Morse theory

As mentioned earlier, the paper on impedance so impressed Hermann Weyl that he invited Bott to the Institute for Advanced Study at Princeton in 1949. There Bott came into contact with Marston Morse. Morse's theory of critical points would play a decisive role throughout Bott's career, notably in his work on homogeneous spaces, the Lefschetz hyperplane theorem, the periodicity theorem, and the Yang-Mills functional on a moduli space.

In the Twenties Morse had initiated the study of the critical points of a function on a space and its relation to the topology of the space. A smooth function $f$ on a smooth manifold $M$ has a critical point at $p$ in $M$ if there is a coordinate system $(x_1, \ldots, x_n)$ at $p$ such that all the partial derivatives of $f$ vanish at $p$ :

$$\dfrac{\partial f}{\partial x_i} (p) =0$$   for all $$i= 1,\ldots, n.$$

Such a critical point is nondegenerate if the matrix of second partials, called the Hessian of $f$ at $p$ ,

$$H_p f= \left[ \dfrac{\partial ^2 f}{\partial x_i \partial x_j}(p) \right],$$

is nonsingular. The index $\lambda (p)$ of a nondegenerate critical point $p$ is the number of negative eigenvalues of the Hessian $H_p f$ ; it is the number of independent directions along which $f$ will decrease from $p$ .

If a smooth function has only nondegenerate critical points, we call it a Morse function. The behavior of the critical points of a Morse function can be summarized in its Morse polynomial:

$$\mathcal{M}_t(f) := \sum t^{\lambda (p)},$$

where the sum runs over all critical points $p$ .

A typical example of a Morse function is the height function $f$ of a torus standing vertically on a table top (Figure 1).

Figure 1: Critical points of the height function

The height function on this torus has four critical points of index 0, 1, 1, 2 respectively. Its Morse polynomial is

$$\mathcal{M}_t (f) = 1 + 2t+t^2.$$

For a Morse function $f$ on a compact manifold $M$ , the fundamental results of Morse theory hinge on the fact that $M$ has the homotopy type of a CW complex with one cell of dimension $\lambda$ for each critical point of $f$ of index $\lambda$ . This realization came about in the early Fifties, due to the work of Pitcher, Thom, and Bott.

Two consequences follow immediately:

i)

The weak Morse inequalities:

$$\char93 \$$   critical points of index $$i \ge i$$-th Betti number$$.$$

If

$$P_t (M) = \sum \dim H_i(M) t^i$$

is the Poincaré polynomial of $M$ , the Morse inequalities can be restated in the form

$$\mathcal{M}_t(f) \ge P_t(M),$$

meaning that their difference $\mathcal{M}_t(f) - P_t(M)$ is a polynomial with nonnegative coefficients. This inequality provides a topological constraint on analysis, for it says that the $i$ -th Betti number of the manifold sets a lower bound on the number of critical points of index $i$ that the function $f$ must have.

ii)

The lacunary principle: If no two critical points of the Morse function $f$ have consecutive indices, then

$$\mathcal{M}_t(f) = P_t(M).$$ (1)

The explanation is simple: since in the CW complex of $M$ there are no two cells of consecutive dimensions, the boundary operator is automatically zero. Therefore, the cellular chain complex is its own homology.

A Morse function $f$ on $M$ satisfying (1) is said to be perfect. The height function on the torus above is a perfect Morse function.

Classical Morse theory deals only with functions all of whose critical points are nondegenerate; in particular, the critical points must all be isolated points. In many situations, however, the critical points form submanifolds of $M$ . For example, if the torus now sits flat on the table, as a donut usually would, then the height function has the top and bottom circles as critical manifolds (Figure 2).

Figure 2: Critical manifolds of the height function

One of Bott's first insights was to see how to extend Morse theory to this situation. In [9] he introduced the notion of a nondegenerate critical manifold: a critical manifold $N$ is nondegenerate if at any point $p$ in $N$ the Hessian of $f$ restricted to the normal space to $N$ is nonsingular. The index $\lambda (N)$ of the nondegenerate critical manifold $N$ is then defined to be the number of negative eigenvalues of this normal Hessian; it represents the number of independent normal directions along which $f$ is decreasing. For simplicity, assume that the normal bundles of the nondegenerate critical manifolds are all orientable. To form the Morse polynomial of $f$ , each critical manifold $N$ is counted with its Poincaré polynomial; thus,

$$\mathcal{M}_t(f) := \sum P_t(N) t^{\lambda (N)},$$

summed over all critical manifolds.

With this definition of the Morse polynomial, Bott proved in [9] that if a smooth function $f$ on a smooth manifold $M$ has only nondegenerate critical manifolds, then the Morse inequality again holds:

$$\mathcal{M}_t(f) \ge P_t(M).$$