a Morse function
/ MORS /
Picture a landscape — a manifold with a height function — and ask how its shape is revealed by water slowly rising. New lakes appear at valley bottoms, lakes merge at passes, and islands drown at peaks. A Morse function is a height function whose critical points (peaks, passes, pits) are all of the simplest possible non-degenerate type, and Morse theory reads off the topology of the manifold from the list of these critical points alone.
A smooth function f: M -> R has a critical point at p when df_p = 0. The critical point is non-degenerate if the Hessian — the matrix of second partial derivatives in any chart — is invertible there; f is a Morse function if every one of its critical points is non-degenerate. By the Morse lemma, near such a point there are coordinates in which f takes the exact normal form f(x) = f(p) - x_1^2 - ... - x_k^2 + x_{k+1}^2 + ... + x_n^2, and the number k of minus signs, the Morse index, is the dimension of the directions in which f decreases — index 0 is a local minimum, index n a local maximum, intermediate indices are saddles. Non-degeneracy makes critical points isolated, and by Sard almost every function (e.g. almost every height function on an embedded manifold) is Morse, so Morse functions are abundant.
The payoff is a complete handle-by-handle reconstruction of the manifold. As the level value c increases past a critical point of index k, the sublevel set {f <= c} changes by attaching a k-handle; running through all critical points builds M as a CW complex with one cell of dimension k for each index-k critical point. This forces the Morse inequalities: the number of index-k critical points is at least the k-th Betti number b_k, and the alternating sum of critical-point counts equals the Euler characteristic. The honest caveat: Morse theory needs the function (and, for the gradient flow version, a metric making the flow well-behaved), and degenerate critical points — a monkey saddle, or a flat plateau — are NOT allowed; such functions fall outside classical Morse theory and require Morse-Bott or singularity-theory refinements.
The height function on the torus standing upright in R^3 is Morse with four critical points: a minimum (index 0) at the bottom, two saddles (index 1) at the inner top and bottom of the hole, and a maximum (index 2) at the top. The alternating sum 1 - 2 + 1 = 0 equals the Euler characteristic of the torus.
Four critical points encode the torus: 1 - 2 + 1 = 0 = chi.
Degenerate critical points (the Hessian singular) are excluded — a monkey saddle or a plateau breaks Morse theory and needs Morse-Bott. The Morse index is a property of f at p, defined by the Hessian's signature, independent of coordinates.