# Recent Results on the Size of Critical Sets

## Abstract

In the first part of this survey we review some special cases of \({\varphi }_{\mathcal{F}}\)-category of a pair (*M*, *N*) of manifolds such as φ-category, Morse-Smale characteristic, and Morse-Smale characteristic for circular functions. Section 2 presents examples of pairs with finite φ, and Sect. 3 provides lower estimates for the size of the critical sets in terms of topological dimension. We employ the cardinality when the manifolds admit maps with finitely many critical points and the topological dimension when no such maps exist.

## Keywords

Fundamental Group Boundary Component Hyperbolic Surface Critical Orbit Zero Codimension## Notes

### Acknowledgements

The first author would like to express all his thanks to Professor Themistocles M. Rassias for the strong encouraging to prepare this survey.

The first author is partially supported by the King Saud University D.S.F.P. Program

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