction that determines whether an isolated degenerate complex tangent in a 3-manifold embedding into a3^3 can be removed, linking it to homotopy classes of certain plane spaces.

Contribution

It introduces a topological obstruction criterion for removing degenerate complex tangents, connecting it to homotopy classes of totally real 3-planes in a3^3.

Findings

Obstruction is a homotopy class in the space a3 of totally real 3-planes.

Vanishing of the obstruction is necessary and sufficient for local removal.

Computed homotopy and homology groups of a3 and its complement a3.

Abstract

In this article, we derive a topological obstruction to the removal of a isolated degenerate complex tangent to an embedding of a 3-manifold into $\mathbb{C}^3$ (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 $\mathbb{Y}$ consisting of totally real 3-planes in the Grassmanian of real 3-planes in $\mathbb{C}^3$ (=$\mathbb{R}^6$). We further compute additional homotopy and homology groups for the space $\mathbb{Y}$ and of its complement $\mathbb{W}$ consisting of "partially complex" 3-planes in $\mathbb{C}^3$.