Morse functions and handle attachments

A critical point p satisfies dfₚ=0. At such a point the second derivative defines a symmetric bilinear form on the tangent space independently of the chosen coordinates: the extra coordinate-change terms involving first derivatives vanish. Nondegeneracy means this Hessian is invertible. Its index λ is the number of negative eigenvalues counted with multiplicity. A minimum has index zero, a maximum has index n, and intermediate indices describe saddle behavior. Eigenvalue magnitudes can change under coordinates, while the numbers of positive and negative directions do not.

A smooth function is Morse when every critical point is nondegenerate. The Morse lemma gives smooth coordinates in which the function is exactly a constant plus negative squares in λ directions and positive squares in the remaining directions, rather than merely having that quadratic Taylor approximation. This is a theorem about a neighborhood of one critical point. The gradient flow also depends on a chosen Riemannian metric, so these coordinates do not automatically make every global flow line a straight coordinate trajectory.

Let a<b be regular values and assume f⁻¹([a,b]) is compact, is contained in the source interior, and contains no critical points. A normalized gradient construction gives a product structure across that band and identifies the sublevels at its ends up to diffeomorphism. Compactness prevents a trajectory from escaping before it reaches the other level. On a closed manifold this protection is automatic; for a noncompact source it must be supplied, for example by a proper function and a bounded value interval.

If the compact band contains exactly one critical point of index λ, with critical value strictly between its regular endpoints, the upper sublevel is obtained from the lower by attaching an n-dimensional λ-handle and smoothing corners. The handle is D^λ×D^(n−λ), attached along S^(λ−1)×D^(n−λ). Up to homotopy this adds a λ-cell, so Euler characteristic changes by (−1)^λ. The local model does not eliminate the need for compactness, regular endpoints, or control of the rest of the band.

On a closed manifold a Morse function has finitely many critical points, and its handle decomposition yields the Euler characteristic as the alternating sum of their index counts. The weak Morse inequalities say that each count is at least the corresponding homology rank over a field. Extra critical points can cancel algebraically or support attachments that do not create independent homology classes. An index count is therefore a constraint on topology, not a direct list of Betti numbers.

Attaching maps and framings can distinguish manifolds even when the numbers of handles agree. Morse functions themselves can be perturbed and compared, and genericity theorems establish their existence, but cancellation of a particular handle pair has further geometric conditions. Keep a computational workflow modest and precise: locate critical points, verify nondegeneracy, determine index and value, isolate compact regular bands, and only then infer the corresponding handle changes. A Hessian sign table alone is not a complete classification theorem.