Stability of fold maps of manifolds with boundary
Let \(X^m\) be a compact orientable smooth manifold with nonempty boundary and consider \(f:X^m \to Y^n\), \(m \geq n\), a stable smooth map. The singular set is allowed to intersect the boundary. The author characterizes stability of immersions on manifolds with boundary, gives a local form about a fold point (in particular on the boundary) and provides necessary and sufficient conditions for \(f\) with only fold singularities to be stable. Other examples of results about stable maps on manifolds with boundary appeared in \textit{E. von Essen} [Math. Scand. 43, 197--203 (1978; Zbl 0408.58025)], \textit{N. Shibata} [Hiroshima Math. J. 30, No. 3, 415--435 (2000; Zbl 0971.57035) and Kumamoto J. Math. 16, 43--49 (2003; Zbl 1051.58017)]. Examples of results (when \(n=2\) and \(Y^2\) is the plane) in which a requirement is present that the singular set does not intersect the boundary appeared in \textit{H. Levine} [Proc. Am. Math. Soc. 123, No. 8, 2563--2567 (1995; Zbl 0824.57023)].
scientific article; zbMATH DE number 2048547