## 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

### Keywords:

critical points; homotopy groups
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