On special generic maps into \(\mathbb{R}^3\) (Q1366652)

Let \(f:M\to\mathbb{R}^p\) be a smooth map of a closed \(n\)-dimensional manifold \(M\) into \(\mathbb{R}^p\) \((n\geq p)\) which has only definite fold singularities as its singular points. Such a map is called a special generic map, which was first defined by Burlet and de Rham for \((n,p)=(3,2)\) and later extended to general \((n,p)\) by Porto, Furuya, Sakuma, and Saeki. In this paper, we study the global topology of such maps for \(p=3\) and give various new results, among which are a splitting theorem for manifolds admitting special generic maps into \(\mathbb{R}^3\) and a classification theorem of 4- and 5-dimensional manifolds with free fundamental groups admitting special generic maps into \(\mathbb{R}^3\). Furthermore, we study the topological structure of the surfaces which arise as the singular set of a special generic map into \(\mathbb{R}^3\) on a given manifold.