Morse inequalities

Since the usual Morse inequality gives the opposite inequality, we have... 

For a later purpose (Chapter 7), we shall explain the perfectness of the Morse function given by the moment map of a torus action on a general symplectic manifold. However, when the fixed points of a torus action are all isolated, such as the case of (C2), the perfectness follows easily from the Morse inequality since they all have even indices. The reader who has interests only in (C2) could skip 5.1. 

The number of points of a potential energy hypersurface E(K) where the energy gradient vanishes (the critical points of E) and the relative location of these points are of special importance in the study of reaction paths and reaction mechanisms. The Morse inequalities of algebraic topology provide lower bounds for the number of critical points of various types, e.g., minima, saddle points of one or more negative canonical curvatures [30-32]. 

Alternative relations to the Morse inequalities have been proposed in ref. [ 14] for both lower and upper bounds on the number of critical points along potential energy hypersurfaces. These bounds for the number of critical points of various types are important for the analysis of catchment regions [13], in the spirit of the mathematical model of "watersheds", originally derived for geographical terrains [28,29], and the differential geometric model [1-4]. 

Both, the continuous and discrete, Morse-Smale systems on compact smooth manifolds were singled out by Smale in his article Morse inequalities for a dynamical system [142]. The title itself reveals that the work deals... 

Smale, S. [1960] Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66, 43-49. 

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