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

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 \(\aleph_1\).

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
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