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Conley pairs in geometry – Lusternik-Schnirelmann theory and more. (English) Zbl 1475.55003

The paper under review is an expository article investigating Conley pairs and their applications. While Conley theory can be used to study the gradient flow in general, the paper looks at the special case where the critical points are only either non-degenerate or isolated. This leads to the study of Morse theory and Lusternik-Schnirelmann category, respectively. A relationship between these two topics is proved in the celebrated Lusternik-Schnirelmann theorem, which says that if \(f\) is a \(C^2\) function on a closed manifold, then the number of critical points of \(f\) bounds from above the (non-reduced) Lusternik-Schnirelmann category. Other tools for detecting critical points, such as cup-length, are studied in this paper.

MSC:

55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37B30 Index theory for dynamical systems, Morse-Conley indices
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
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References:

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