Morse Theory — An Introduction

§ 1.1 — Critical Points of Functions

Definition

A point $x_0$ satisfying $f'(x_0) = 0$ is called a critical point of the function $f$. Critical points fall into two categories according to the value of $f''(x_0)$:

• $x_0$ is a non-degenerate critical point if $f''(x_0) \neq 0$.
• $x_0$ is a degenerate critical point if $f''(x_0) = 0$.

Observation: Non-degenerate critical points are “stable,” while degenerate critical points are “unstable.”

§ 1.2 — The Hessian Matrix

We say that a point $p_0 = (x_0, y_0)$ in the $xy$-plane is a critical point of a function $z = f(x,y)$ if:

$\frac{\partial f}{\partial x}(p_0) = 0, \qquad \frac{\partial f}{\partial y}(p_0) = 0$

We assume $f(x,y)$ is of class $C^\infty$. We further assume $\frac{\partial^2 f}{\partial x^2}(p_0) \neq 0$ and $\frac{\partial^2 f}{\partial y^2}(p_0) \neq 0$, though after some coordinate changes this won’t necessarily hold.

Definition — Hessian

Suppose $p_0 = (x_0, y_0)$ is a critical point of $z = f(x,y)$. The Hessian of $f$ at $p_0$, denoted $H_f(p_0)$, is the matrix of second derivatives evaluated at $p_0$:

$H_f(p_0) = \begin{pmatrix} \dfrac{\partial^2 f}{\partial x^2}(p_0) & \dfrac{\partial^2 f}{\partial x \, \partial y}(p_0) \\[10pt] \dfrac{\partial^2 f}{\partial y \, \partial x}(p_0) & \dfrac{\partial^2 f}{\partial y^2}(p_0) \end{pmatrix}$

A critical point $p_0$ is non-degenerate if the determinant of the Hessian is nonzero:

$\det H_f(p_0) = \frac{\partial^2 f}{\partial x^2}(p_0) \cdot \frac{\partial^2 f}{\partial y^2}(p_0) - \left(\frac{\partial^2 f}{\partial x \, \partial y}(p_0)\right)^2 \neq 0$

Note that $H_f(p_0)$ is symmetric, since $\dfrac{\partial^2 f}{\partial x \, \partial y} = \dfrac{\partial^2 f}{\partial y \, \partial x}$.

Hessian Under Coordinate Change

Lemma

Let $p_0$ be a critical point of $z = f(x,y)$. Denote by $H_f(p_0)$ the Hessian in coordinates $(x,y)$ and by $\overline{H}_f(p_0)$ the Hessian in coordinates $(X,Y)$. Then:

$\overline{H}_f(p_0) = {}^t\!J(p_0) \cdot H_f(p_0) \cdot J(p_0)$

where $J(p_0)$ is the Jacobian matrix:

$J(p_0) = \begin{pmatrix} \dfrac{\partial x}{\partial X}(p_0) & \dfrac{\partial x}{\partial Y}(p_0) \\[8pt] \dfrac{\partial y}{\partial X}(p_0) & \dfrac{\partial y}{\partial Y}(p_0) \end{pmatrix}$

Corollary

The property that $p_0$ is a non-degenerate critical point does not depend on the choice of coordinates. The same is true for degenerate critical points.

§ 1.3 — The Morse Lemma

Morse Lemma

Let $p_0$ be a non-degenerate critical point of a function $f$ of two variables. Then we can choose appropriate local coordinates $(X, Y)$ such that $f$ takes one of the following three standard forms:

\text{(i)} \quad & f = X^2 + Y^2 + C \\[4pt] \text{(ii)} \quad & f = X^2 - Y^2 + C \\[4pt] \text{(iii)} \quad & f = -X^2 - Y^2 + C \end{aligned}$$ where $C$ is a constant and $p_0$ is the origin.

The theorem states that a function looks extremely simple near a non-degenerate critical point — up to a coordinate change, it is purely quadratic.

Corollary

A non-degenerate critical point of a function of two variables is isolated.

§ 1.4 — Index of a Non-Degenerate Critical Point

Definition

Let $p_0$ be a non-degenerate critical point of $f$. Using the coordinate system given by the Morse Lemma, we define the index of $p_0$ as:

$\operatorname{index}(p_0) = \begin{cases} 0 & \text{if } f = x^2 + y^2 + C \quad \text{(local minimum)} \\ 1 & \text{if } f = x^2 - y^2 + C \quad \text{(saddle point)} \\ 2 & \text{if } f = -x^2 - y^2 + C \quad \text{(local maximum)} \end{cases}$

In other words, the index is the number of minus signs in the standard form.

The index of a non-degenerate critical point $p_0$ is determined by the behaviour of $f$ near $p_0$.

§ 2.1 — Morse Functions on Surfaces

We now move from functions on $\mathbb{R}^2$ to functions on surfaces. Recall that a closed surface is a compact surface without boundary. The genus of a closed surface is the number of “holes” in it.

Let $M$ be a surface. A function $f\colon M \to \mathbb{R}$ assigns a real number to each point $p \in M$. We say $f$ is of class $C^\infty$ if it is smooth with respect to any smooth local coordinates at each point of $M$.

Definition — Morse Function

Suppose that every critical point of $f\colon M \to \mathbb{R}$ is non-degenerate. Then we say that $f$ is a Morse function.

Example

Consider the sphere $S^2 \subset \mathbb{R}^3$ defined by $x^2 + y^2 + z^2 = 1$. Let $f\colon S^2 \to \mathbb{R}$ be the height function $f(x,y,z) = z$. Then $f$ has exactly two critical points — the north pole $p_0$ (a maximum) and the south pole $q_0$ (a minimum) — and both are non-degenerate. Hence $f$ is a Morse function.

Lemma

A Morse function $f\colon M \to \mathbb{R}$ defined on a closed surface $M$ has only a finite number of critical points.

§ 2.2 — Diffeomorphism and the Reeb Theorem

Definition — Homeomorphism

Suppose there is a one-to-one and onto map $h\colon X \to Y$ between two topological spaces $X$ and $Y$ with inverse $h^{-1}\colon Y \to X$. If both maps are continuous, we say $X$ and $Y$ are homeomorphic and $h$ is a homeomorphism — intuitively, they have the same shape.

Definition — Diffeomorphism

A homeomorphism $h\colon M \to N$ between surfaces is a diffeomorphism if both $h$ and $h^{-1}$ are of class $C^\infty$. Two surfaces are called diffeomorphic if there is a diffeomorphism between them.

Theorem — Maximum-Value Theorem

Let $f\colon X \to \mathbb{R}$ be a continuous function on a compact space $X$. Then $f$ takes its maximum value at some point $p_0$ and its minimum value at some point $q_0$.

Theorem (Reeb)

Let $M$ be a closed surface. Suppose there exists a Morse function $f\colon M \to \mathbb{R}$ with exactly two non-degenerate critical points. Then $M$ is diffeomorphic to the sphere $S^2$.

Proof Sketch

By the maximum-value theorem, $f\colon M \to \mathbb{R}$ takes its maximum at $p_0$ and its minimum at $q_0$. Since $f$ is a Morse function, both are non-degenerate critical points. By the Morse Lemma, $f$ takes standard form near each:

$f = \begin{cases} -x^2 - y^2 + A & \text{near } p_0 \\ X^2 + Y^2 + a & \text{near } q_0 \end{cases}$

Here $A$ and $a$ are the maximum and minimum values. For small $\varepsilon > 0$, the set $D(p_0) = \{A - \varepsilon \leq f(p) \leq A\}$ satisfies $x^2 + y^2 \leq \varepsilon$, making it diffeomorphic to a 2-disk. Similarly for $D(q_0)$.

Removing the interiors of $D(p_0)$ and $D(q_0)$ from $M$ yields a surface $M_0$ with boundary $\partial M_0 = C(p_0) \cup C(q_0)$.

Lemma

Let $f\colon M_0 \to \mathbb{R}$ be a $C^\infty$ function taking constant values on the boundary circles $C(p_0)$ and $C(q_0)$. If $f$ has no critical points on $M_0$, then $M_0$ is diffeomorphic to $C(q_0) \times [0,1]$.

Since $C(q_0) \cong S^1$, we conclude $M_0 \cong S^1 \times [0,1]$ — an annulus. Gluing $D(p_0)$ and $D(q_0)$ back along the boundary circles reconstructs $M$ and shows it is diffeomorphic to $S^2$.

§ 1.5 — Handle Decomposition

Start with a Morse function $f\colon M \to \mathbb{R}$ on a closed, connected surface. Define the sublevel set:

$M_t = \{p \in M \mid f(p) \leq t\}$

Let $L_t$ be the level curve $\{f = t\}$. Then $M_t$ is everything below $L_t$, and $L_t = \partial M_t$.

Since $f$ has maximum $A$ and minimum $a$: $M_t = \varnothing$ for $t < a$, and $M_t = M$ for $t \geq A$.

The fundamental idea of Morse Theory is to trace the change of shape of $M_t$ as $t$ increases from below $a$ to above $A$.

Definition — Critical Values

A real number $c_0$ is a critical value if $f(p_0) = c_0$ for some critical point $p_0$.

Lemma

Let $b < c$ be real numbers such that $f$ has no critical values in $[b,c]$. Then $M_b$ and $M_c$ are diffeomorphic.

The topology of $M_t$ only changes when $t$ passes through a critical value. The change depends on the index of the critical point.

Index 0 — Attaching a 0-handle

When the index of $p_0$ is zero, $f = x^2 + y^2 + c_0$ near $p_0$. The sublevel set gains a new connected component — a disk $D^2$:

$M_{c_0 + \varepsilon} \cong M_{c_0 - \varepsilon} \;\sqcup\; D^2$

Index 1 — Attaching a 1-handle

When the index of $p_0$ is one, $f = x^2 - y^2 + c_0$ near $p_0$. The sublevel set gains a bridge $D^1 \times D^1$:

$M_{c_0 + \varepsilon} \cong M_{c_0 - \varepsilon} \;\cup\; (D^1 \times D^1)$

Index 2 — Attaching a 2-handle

When the index of $p_0$ is two, $f = -x^2 - y^2 + c_0$ near $p_0$. The sublevel set is capped off by a disk $D^2$:

$M_{c_0 + \varepsilon} \cong M_{c_0 - \varepsilon} \;\cup\; D^2$

Theorem — Handle Decomposition

If a closed surface $M$ admits a Morse function $f\colon M \to \mathbb{R}$, then $M$ can be described as a union of finitely many 0-handles, 1-handles, and 2-handles.

This is the fundamental result of Morse theory on surfaces: any closed surface equipped with a Morse function can be built up, piece by piece, by attaching handles of index 0, 1, and 2 as we sweep through the critical values from the minimum to the maximum.

Notes on Morse Theory — transcribed from handwritten lecture notes.