Morse Theory

This chapter is optional: nothing later depends on it that isn't recalled where it is used. It is here because it answers a question that the rest of the book raises and doesn't answer: how can a manifold be understood by cutting it into simple pieces? Morse theory does it with a single function. Take the height of a surface standing upright and slice it at increasing levels: the slices change shape only when they pass a critical point of the height, a peak, a pass or a pit, and each critical point adds a standard piece, a handle. Counting critical points by type recovers the Euler characteristic, and bounds the Betti numbers.

In three dimensions the same construction splits every closed three-manifold into two handlebodies glued along a surface, a Heegaard splitting, which is the basic way three-manifolds are described (10A.2 Building Three-Manifolds). And the search for critical points of other kinds, the saddles of an energy rather than its minima, leads to the min–max methods behind finite extinction (10A.8 Min–Max and Width, 12C.2 Finite Extinction).

By the end of this chapter you will be able to:

• define Morse functions, their critical points and indices, and state the Morse lemma;
• describe how sublevel sets change at critical points, by attaching handles;
• state the Morse inequalities and the relation $\sum(-1)^kc_k = \chi(M)$;
• count peaks, passes and pits on a landscape;
• explain how a Morse function on a three-manifold gives a Heegaard splitting, and prove Reeb's sphere theorem.

10.1Peaks, passes and pits

10.2Morse functions

Let $M$ be a smooth manifold (a submanifold of some $\mathbb{R}^N$, as in 7A.7 Smooth Topology) and $f : M \to \mathbb{R}$ smooth. A critical point is a point $p$ where $df_p = 0$. It is non-degenerate if the Hessian of $f$ at $p$, computed in any chart, is an invertible matrix (2B.8 Calculus in Several Variables). The index of a non-degenerate critical point is the number of negative eigenvalues of the Hessian: $0$ at a local minimum, $n$ at a local maximum, and in between at saddles. A Morse function is one all of whose critical points are non-degenerate.

So a non-degenerate critical point looks, up to a change of coordinates, exactly like its quadratic part: the second-derivative test of 2B.8 Calculus in Several Variables made exact. Non-degenerate critical points are isolated, so a Morse function on a compact manifold has finitely many.

Morse functions are common. For almost every unit vector $v$, the height function $x \mapsto \langle x, v\rangle$ restricted to a submanifold $M \subseteq \mathbb{R}^N$ is a Morse function (by Sard's theorem, 7A.7 Smooth Topology, applied to the Gauss map). Any smooth function can be made Morse by an arbitrarily small perturbation.

The torus. Stand a torus on its end and take $f$ = height (Figure 10.2). There are four critical points: the bottom (index $0$), the bottom of the hole (index $1$, a saddle), the top of the hole (index $1$), and the top (index $2$).

10.3Sublevel sets and handles

Write $M^c = \{x \in M : f(x) \leq c\}$ for the sublevel sets.

So a Morse function builds the manifold from nothing: start with the empty set below the minimum, add a $0$-handle (a disc) at each minimum, then handles of increasing index, and finish with an $n$-handle at each maximum. Every closed manifold has a handle decomposition, and is homotopy equivalent to a cell complex with one $k$-cell for each critical point of index $k$.

The relation is Poincaré–Hopf (7A.7 Smooth Topology) for the gradient field $\nabla f$, whose zero at a critical point of index $k$ has index $(-1)^k$. On a sphere, peaks and pits have index $2$ and $0$ and count $+1$, passes have index $1$ and count $-1$: Maxwell's formula. On a surface of genus $g$, a Morse function has at least $2g$ saddles, since $c_1 \geq b_1 = 2g$; the upright surface has exactly one minimum, one maximum and $2g$ saddles.

10.4Heegaard splittings

On a closed three-manifold $M$, choose a Morse function with one minimum and one maximum, all index-1 critical points at level $1$ and all index-2 points at level $2$ (such self-indexing Morse functions exist; Milnor's Lectures on the h-cobordism theorem). Then:

• $M^{3/2}$ is a $0$-handle with $g$ one-handles attached: a handlebody of genus $g$, a solid ball with $g$ solid handles, like a thickened graph;
• $\{f \geq \frac32\}$ is, by the same argument applied to $-f$, another handlebody of genus $g$;
• they are glued along the surface $f^{-1}(\frac32)$, of genus $g$.

This is a Heegaard splitting of $M$: every closed orientable three-manifold is two handlebodies of the same genus glued along their boundary surface (10A.2 Building Three-Manifolds). Genus $0$: two balls, giving $S^3$. Genus $1$: two solid tori, giving $S^3$, $S^2\times S^1$ or a lens space depending on the gluing (7A.2 Compactness and Compactification, 7A.5 Computing π₁). The gluing map is recorded by the curves on the Heegaard surface that bound discs in the second handlebody, a Heegaard diagram, and the fundamental group can be read off it (7A.5 Computing π₁).

10.5History

Cayley's paper on contour and slope lines appeared in 1859 and Maxwell's "On hills and dales" in 1870. Marston Morse developed the theory of critical points in a series of papers from 1925, culminating in The Calculus of Variations in the Large (1934). Georges Reeb proved his sphere theorem and introduced the graph named after him in 1946. Raoul Bott used Morse theory to prove his periodicity theorem in 1959, and Stephen Smale used handle decompositions to prove the h-cobordism theorem and the high-dimensional Poincaré conjecture in 1961. Milnor's Morse Theory (1963) remains the standard introduction. Poul Heegaard introduced his splittings in his 1898 thesis.

10.6Exercises

Solution

$f_\theta = (2 + \cos\phi)\cos\theta$ and $f_\phi = -\sin\phi\sin\theta$ vanish at $\theta = \pm\frac\pi2$, $\sin\phi = 0$. At $(\theta, \phi) = (\frac\pi2, 0)$: $f_{\theta\theta} = -3$, $f_{\phi\phi} = -1$: index $2$ (top, $f = 3$). At $(\frac\pi2, \pi)$: $f_{\theta\theta} = -1$, $f_{\phi\phi} = +1$: index $1$ ($f = 1$). At $(-\frac\pi2, \pi)$: $f_{\theta\theta} = 1$, $f_{\phi\phi} = -1$: index $1$ ($f = -1$). At $(-\frac\pi2, 0)$: $f_{\theta\theta} = 3$, $f_{\phi\phi} = 1$: index $0$ ($f = -3$).

Solution

peaks − passes + inland pits $= 1$, so passes $= 5 + 2 - 1 = 6$. Gluing a disc (the sea, with one minimum at its centre) to the island along the coastline gives a sphere with a Morse function, so peaks − passes + (inland pits $+ 1$) $= 2$.

Solution

Up to homotopy, $k$-handles are $k$-cells, and 7A.5 Computing π₁ shows that 1-cells attached to a connected complex add free generators, 2-cells add relations, and cells of dimension $\geq 3$ don't change $\pi_1$. A genus-$g$ Heegaard splitting comes from one $0$-handle, $g$ one-handles, $g$ two-handles and one $3$-handle, so $\pi_1$ has a presentation with $g$ generators and $g$ relations.

Solution

(a) A function on a compact manifold attains its minimum and maximum, which are critical points, and they are different unless $f$ is constant (not Morse). (b) Near a minimum the Morse lemma gives $f = f(p) + |x|^2$, so small sublevel sets are discs; similarly near the maximum. Between the two levels there are no critical values, so the region is a product of a level set (the boundary sphere of the small disc) and an interval. (c) $M$ is two discs glued along a homeomorphism of their boundary spheres, and any such gluing gives $S^n$ (extend the boundary homeomorphism radially over the disc, the "Alexander trick", which works for homeomorphisms; for diffeomorphisms this is where exotic spheres come from).