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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.

Stable) fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of algebraic and differentail topological properties of smooth manifolds. Round fold maps were introduced as stable fold maps with singular value sets, or 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. To construct explicit fold maps has been a fundamental and difficult problem. In 2014, the author succeeded in constructing explicit round fold maps into a Euclidean space of dimension larger than 1 on bundles over the standard sphere whose dimension is equal to that of the Euclidean space and connected sums of bundles over the standard sphere with fibers diffeomorphic to standard spheres by constructing maps locally and gluing them together properly. Later, the author constructed such maps on bundles over spheres or more general manifolds including families of bundles whose fibers are circles over given manifolds by applying operations compatiable with the structures of bundles (P-operations). In this paper, we obtain new round fold maps on closed smooth manifolds having the structures of bundles whose fibers are circles over given closed manifolds by applying P-operations under weaker condtions. We also study algebraic and differential topological properties of the resulting maps and the source manifolds.2010 Mathematics Subject Classification. Primary 57R45. Secondary 57N15.

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We show that a closed orientable 3-dimensional manifold admits a round fold map into the plane, i.e. a fold map whose critical value set consists of disjoint simple closed curves isotopic to concentric circles, if and only if it is a graph manifold, generalizing the characterization for simple stable maps into the plane. Furthermore, we also give a characterization of closed orientable graph manifolds that admit directed round fold maps into the plane, i.e. round fold maps such that the number of regular fiber components of a regular value increases toward the central region in the plane.