Constructions of round fold maps on smooth bundles

In this paper, we construct round fold maps or stable fold maps with singular value sets of concentric spheres introduced by the author [11] on smooth bundles over spheres and bundles over more general manifolds. The class of round fold maps includes some special generic maps on spheres (see [20] for example) and such maps have been constructed on smooth bundles over the standard sphere S^k with k ≥ 2 and connected sums of smooth bundles over S^k with k ≥ 2 with fibers diffeomorphic to standard spheres, for example, in previous studies by the author ([10], [12]). In this paper, we obtain round fold maps which do not appear in these studies with information on the diffeomorphism types of their source manifolds in new manners.