Morse function

A Morse function is a smooth real-valued function whose critical points are all nondegenerate. Near a critical point of index $k$, the Morse lemma gives coordinates in which the function is a constant minus $k$ squares plus the remaining positive squares.

A descending gradient-like field decreases $f$ away from its critical points and agrees with a negative gradient in chosen Morse lemma coordinates near those points. It transports attaching spheres and belt spheres across regular levels. Perturbations away from the critical points can arrange the transverse intersections used by the Morse-Smale complex, while preserving the prescribed local models.

The incoming and outgoing boundaries are the regular levels zero and one, and all critical points lie in the interior. Near the boundary use a collar coordinate as the function. The Morse handle-attachment theorem produces a relative chain group freely generated by the interior critical points. Thus $\chi (W,M_0)$ is the alternating sum of their Morse indices, regardless of how their critical values are ordered.

If a cobordism Morse function on a trivial cobordism has exactly two critical points, its relative Morse chain complex has two generators and must be acyclic. Their indices therefore differ by one. The only differential is multiplication by the signed intersection of the lower point's belt sphere with the higher point's attaching sphere in a common regular level. A two-term complex of copies of $\mathbb{Z}$ is acyclic exactly when that integer is a unit. Algebraic intersection is therefore a necessary condition; it is not by itself a geometric handle-cancellation statement.

The critical points of $F$ are exactly pairs of critical points of its factors. Its Hessian matrix is their block direct sum, so their Morse indices add and nondegeneracy is preserved. The critical-count polynomial is the product of those of the factors. This construction also gives products of arbitrary smooth functions, with the critical-point count still multiplying even when some Hessians are degenerate.

For a closed connected orientable surface of genus $g$, the Künneth theorem gives the Betti numbers of $S^1\times {\Sigma }_g$ as $(1,2g+1,2g+1,1)$. The weak Morse inequalities give the displayed lower bound. A surface Morse function with one minimum, $2g$ saddles and one maximum exists from its standard handle decomposition. Adding a positive multiple of the cosine Morse function on the circle makes a product of Morse functions, whose critical counts are precisely these four Betti numbers. The lower bound is attained.

A Morse function is perfect over a specified field when its numbers of critical points equal the Betti numbers over that field. Over the rationals this forces every integer matrix in its Morse chain complex to have rank zero, hence to vanish. Consequently its integral homology is free of rank $c_i$ in every degree; a manifold with integral homology torsion cannot admit such an integral-critical-count perfect function.

The Morse lemma gives coordinates $f=f(z)-|u{|}^2+|v{|}^2$. Choose a Riemannian metric Euclidean in these coordinates. A downward gradient flow converging to $z$ has $u=0$ eventually, while $v(t)=e^{-2(t-t_0)}v(t_0)$. Thus its Riemannian distance from $z$ decays exponentially. A partition of unity makes this a global metric choice; changing the metric may change the trajectories.

The Morse index of a nondegenerate critical point is the number of negative directions of its Hessian matrix, counted with multiplicity. Equivalently, it is the maximal dimension of a subspace on which the second variation is negative definite. The same definition applies to a finite-index critical point of an infinite-dimensional energy.

The Morse lemma puts a smooth function near a nondegenerate critical point into a quadratic normal form.

At a nondegenerate critical point placed at the origin, integral Taylor theorem gives $f(x)-f(0)=x^{T}A(x)x$, where $A(x)$ is a smooth symmetric matrix and $A(0)=\frac{1}{2}\mathrm{Hess}f(0)$. A constant linear change makes $A(0)$ diagonal with entries $\pm 1$. Successive completion of squares, with the Schur complement at each step, produces a smooth invertible triangular matrix $B(x)$ such that $x^{T}A(x)x={\sum }_i{ϵ}_i(B(x)x{)}_i^2$. The pivots keep their nonzero signs near zero. The map $x\mapsto B(x)x$ has invertible derivative at zero, so the inverse function theorem makes it a coordinate system. The signs are the Hessian matrix signature by Sylvester's law of inertia.