Round fold maps and the topologies and the differentiable structures of manifolds admitting explicit ones

Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of geometric properties of smooth manifolds. Round fold maps were introduced as stable fold maps such that the sets of all of the singular values of them are concentric spheres by the author in 2013-4. Topological properties of such maps and topological information of their source manifolds such as homology and homotopy groups have been studied under appropriate conditions by the author. In this paper, we redefine round fold maps respecting the definition. As more precise information of manifolds admitting round fold maps, we study the topologies and differentiable structures of manifolds admitting such maps under appropriate differential topological conditions.