The images of special generic maps of an elementary class

Abstract

The class of special generic maps contains Morse functions with exactly two singular points, characterizing spheres topologically which are not $4$-dimensional and the $4$-dimensional unit sphere. This class is for higher dimensional versions of such functions. Canonical projections of unit spheres are simplest examples and suitable manifolds diffeomorphic to ones represented as connected sums of products of spheres admit such maps. They have been found to restrict the topologies and the differentiable structures of the manifolds strongly. The present paper focuses on images of special generic maps on closed manifolds. They are smoothly immersed compact manifolds whose dimensions are same as those of the targets. Some studies imply that they have much information on homology groups and cohomology rings. We present new construction and explicit examples of special generic maps by investigating the images and the present paper is essentially on construction and explicit algebraic topological and differential topological studies of immersed compact manifolds.