Morse theory
In the mathematical field of differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold.
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Description
According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics (critical points of the energy functional on paths). These techniques were used in Raoul Bott's proof of his periodicity theorem.
The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory.
See also
- Digital Morse theory
- Discrete Morse theory
- Jacobi set
- Lagrangian Grassmannian
- Lusternik–Schnirelmann category
- Mathematics
- Morse–Smale system
- Sard's lemma
- Stratified Morse theory
External links
- Morse theory @ Wikipedia
