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Stable fold maps are fundamental tools in a generalization of the theory ofMorse functions on smooth manifolds and its application to studies oftopological properties of smooth manifolds. Round fold maps were introduced asstable fold maps with singular value sets, defined as the set consisting of allthe singular values, of concentric spheres by the author in 2013; for example,some special generic maps on spheres are regarded as round fold maps whosesingular value sets are connected. Algebraic invariants such as homology andhomotopy groups of manifolds admitting round fold maps and more precisely, thehomeomorphism and diffeomorphism types of manifolds admitting such maps havingappropriate differential topological structures were studied. Moreover,explicit round fold maps into the Eucidean space of dimension larger than $1$are constructed on some fundamental manifolds such as manifolds having thestructures of bundles over the standard sphere of dimension equal to theEuclidean space whose fibers are closed smooth manifolds and manifolds ofdimension not smaller than twice the dimension of the Euclidean spacerepresented as the connected sum of manifolds having the structures of bundlesover the standard sphere of dimension equal to the Euclidean space whose fibersare diffeomorphic to standard spheres. In this paper, we construct new explicitfold maps on some fundamental manifolds including the manifolds before by usingextended methods of ones used in the constructions before.

Explicit round fold maps on some fundamental manifolds