Simplifying \(3\)-manifolds in \(\mathbb R^4\) (Q318701)

The work is in the smooth category and concerns smooth embedding of connected closed 3-manifolds \(M\) into \(\mathbb R^4\) and \(S^3 \times \mathbb R\). First the authors prove that any surface in Morse position in \(\mathbb R^3\) with a unique local maximum must be a Heegaard surface of \(S^3 \supset \mathbb R^3\). Then they discuss some examples in \(\mathbb R^3\) and \(\mathbb R^4\) that must have multiple local maxima in any Morse embedding.NEWLINENEWLINEPassing to 3-manifolds they prove: an embedding \(e:M \to \mathbb R^4\) whose fourth coordinate is a Morse function with one local maximum is isotopic to a Heegaard embedding \(f:M \to \mathbb R^4\). If the ambient space is \(S^3 \times \mathbb R\), then for all generic levels \(t\), \(f(M) \times (S^3 \times {t})\) is a Heegaard surface for \(S^3 \times {t}\), i.e. \(f(M) \times (S^3 \times {t})\) cuts \(S^3 \times {t}\) into two handlebodies. In addition the uniqueness of the embedding via the Goeritz group of the middle level Heegaard surface is studied.