Lifting a generic map of a surface into the plane to an embedding into 4-space

Abstract

Let $f:M \to \mathbf{R}^2$ be a stable map of a closed surface $M$ into the plane and $\pi^2_2:\mathbf{R}^4 \to \mathbf{R}^2$ the orthogonal projection. In this paper, we will show that for any such $f$ there exists an embedding $F:M \to \mathbf{R}^4$ such that $f=\pi^2_2 \circ F$ is satisfied.