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Generic smooth maps with sphere fibers. (English) Zbl 1091.57021
The authors study various topological properties of generic smooth maps between manifolds whose fibers are disjoint unions of homotopy spheres. This is a generalization of the class of special generic maps. They have fold points and cusp points as their singularities. In particular it is shown that if a closed 4-manifold admits such a generic map into a surface, then it bounds a 5-manifold with nice properties using the method of Stein factorization. As a corollary, the authors show that each regular fiber of such a generic map of the 4-sphere into the plane is a homotopy ribbon 2-link and that any spun 2-knot of a classical knot can be realized as a component of a regular fiber of such a map. They also get a characterization of the standard 4-sphere in terms of sphere maps and derive that such 4-manifolds which admit the spherical maps should be null-cobordant.

57R45 Singularities of differentiable mappings in differential topology
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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