Constructions of round fold maps on circle bundles

{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to construct explicit fold maps, which are often difficult. In this paper, we construct new examples of {\it round fold maps}, which are defined as {\it stable fold maps} with singular value sets of concentric spheres introduced by the author on manifolds having the structures of circle bundles. The class of round fold maps includes some {\it special generic} maps on homotopy spheres and such maps have been constructed on manifolds having the structures of smooth bundles over standard spheres and manifolds represented as connected sums of manifolds admitting bundle structures over a standard sphere with fibers diffeomorphic to a standard sphere, for example, in previous studies by the author in the 2010s. Furthermore, such maps on manifolds admitting the structures of smooth bundles over spheres or more general manifolds including families of circle bundles over given manifolds were constructed by applying operations derived from the theory of bundles ({\it P-operations}), and in this paper, we use the operations to obtain new round fold maps.