Morse theory

In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions 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 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 the space of 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.

To illustrate, consider a mountainous landscape surface ${\displaystyle M}$ (more generally, a manifold). If ${\displaystyle f}$ is the function ${\displaystyle M\to \mathbb {R} }$ giving the elevation of each point, then the inverse image of a point in ${\displaystyle \mathbb {R} }$ is a contour line (more generally, a level set). Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with double point(s). Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines occur at saddle points, or passes, where the surrounding landscape curves up in one direction and down in the other.

Imagine flooding this landscape with water. When the water reaches elevation ${\displaystyle a}$, the underwater surface is ${\displaystyle M^{a}\,{\stackrel {\text{def}}{=}}\,f^{-1}(-\infty ,a]}$, the points with elevation ${\displaystyle a}$ or below. Consider how the topology of this surface changes as the water rises. It appears unchanged except when ${\displaystyle a}$ passes the height of a critical point, where the gradient of ${\displaystyle f}$ is ${\displaystyle 0}$ (more generally, the Jacobian matrix acting as a linear map between tangent spaces does not have maximal rank). In other words, the topology of ${\displaystyle M^{a}}$ does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a mountain pass), or (3) submerges a peak.

To these three types of critical points—basins, passes, and peaks (i.e. minima, saddles, and maxima)—one associates a number called the index, the number of independent directions in which ${\displaystyle f}$ decreases from the point. More precisely, the index of a non-degenerate critical point ${\displaystyle p}$ of ${\displaystyle f}$ is the dimension of the largest subspace of the tangent space to ${\displaystyle M}$ at ${\displaystyle p}$ on which the Hessian of ${\displaystyle f}$ is negative definite. The indices of basins, passes, and peaks are ${\displaystyle 0,1,}$ and ${\displaystyle 2,}$ respectively.

Considering a more general surface, let ${\displaystyle M}$ be a torus oriented as in the picture, with ${\displaystyle f}$ again taking a point to its height above the plane. One can again analyze how the topology of the underwater surface ${\displaystyle M^{a}}$ changes as the water level ${\displaystyle a}$ rises.

Starting from the bottom of the torus, let ${\displaystyle p,q,r,}$ and ${\displaystyle s}$ be the four critical points of index ${\displaystyle 0,1,1,}$ and ${\displaystyle 2}$ corresponding to the basin, two saddles, and peak, respectively. When ${\displaystyle a}$ is less than ${\displaystyle f(p)=0,}$ then ${\displaystyle M^{a}}$ is the empty set. After ${\displaystyle a}$ passes the level of ${\displaystyle p,}$ when ${\displaystyle 0<a<f(q),}$ then ${\displaystyle M^{a}}$ is a disk, which is homotopy equivalent to a point (a 0-cell) which has been "attached" to the empty set. Next, when ${\displaystyle a}$ exceeds the level of ${\displaystyle q,}$ and ${\displaystyle f(q)<a<f(r),}$ then ${\displaystyle M^{a}}$ is a cylinder, and is homotopy equivalent to a disk with a 1-cell attached (image at left). Once ${\displaystyle a}$ passes the level of ${\displaystyle r,}$ and ${\displaystyle f(r)<a<f(s),}$ then ${\displaystyle M^{a}}$ is a torus with a disk removed, which is homotopy equivalent to a cylinder with a 1-cell attached (image at right). Finally, when ${\displaystyle a}$ is greater than the critical level of ${\displaystyle s,}$ ${\displaystyle M^{a}}$ is a torus, i.e. a torus with a disk (a 2-cell) removed and re-attached.