Generic immersions and totally real embeddings

TL;DR

This paper establishes conditions under which CR-regular embeddings imply totally real embeddings for certain manifolds, revealing topological obstructions and special cases like spheres and 5-manifolds.

Contribution

It proves that for most closed orientable manifolds, CR-regular embeddings into complex space guarantee totally real embeddings, highlighting topological constraints.

Findings

CR-regular embedding implies totally real embedding for certain manifolds

Obstructions exist for (4k+1)-manifolds with non-vanishing Kervaire semi-characteristic

Special cases analyzed include spheres and simply-connected 5-manifolds

Abstract

We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed orientable (4k+1)-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex 4k-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.