Morse Theory

Marine Fontaine

Frank Adams 1,

Morse theory draws a relationship between the critical points of a well-chosen smooth real-valued function on a manifold and the global topology of this manifold. Moving upwards along the values of this function, all its level sets will have the same topology as long as we don't cross a critical value. In that case, the topology of the level set will change and subsequently remains the same until the next critical value. I will introduce the basic notions of the theory, specifically how such functions with a number of critical points can be constructed on a manifold.
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