Morse Theory - Basic Concepts

Basic Concepts

Consider, for purposes of illustration, a mountainous landscape M. If f is the function sending each point to its elevation, then the inverse image of a point in (a level set) is simply a contour line. Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. 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. Saddle points are points where the surrounding landscape curves up in one direction and down in the other.

Imagine flooding this landscape with water. Then, assuming the ground is not porous, the region covered by water when the water reaches an elevation of a is f−1 (-∞, a], or the points with elevation less than or equal to a. Consider how the topology of this region changes as the water rises. It appears, intuitively, that it does not change except when a passes the height of a critical point; that is, a point where the gradient of f is 0. In other words, it 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 each of these three types of critical points - basins, passes, and peaks (also called minima, saddles, and maxima) - one associates a number called the index. Intuitively speaking, the index of a critical point b is the number of independent directions around b in which f decreases. Therefore, the indices of basins, passes, and peaks are 0, 1, and 2, respectively.

Define Ma as f−1(-∞, a]. Leaving the context of topography, one can make a similar analysis of how the topology of Ma changes as a increases when M is a torus oriented as in the image and f is projection on a vertical axis, taking a point to its height above the plane.

Starting from the bottom of the torus, let p, q, r, and s be the four critical points of index 0, 1, 1, and 2, respectively. When a is less than 0, Ma is the empty set. After a passes the level of p, when 0<a<f(q), then Ma is a disk, which is homotopy equivalent to a point (a 0-cell), which has been "attached" to the empty set. Next, when a exceeds the level of q, and f(q)<a<f(r), then Ma is a cylinder, and is homotopy equivalent to a disk with a 1-cell attached (image at left). Once a passes the level of r, and f(r)<a<f(s), then Ma is a torus with a disk removed, which is homotopy equivalent to a cylinder with a 1-cell attached (image at right). Finally, when a is greater than the critical level of s, Ma is a torus. A torus, of course, is the same as a torus with a disk removed with a disk (a 2-cell) attached.

We therefore appear to have the following rule: the topology of Mα does not change except when α passes the height of a critical point, and when α passes the height of a critical point of index γ, a γ-cell is attached to Mα. This does not address the question of what happens when two critical points are at the same height. That situation can be resolved by a slight perturbation of f. In the case of a landscape (or a manifold embedded in Euclidean space), this perturbation might simply be tilting the landscape slightly, or rotating the coordinate system.

This rule, however, is false as stated. To see this, let M equal R and let f(x)=x3. Then 0 is a critical point of f, but the topology of Mα does not change when α passes 0. In fact, the concept of index does not make sense. The problem is that the second derivative is also 0 at 0. This kind of situation is called a degenerate critical point. Note that this situation is unstable: by rotating the coordinate system under the graph, the degenerate critical point either is removed or breaks up into two non-degenerate critical points.