Some pairs of manifolds with infinite uncountable \(\varphi\)-category. (English) Zbl 1042.57020

Pintea, Cornel

Topol. Methods Nonlinear Anal. 21, No. 1, 101-113 (2003).

The author studies the cardinality of the sets of critical points of smooth maps. A typical result is the following (Corollary 3.2): Let \(M,N\) be connected smooth manifolds such that \(\dim M \geq \dim N\). If \(f : M \to N\) is a non-surjective closed smooth mapping, then either all the points of \(M\) are critical or \(f\) has infinite uncountable number of critical values. In particular, if \(M\) is compact and \(N\) is non-compact then the critical point set has cardinality \(\aleph_1\).

Reviewer: Yuli Rudyak (Gainesville)

MSC:

57R70	Critical points and critical submanifolds in differential topology
55M30	Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55Q05	Homotopy groups, general; sets of homotopy classes

Keywords:

critical points; homotopy groups