Constructions of Round Fold Maps on Smooth Bundles

Abstract

In this paper, we construct {\it round fold maps} or {\it 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 {\it 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 $k \geq 2$ and connected sums of smooth bundles over $S^k$ with $k \geq 2$ 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.