Editing Morse theory (section)

{{short description|Analyzes the topology of a manifold by studying differentiable functions on that manifold}}
{{redirect|Morse function|anharmonic oscillators|Morse potential}}

In [[mathematics]], specifically in [[differential topology]], '''Morse theory''' enables one to analyze the [[Topological space|topology]] of a [[manifold]] by studying [[differentiable function]]s on that manifold. 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 complex|CW structures]] and [[handle decomposition]]s on manifolds and to obtain substantial information about their [[Homology (mathematics)|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 [[geodesic]]s ([[Critical point (mathematics)|critical points]] of the [[Hamiltonian mechanics|energy]] [[Functional (mathematics)|functional]] on the space of paths). These techniques were used in [[Raoul Bott]]'s proof of his [[Bott periodicity theorem|periodicity theorem]].

The analogue of Morse theory for complex manifolds is [[Picard–Lefschetz theory]].