Explicit round fold maps on some fundamental manifolds

Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of topological properties of smooth manifolds. Round fold maps were introduced as stable fold maps with singular value sets, defined as the set consisting of all the singular values, of concentric spheres by the author in 2013; for example, some special generic maps on spheres are regarded as round fold maps whose singular value sets are connected. Algebraic invariants such as homology and homotopy groups of manifolds admitting round fold maps and more precisely, the homeomorphism and diffeomorphism types of manifolds admitting such maps having appropriate differential topological structures were studied. Moreover, explicit round fold maps into the Eucidean space of dimension larger than $1$ are constructed on some fundamental manifolds such as manifolds having the structures of bundles over the standard sphere of dimension equal to the Euclidean space whose fibers are closed smooth manifolds and manifolds of dimension not smaller than twice the dimension of the Euclidean space represented as the connected sum of manifolds having the structures of bundles over the standard sphere of dimension equal to the Euclidean space whose fibers are diffeomorphic to standard spheres. In this paper, we construct new explicit fold maps on some fundamental manifolds including the manifolds before by using extended methods of ones used in the constructions before.