Morse Theory and Global Geometric Constructions

Morse theory reads global topology from the critical points of a smooth real function. Near a nondegenerate critical point, the function has a universal quadratic form; between critical values, suitable sublevel sets are diffeomorphic; crossing a critical value attaches a handle whose index is the number of negative Hessian directions. Compactness or properness is what turns these local statements into controlled global constructions. The statements below concern smooth manifolds without boundary; pre-existing boundaries require additional conditions on the function and its flow.

Required background. Tensor Fields and Natural Operations supplies the tensorial setting for Hessians; Tangent and Cotangent Spaces supplies tangent-space calculus, and Vector Fields, Flows, and Lie Brackets supplies flows. Exterior Algebra and Differential Forms supplies forms and orientation algebra; Orientation, Densities, and Volume Elements supplies the integration language used in global constructions.

Helpful background. Calculus and Linear Algebra supplies Hessians, eigenvalues, and quadratic forms; The Riemann Curvature Tensor motivates global constructions on manifolds but is not required by Morse theory.

Critical points and the Morse lemma

A point $p\in M$ is critical for $f:M\to\mathbb R$ when $df_p=0$. At a critical point, the second derivative defines an intrinsic symmetric bilinear form on $T_pM$, the Hessian. The point is nondegenerate when this form is nonsingular. A Morse function is one whose critical points are all nondegenerate.

The Morse lemma states that near a nondegenerate critical point there are coordinates $x^1,\ldots,x^n$ centered at $p$ such that

$$f(x)=f(p) -(x^1)^2-\cdots-(x^\lambda)^2 +(x^{\lambda+1})^2+\cdots+(x^n)^2.$$

The integer $\lambda$ is the Morse index. It is independent of coordinates by Sylvester’s law of inertia. A minimum has index zero, a maximum has index $n$, and a saddle has an intermediate index. Milnor proves the local normal form and isolates its nondegeneracy hypothesis in Milnor 1963, §§1–2, pp. 1–11.

Nondegeneracy is essential. The function $x^4$ has a critical point at zero but zero Hessian, and no smooth coordinate change turns its fourth-order behavior into the Morse quadratic form. Degenerate points can split or disappear under arbitrarily small perturbations.

Sublevel sets and handle attachment

For $a\in\mathbb R$, define the sublevel set

$$M^a=f^{-1}((-\infty,a]).$$

If $a\lt b$ and the compact band $f^{-1}([a,b])$ contains no critical points, then $M^a$ and $M^b$ are diffeomorphic. To see the mechanism, choose an auxiliary Riemannian metric. On the band the vector field $X=\nabla f/|\nabla f|^2$ satisfies $X[f]=1$, so its flow advances between level sets at unit rate. A suitable cutoff extends this deformation to the sublevel sets. Compactness ensures the required flow does not escape to infinity. On a noncompact manifold, properness of $f$, meaning that inverse images of compact sets are compact, supplies the same control. This is Milnor 1963, §3, Theorem 3.1, pp. 12–14, PDF.

Suppose the compact band $f^{-1}([c-\epsilon,c+\epsilon])$ contains exactly one critical point, with value $c$ and index $\lambda$, and its endpoints are regular values. Then, after smoothing corners, $M^{c+\epsilon}$ is obtained from $M^{c-\epsilon}$ by attaching a $\lambda$-handle

$$D^\lambda\times D^{n-\lambda}$$

along $S^{\lambda-1}\times D^{n-\lambda}$. At the level of homotopy type, this is attachment of a $\lambda$-dimensional cell $D^\lambda$ along its boundary; the extra $D^{n-\lambda}$ thickens the cell to keep the sublevel set $n$-dimensional. The negative Hessian directions determine the core disk. The global construction joins the local quadratic model to the regular-level flow away from the critical point, so its compact-band hypotheses are essential. Milnor gives the precise cell-attachment statement and proof in Milnor 1963, §3, Theorem 3.2, pp. 14–20, PDF.

Morse counts and Euler characteristic

For a Morse function on a compact manifold without boundary, let $c_\lambda$ count critical points of index $\lambda$, and let $b_\lambda$ be the Betti number over a field. The weak Morse inequalities state

$$c_\lambda\ge b_\lambda.$$

The stronger alternating inequalities contain more information, while their final equality is

$$\chi(M)=\sum_{\lambda=0}^n(-1)^\lambda c_\lambda.$$

For the height function on $S^2$, there is a minimum of index zero and a maximum of index two. Thus $\chi(S^2)=1+1=2$. An explicit torus example avoids relying on a picture of an embedded surface. On $T^2$ with angular coordinates $(\theta,\varphi)$, take

$$f(\theta,\varphi)=-\cos\theta-\epsilon\cos\varphi, \qquad 0<\epsilon<1.$$

Its four critical points are $(0,0)$, $(0,\pi)$, $(\pi,0)$, and $(\pi,\pi)$. Their respective critical values are $-1-\epsilon$, $-1+\epsilon$, $1-\epsilon$, and $1+\epsilon$, so the levels are distinct. Since the Hessian at a critical point is

$$\operatorname{Hess}f =\operatorname{diag}(\cos\theta,\epsilon\cos\varphi),$$

the indices in increasing critical-value order are $0,1,1,2$. The alternating count is $1-2+1=0=\chi(T^2)$. The sphere’s handle sequence is a zero-handle followed by a two-handle. The torus acquires a zero-handle, then two one-handles, and finally a two-handle that caps the remaining boundary.

These examples independently check the index signs against known homology. The critical-point count itself depends on the chosen function; only the constrained inequalities and alternating total are topological.

Partitions of unity and exhaustion functions

Morse functions are one member of a broader toolkit for turning local data into global objects. On a paracompact smooth manifold, a locally finite partition of unity $\{\rho_i\}$ subordinate to an open cover satisfies $\rho_i\ge0$, $\operatorname{supp}\rho_i\subset U_i$, and $\sum_i\rho_i=1$. It can glue local Riemannian metrics because positive-definite forms are closed under positive weighted sums. Connections on a fixed bundle can also be combined because they form an affine space and the weights sum to one.

Arbitrary Lorentzian metrics cannot be glued by the same unchecked recipe. For example, the two forms $-dt^2+dx^2$ and $dt^2-dx^2$ are both Lorentzian in two dimensions, yet their equally weighted average is zero. Preserving Lorentzian signature or a nonlinear equation requires a separate construction. Local finiteness makes the sum smooth; it does not preserve every geometric property.

A proper smooth exhaustion $f:M\to\mathbb R$, bounded below and tending to infinity along every escaping sequence, organizes a noncompact manifold into compact sublevel sets. Under standard transversality results it can often be perturbed to a Morse exhaustion. This gives a controlled sequence of handle attachments, but conclusions about finitely many handles require additional finite-type assumptions.

Morse theory does not by itself establish causal properties, geodesic completeness, or spacetime singularity theorems. A time function in Lorentzian geometry has causal-gradient requirements beyond being Morse, and a Morse function need not be temporal.

Exercise: sphere and torus checks

For the standard height $h=z$ on $S^2$ and the function $f(\theta,\varphi)=-\cos\theta-\epsilon\cos\varphi$ on $T^2$, with $0<\epsilon<1$, find the critical points, critical values, and Morse indices. Compute each alternating critical-point count, compare it with the homology-based Euler characteristic, and describe the torus sublevel-set handle sequence.

Solution

For the unit sphere $S^2$, the critical points are the south pole $(0,0,-1)$ with height $-1$ and index $0$, and the north pole $(0,0,1)$ with height $+1$ and index $2$. Thus

$$\sum(-1)^\lambda c_\lambda=1+1=2.$$

Its Betti numbers are $1,0,1$, giving the same Euler characteristic. For the torus,

$$df=\sin\theta\,d\theta+\epsilon\sin\varphi\,d\varphi,$$

so $df=0$ exactly when each angle is $0$ or $\pi$. At $(0,0)$ the value is $-1-\epsilon$ and the Hessian has signs $(+,+)$, giving index $0$. At $(0,\pi)$ the value is $-1+\epsilon$ and the signs are $(+,-)$, giving index $1$. At $(\pi,0)$ the value is $1-\epsilon$ and the signs are $(-,+)$, again giving index $1$. At $(\pi,\pi)$ the value is $1+\epsilon$ and the signs are $(-,-)$, giving index $2$. Hence the critical-point count is $1-1-1+1=0$. The torus Betti numbers are $1,2,1$, whose alternating sum is also zero.

Starting below the minimum, the sublevel set is empty. Crossing the minimum adds a disk. The first one-handle turns it into an annulus, the second makes a once-punctured torus, and the final two-handle caps its boundary to give $T^2$. Thus the two index-one points create the two independent one-cycles before the maximum closes the surface.

References

• Milnor, John. Morse Theory. Annals of Mathematics Studies 51, Princeton University Press, 1963. Open PDF, §§1–6, pp. 1–38.