髙瀨 将道

This note corrects an error in Theorem 5.2(c) of our paper "Generic immersions and totally real embeddings".

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.

It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the 3-dimensional complex space. We show in fact that a 1-dimensional submanifold of a closed orientable 3-manifold can be realised as the set of complex tangents of a smooth embedding of the 3 manifold into the 3-dimensional complex space if and only if it represents the trivial integral homology class in the 3-manifold. The proof involves a new application of singularity theory of differentiable maps.

For each diagram D of a 2-knot, we provide a way to construct a new diagram D' of the same knot such that any sequence of Roseman moves between D and D' necessarily involves branch points. The proof is done by developing the observation that no sphere eversion can be lifted to an isotopy in 4-space.