Morse theory

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically in <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>, Morse theory enables one to analyze the <a href="/facts/Topological_space/v2ut7CbK">topology</a> of a <a href="/facts/Manifold/rXPERJtu">manifold</a> by studying <a href="/facts/Differentiable_function/jQqTgLk1">differentiable functions</a> on that manifold. According to the basic insights of <a href="/facts/Marston_Morse/qYfc9fjk">Marston Morse</a>, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find <a href="/facts/CW_complex/rN7buLMo">CW structures</a> and <a href="/facts/Handle_decomposition/NFd5F0zl">handle decompositions</a> on manifolds and to obtain substantial information about their <a href="/facts/Homology_(mathematics)/VFJmryEW">homology</a>.
Before Morse, <a href="/facts/Arthur_Cayley/NBd7QkeE">Arthur Cayley</a> and <a href="/facts/James_Clerk_Maxwell/jtYyze61">James Clerk Maxwell</a> had developed some of the ideas of Morse theory in the context of <a href="/facts/Topography/SS8Bx4t9">topography</a>. Morse originally applied his theory to <a href="/facts/Geodesic/fbWR6l80">geodesics</a> (<a href="/facts/Critical_point_(mathematics)/MrEk9PnY">critical points</a> of the <a href="/facts/Hamiltonian_mechanics/2ReerlNA">energy</a> <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a> on the space of paths). These techniques were used in <a href="/facts/Raoul_Bott/5JHkhPx8">Raoul Bott</a>'s proof of his <a href="/facts/Bott_periodicity_theorem/GqgJRllU">periodicity theorem</a>.
The analogue of Morse theory for complex manifolds is <a href="/facts/Picard%25E2%2580%2593Lefschetz_theory/gT57XAnS">Picard–Lefschetz theory</a>.

Morse theory open-in-new

Morse theory