Morse theory relates the critical points of a Morse function to changes in the homotopy type of its sublevel sets. Crossing a nondegenerate critical value attaches a cell or handle of dimension equal to the Morse index. The theory also applies to geodesic energy through broken-geodesic approximations.
Morse theory is a branch of differential topology that studies the topology of manifolds using the analysis of smooth functions on them. Developed by the mathematician Marston Morse in the early 20th century, this theory connects critical points of smooth functions defined on manifolds with the topology of those manifolds.
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