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A smooth function from a manifold to the real numbers which does not have any degenerate critical points. Practical application: used to analyze the topology of the manifold by studying these functions.

The number of negative eigenvalues of the Hessian matrix of a Morse function at a critical point. Practical application: indicates the type of critical point (e.g., maximum, minimum, saddle).

A result stating that near any non-degenerate critical point, a Morse function can be locally expressed as a sum of squared variables. Practical application: simplifies the analysis of critical points and the topology near them.

A homology theory constructed from a Morse function, yielding the same information as singular homology for smooth manifolds. Practical application: provides an alternative approach to computing homology groups.

A point on a manifold where the differential of a Morse function vanishes. Practical application: critical points correspond to topological features of the manifold, such as holes or handles.

A decomposition of a manifold given by the stable and unstable manifolds of a Morse-Smale function. Practical application: used in computer graphics and data analysis for feature extraction and shape understanding.

Points where the topology of the level sets of the Morse function changes, corresponding to a pair of critical points that cancel each other out. Practical application: used for tracking changes in topological features.

The process of breaking a manifold into simple pieces called handles, using a Morse function. Practical application: allows for the construction and understanding of manifolds via simpler building blocks.

Inequalities relating the number of critical points of a Morse function with the Betti numbers of the manifold. Practical application: provides bounds on the topology of the manifold such as the number of holes.