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
on a smooth manifold
has a *critical point*
at
in
if there is a coordinate system
at
such that all the partial derivatives of
vanish at
:

for all

Such a critical point is

is nonsingular. The

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*:

where the sum runs over all critical points .

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

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

For a Morse function on a compact manifold , the fundamental results of Morse theory hinge on the fact that has the homotopy type of a CW complex with one cell of dimension for each critical point of of index . 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:
critical points of index -th Betti number
- ii)
- The lacunary principle: If no two
critical points of the Morse function
have consecutive indices, then

The explanation is simple: since in the CW complex of 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
on
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 . 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).

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
is *nondegenerate* if at any point
in
the Hessian of
restricted to the normal space to
is
nonsingular. The index
of the nondegenerate critical
manifold
is then defined to be the number of negative eigenvalues of
this normal Hessian; it represents the number of independent normal
directions along which
is decreasing.
For simplicity, assume that the normal bundles of the nondegenerate
critical manifolds are all orientable.
To form the Morse polynomial of
, each critical manifold
is counted with its Poincaré polynomial;
thus,

summed over all critical manifolds.

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