On the cusped fan in a planar portrait of a manifold

A planar portrait of a manifold is the pair of the image and the critical values of the manifold through a stable map into the plane. It can be considerd a geometric representation of the manifold drawn in the plane. The cusped fan is its basic local configuration. In this article, we focus on the fibreing structure over the cusped fan, and give its characterisation. As application, planar portraits of the real, complex, and quaternion projective plane, regular toric surfaces, and some sphere bundles over spheres etc. are constructed. Conversely, the source manifolds of certain planar portraits are detected.