Constructions of Round Fold Maps on Smooth Bundles

Abstract

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.

Round fold maps on $3$--manifolds

“…On the other hand, the first author introduced the notion of a round fold map [3,4]: a smooth map f : M → R 2 is a round fold map if it is a fold map and f | S(f ) is an embedding onto the disjoint union of some concentric circles in R 2 (for details, see §2). As has been studied by the first author, round fold maps have various nice properties.…”

On simple classes of special generic maps and round fold maps and fold maps obtained by composing projections

“…In the third section, we prove Main Theorem 2. Some of [11]- [13] are closely related to this. In the fourth section, we prove Main Theorem 3.…”

“…Definition 4 (In [11] these notions are defined first essentially and we change the names of these notions). Let f : M → R n be a round fold map and φ be the diffeomorphism in Definition 3.…”

Round fold maps of n--dimensional manifolds into ${\mathbb{R}}^{n-1}$

“…On the other hand, the first author introduced the notion of a round fold map [10,11]: a smooth map f : M → R p is a round fold map if it is a fold map and f | S(f ) is an embedding onto the disjoint union of some concentric (p − 1)dimensional spheres in R p (for details, see §2). Round fold maps are naturally simple.…”